1.2.1 Meaning and Definition of Statistics

Statistics has different meanings for different people and the purpose. Statistics has been defined also in different ways by different writers. This is due to changes in the scope of statistics with the passage of time.

Statistics is used in two senses:

• In plural sense meaning a collection of facts or estimates – the figure themselves (numerical data).
• As a singular noun meaning Statistics is the scientific method of collecting, organizing, summarizing, presenting and analyzing data, as well as interpreting data. (Interpretation means drawing valid conclusions and making reasonable decisions on the basis of such analysis).

1. Collection of data: Once an investigator has collected data through a survey, it is necessary to edit these data in order to correct any apparent inconsistencies, ambiguities, recording errors or for that matter any mistake that can enter into the actual computations. But even before the data has been collected and edited, it is assumes that these can be suitably classed according to some common characteristic of the population sampled.
2. Description of data: The organized data can now be presented in the form of tables or diagrams or graphs. This presentation in an orderly manner facilitates the understanding as well as analysis of data.
3. Analysis of data: The basic purpose of data analysis is to make it useful for certain conclusions. This analysis may simply be a critical observation of data to draw some meaningful conclusions about it or it may involve highly complex and sophisticated mathematical techniques. Some simple statistical tools such as calculations of averages, dispersion of data around averages and percentages are commonly used to analyze data.
4. Interpretation of data: Interpretation means drawing conclusions from the data which form the basis of decision making. Correct interpretation requires a high degree of skill and experience and is necessary in order to draw valid conclusions.

1.2.2 Uses of statistics

Statistics is an increasingly important subject which is useful in many types of scientific investigations. Statistics is particularly useful in situations where there is experimental uncertainty and may be defined as ‘the science of making decisions in the face of uncertainty’. It is applicable in various fields including education, business, agriculture, engineering.

• To present data in a concise and definite form – helps in classifying and tabulating raw data for processing and further tabulation for other users.
• To make it easy to understand complex and large data - permits summarization and presentation of large quantities of information. i.e. It condenses and summarizes voluminous data into a few presentable, understandable and precise figures. For example, stock market prices of individual stocks and their trends are highly complex to comprehend, but a graph of prices trends gives us the overall picture at a glance.
• To undertake and understand research in our areas of interest such as It helps in determining functional relationship between two or more phenomenon. Statistical techniques such as correlational analysis assist in establishing the degree of association between two or more independent variables. For example, the coefficient of correlation between literacy and employment gives us the degree of association between extent of training and industrial productivity.
• Used in government and other organizations to formulate new programmes and policies as well as in administration ie It helps the central management and the government in formulating policies. Example, the recently conducted census, will be used as a source of information for planning by the government for the next 10 years until another census is conducted in 2019.
• For comparison of variables in different sets of data - Arrangement of data with respect to different characteristics facilitates comparison and interpretation. For example, data on age, height, gender, and family income of college students gives us a much better picture of students when the data is categorized relative to these characteristics.
• Aids in forecasting outcomes of future events- Statistical methods are highly useful tools in analyzing the past data and predicting some future trends. Eg Helps businesses in decision making by making future estimates and expectations . For example, the sales for a particular product for the next year can be computed by knowing the sales for the same product over the previous years, the current market trends and the possible changes in the variable that affect the demand of the product.

1.2.3 Scope of Statistics

Some of the important areas where the knowledge of statistics is usefully applied are as follows:

1. Government. Various departments of the government collect and interpret vast amount of data and information for efficient functioning and decision making.
2. Economics. Statistics are widely used in economics study and research. The subject of economics is mainly concerned with production and distribution of wealth as well as savings and investments. Some of the areas of economic interest in which statistical tools are used are as follows:
   • Statistical methods are extensively used in measuring and forecasting Gross National Product ( GNP ).
   • Economic stability is primarily judged by statistical studies of business cycles.
   • Statistical analyzes of population growth, unemployment figures, rural or urban population shifts and so on influence much of the economic policy making.
   • Econometric models which involve application of statistical methods and used for optimum utilization of resources available.
   • Financial statistics are necessary in the fields of money and banking including consumer savings and credit availability.
3. Physical, Natural and Social Sciences. In physical sciences, as an example, the science of meteorology uses statistics in analyzing the data gathered by satellites in predicting weather conditions.
4. Statistics and Research. There is hardly any advanced research going on without the use of statistics in one form or another. Statistics are used extensively in medical, pharmaceutical and agricultural research. The effectiveness of a new drug is determined by statistical experimentation and evaluation.
5. Other Areas. Statistics are commonly used by insurance companies, stock brokerage firms, banks, public utility companies and so on. Statistics are also immensely useful to politicians since they can predict their chance of winning through the use of sampling techniques in random selection of voters sampled and studying their attitude on issues and policies.

1.2.4 Limitations of Statistics

Statistics has a number of limitations, pertinent among them are as follows:

• It does not deal with individual values. Statistics only deals with aggregate values. For example, the marks obtained by one student in a class does not carry any meaning in itself, unless it can be compared with a set standard or with other students in the same class or with his own marks obtained earlier.
• It cannot deal with qualitative characteristics. Statistics is not applicable to qualitative characteristics such as honesty, kindness, goodness, colour, poverty, beauty, and so on, since these cannot be expressed in quantitative terms. The characteristics, however, can be statistically dealt with if some quantitative values can be assigned to these with logical criterion.
• Statistical conclusions are not universally true. Since statistics is not an exact science, as is the case with natural sciences, the statistical conclusions are true only under certain assumptions.
• Statistical interpretation requires a high degree of skill and understanding of the subject. In order to get meaningful results, it is necessary that the data be properly and professionally collected and critically interpreted. It requires extensive training to read and analyze statistics in its proper context.
• Statistics can be misused. The famous statement that ‘figures don’t lie but the liars can figure’ is a testimony to the misuse of statistics. Thus, inaccurate or incomplete figures can be manipulated to get desirable references. Example: advertising slogans such as “4 out of 5 dentists recommend brand X toothpaste” give the impression that 80% of all dentists recommend this brand. This may not be true since we don’t know how big the sample is or whether the sample represents the entire population. Another example is opinion polls on the news where percentages are given without sample size or representativeness.
• There are certain phenomena or concepts where statistics cannot be used. This is because these phenomena or concepts are not amenable to measurement. For example, beauty, intelligence, and courage cannot be quantified. Statistics has no place where quantification is not possible.
• Statistics reveal the average behaviour—the normal or general trend. Applying an ‘average’ to an individual may lead to wrong or dangerous conclusions. For example, an average river depth of four feet does not mean it is safe throughout; some points may be much deeper.
• Since statistics are collected for a particular purpose, such data may not be relevant or useful in other situations. For example, secondary data (i.e., data originally collected by someone else) may not be useful for another person.
• Statistics are not 100 per cent precise as Mathematics or Accountancy. Users should be aware of this limitation.
• In statistical surveys, sampling is generally used as it is not physically possible to cover the whole universe. The results may not fully represent the universe. Moreover, surveys with identical sample sizes but different sample units may give different outcomes.
• At times, association or relationship between two or more variables is studied, but this does not indicate a cause-and-effect relationship. It only shows similarity or dissimilarity in movement. Interpretation requires care.
• A major limitation of statistics is that it does not reveal everything about a phenomenon. Some background information or other relevant aspects may not be covered. The user of statistics must interpret results while considering other relevant information.

1.2.5 Misuses

Sometimes people, knowingly or unknowingly, use statistical data wrongly. Such forms of misuse include:

• Failure to give the sources of data: this may compromise the reliability of the data because the user of such data will not know how far this data will fit his/her situation including if he/she wants to refer to the original source.
• Defective data: This may be done knowingly in order to defend one’s position or to prove a particular point. This apart, the definition used to denote a certain phenomenon may be defective. For example, in case of data relating to unemployed persons, the definition may include even those who are employed, though partially. The question here is how far it is justified to include partially employed persons amongst unemployed ones.
• Unrepresentative sample: In statistics, several times one has to conduct a survey, which necessitates to choose a sample from the given population or universe. The sample may turn out to be unrepresentative of the universe. One may choose a sample just on the basis of convenience. He may collect the desired information from either his friends or nearby respondents in his neighbourhood even though such respondents do not constitute a representative sample.
• Inadequate sample: At times one may conduct a survey based on an extremely inadequate sample. For example, in a city we may find that there are 100,000 households. When we have to conduct a household survey, we may take a sample of merely 100 households comprising only 0.1 per cent of the universe. A survey based on such a small sample may not yield right information.
• Unfair Comparisons: For instance, one may construct an index of production choosing the base year where the production was much less. Then he may compare the subsequent year’s production from this low base. Such a comparison will undoubtedly give a wrong picture of the production though in reality it is not so. Another source of unfair comparisons could be when one makes absolute comparisons instead of relative ones. An absolute comparison of two figures, say, of production or export, may show a good increase, but in relative terms it may turn out to be very negligible. Another example of unfair comparison is when the population in two cities is different, but a comparison of overall death rates and deaths by a particular disease is attempted. Such a comparison is wrong. Likewise, when data are not properly classified or when changes in the composition of population in the two years are not taken into consideration, comparisons of such data would be unfair as they would lead to misleading conclusions.
• Unwanted conclusions: This may be as a result of making false assumptions. For example, while making projections of population in the next five years, one may assume a lower rate of growth though the past two years indicate otherwise. Sometimes one may not be sure about the changes in business environment in the near future. In such a case, one may use an assumption that may turn out to be wrong. Another source of unwarranted conclusion may be the use of wrong average. Suppose in a series there are extreme values, one is too high while the other is too low, such as 800 and 50. The use of an arithmetic average in such a case may give a wrong idea. Instead, harmonic mean would be proper in such a case.
• Confusion of correlation and causation: In statistics, several times one has to examine the relationship between two variables. A close relationship between the two variables may not establish a cause-and-effect-relationship in the sense that one variable is the cause and the other is the effect. It should be taken as something that measures degree of association rather than try to find out causal relationship.

1.2.6 Branches of statistics

Statistics can be divided into two branches:

• Descriptive: statistics that summarize the characteristics of given data, without trying to extrapolate or make predictions. Utilizes numerical and graphical method to summarize the information, look for patterns in the data set and present the information in a convenient form (Describes or summarizes things you definitely know).

• Inferential: statistics used to make claims or predictions about the larger population based on a subset (sample) of that population. Utilizes sample data to make estimates, decisions, predictions and other generalizations about a larger set of data. (Compares groups, tests hypothesis or predicts or infers). Conclusions made are called Statistical inference which cannot be absolutely certain hence the need to use probability in drawing conclusions.

Remark:
In this course, you will study numerical and graphical ways to describe and display your data. This area of statistics is what we have called “Descriptive Statistics.” You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs.

1.3.1 Definition of some terms.

• Organization of Data - Data organization, in broad terms, refers to the method of classifying and organizing data sets to make them more useful. Some IT experts apply this primarily to physical records, although some types of data organization can also be applied to digital records.

• Data is a collection of observations from an experiment or a survey.

• A population is a set of units (people, objects, transactions or events). The entire set of all possible outcomes or measurements of interest.

• In collecting data, it’s often not possible to observe the whole group referred to as target group population; hence one observes a smaller representative of the group called a sample (sample - a subset of the population for which we have data, and that we hope is representative of the population).

• If the whole group is observed a census has been conducted.

• If a smaller group is observed a sample survey has been conducted.

• If the sample is a representative of a population, then important conclusions about population can be made from it.

• Target population may be finite or infinite.

• Finite Population: e.g. number of students in ABC University.

• Infinite Population: e.g. number of insects in ABC University.

• Variable: A characteristic or property of an individual population unit. A quantity that can assume prescribed set of values. May be discrete, continuous or constant.

• Discrete Variable - Take on a finite number (values), are countable. E.g. size of a family.

• Continuous Variable - Takes any value within a specified range. E.g. Height of students.

• Constant Variables - Takes one value. E.g. Number of hours in a day.

1.3.2 Levels of Measurement

• Measurement: is the process we use to assign numbers to variables of individual population units according to a set of rules.
• Nominal measurement – classifies data into mutually exclusive (non-overlapping) exhausting categories in which no order, or ranking can be imposed on the data e.g. gender - male & female, bloodgroups O. A, B & AB, eye colour – blue, brown, religion etc.
• Ordinal - classifies data into categories that can be ranked or ordered with respect to each other. For example – guest speaker might be ranked as good, average or poor, health condition of a patient can be good, better or best. The precise difference between ranks does not exist. More examples: Grade A, B… etc, Ranking scale (poor, good, excellent, etc), judging (1st, 2ndetc)
• Interval measurement: classifies and ranks data and precise difference between units of measurement exist. However, there is no meaningful zero. For example – temperature has no meaningful difference between each unit. 0 degrees Celsius does not mean there is no heat, IQ, Exam score.
• Ratio measurement: There is a difference between units and a true zero exists. Examples – height, time, age, salary, etc.

1.3.3 Types of Data

All data can be classified as one of two general types: Quantitative Data and Qualitative Data.

1. Quantitative data (Numerical data – it yields numerical responses, for example, “What is your age?”) They are data that are measured on a naturally occurring numerical scale. They represent a measurable quantity. Observations are numbers representing an amount or count of a certain characteristic like height, weight etc

Examples: The number of patients admitted in the County hospital, the current unemployment rate for each county the scores of a sample of 150 students in an exam, the number of male students in the class.

Ratio and interval measurement fall under the quantitative category.

These data can be classified into two types: discrete and continuous.

• Discrete Data - Discrete data can only take on particular values and thus has clear boundaries. Assumes only countable number of values. Example: You can have 30 students or 31 students, but not 30.5 students, so “number of students” is a discrete variable, family size etc. In fact, any variable based on counting is discrete, whether you are counting the number of books purchased in a year or the number of motor accidents reported in a year.
• Continuous Data - Continuous data can take any value, or any value within a range or an interval. Most data measured by interval and ratio scales, other than that based on counting, is continuous. Example: weight and height of students, distance from town to campus, an income received by an employee are all continuous.

2. Qualitative data (Categorical data – that which yields responses such as Yes or No. for example,” Did you buy the books?”)

Qualitative data cannot be measured on a natural numerical scale; they can only be classified into groups or categories. Take on values that are names or labels. Categories are non - overlapping, may or may not suggest an order or rank.

Examples: The political party affiliations in a sample of 50 chief executive officers, the size of a car (subcompact, compact, mid-size, or full-size) rented by each of a sample of 30 business travelers, a coffee tester’s ranking (best, worst, etc.) of four brands of coffee for a panel of 10 testers.

These data can be classified into three types: Attribute, Nominal and Ordinal.

• Attribute Data: Also known as dichotomous data. These data has only two categories. Example: yes/no, male/female.

• Nominal Data: These data have several unordered categories. Example: type of an insurance policy (motor, medical, fire, burglary, life insurance policies).

• Ordinal or Ranked Data: These data have several ordered categories. Example: Questionnaire response such as Strongly Agree ……… Strongly Disagree to questions like: I am the best student in my class, My classmates are very co-operative, I live in the best hostel, Muscle response (none, partial, complete), Tree vigor (Healthy, sick, dead), Income (less than kSh9999, KSh10,000-KSh19,999, KSh20,000-KSh49,999, Greater than KSh50,000)

• Remark:
  In economics, data is also often categorized by how it relates to time.

• Cross-sectional data.
  In cross-sectional data, all observations come from the same point in time. The observations typically correspond to individuals or groups like states or countries. For instance, a survey of Americans on who they support in the upcoming presidential election is cross-sectional data. So is a data set with the homicide rate for each state in a single year.

• Longitudinal or time-series data.
  In longitudinal or time-series data, each data point corresponds to a particular point in time – usually for a single individual or group. For instance, if you recorded your income every day for a year, that would give me a longitudinal data set. The GDP of the U.S. from 1945 to the present is also a longitudinal data set.

• Panel data.
  Panel data is both cross-sectional and longitudinal. It involves getting cross-sectional data for many time periods (or, alternatively, time-series data for many different individuals or groups). For instance, if you recorded the income for each one of your classmates every year for the next 20 years, that would be a panel data set. One way to think of this is in terms of dimensions. Both cross-sectional and time-series data are one-dimensional; panel data is two-dimensional.

1.3.4 Data Sources and Collection Tools

1.4.1 Objectives

By the end of the lecture the learner should be able to:

• Summarize a set of data using a table or frequency distribution table.
• Display data graphically using bar graphs, histogram, frequency polygon, frequency curve, and Ogive curve and interpret the graphs.

1.4.2 Introduction

When data is collected (raw data), it is usually not organized. After the data have been collected, the next step is to present them in some suitable form. Proper presentation is necessary because statistical data in raw form are difficult to comprehend.

Often, the first stage in presenting data is to produce a table.

• If the data are few, they can be easily presented and understood.
• If the number of figures is large, proper classification is essential for analysis.

Next is to represent the data diagrammatically or graphically.

A statistical graph is a tool that helps you learn about the shape or distribution of a sample or population. Graphs often communicate information more effectively than large sets of numbers.

Common graphs include:

• Dot plot
• Bar graph
• Histogram
• Stem-and-leaf plot
• Frequency curve
• Frequency polygon
• Pie chart
• Box plot
• Cumulative frequency (Ogive) curve

In this course, we will look at:

• Histogram
• Line graphs
• Bar graphs
• Frequency polygons
• Cumulative frequency (Ogive) curve

1.4.3 Frequency Distribution

One method of data presentation is the frequency distribution.

The frequency of a value is the number of times that value appears. When observations are few and values repeat, we can arrange them in a table showing each value and its frequency. This is called a frequency table.

A frequency table/distribution is a listing of possible values for a variable together with the number of observations (or relative frequencies) for each value.

##   Scores (x) Frequency (f)
## 1          3             4
## 2          4             3
## 3          5             4
## 4          6             3
## 5         10             4
## 6         11             1
## 7         29             3
## 8         31             3

1.4.4 Grouped Frequency Distribution Tables (Classification According to Class-Intervals)

If amount of data is large we put it into groups/categories/classes and determine number of units in each category (class frequency).

A grouped frequency distribution table normally has columns which show the class intervals, class mid-points, class frequencies, and cumulative frequencies, the last of these being a running total of the frequencies themselves. There may also be a column of tallied frequencies, if the table is being constructed from the raw data without having first arranged the values in rank order.

1.5.1 Histogram

A histogram consists of a set of adjoining rectangles such that their bases are on the x-axis with centers at class marks and length equals class interval size. The horizontal axis is labeled with what the data represents (for instance, distance from campus to your hostel). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram can give you the shape of the data, the center, and the spread of the data.

The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample or population. If:

• \(f\) = frequency
• \(n\) = total number of data values (or the sum of the individual frequencies), and
• \(RF\) = relative frequency,

Then:

\[RF = \frac{\text{frequency}}{\text{total frequencies}} = \frac{f}{n}\]

The areas of the rectangles are proportional to the class frequencies. If class intervals have equal sizes the histogram is obtained by plotting the frequencies against the true class limits (class boundaries) such that the heights of rectangles are proportional to class frequencies.

But if class intervals are not equal, then plot the frequency density (or relative frequencies) against the class boundaries as illustrated in Example 1.4 (ii).

\[\text{Frequency density} = \frac{\text{frequency } (f)}{\text{class width } (i)}\]

To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. This usually equals the number of intervals/classes in the data set. Choose a starting point to be the lower class boundary of a class lower than the first interval in the data set. For instance if the class intervals were: 10–15, 15–20, … then the first interval will be 5–10 with a height/frequency zero.

Example 1.4 (i): Histogram – Equal Class Widths

Represent the following data by a histogram.

The class intervals are of equal size and class boundaries are given since the exclusive method of data classification has been used.

marks_mid  <- seq(5, 95, by = 10)
marks_freq <- c(5, 11, 19, 21, 16, 10, 8, 6, 3, 1)

hist_df <- data.frame(
  lower = seq(0, 90, by = 10),
  upper = seq(10, 100, by = 10),
  freq  = marks_freq
)

ggplot(hist_df, aes(xmin = lower, xmax = upper, ymin = 0, ymax = freq)) +
  geom_rect(fill = "#2e86c1", color = "white", linewidth = 0.5) +
  scale_x_continuous(breaks = seq(0, 100, by = 10),
                     labels = seq(0, 100, by = 10)) +
  labs(title = "Histogram of student marks",
       x     = "Marks",
       y     = "Frequency") +
  theme_classic(base_size = 13) +
  theme(plot.title = element_text(hjust = 0.5, color = "#1a5276", face = "bold"))

Figure 1.2: Histogram of student marks (equal class widths)

Example 1.4 (ii): Histogram – Unequal Class Widths (Frequency Density)

Construct a histogram to represent the following data set:

The class sizes are unequal and therefore to construct the histogram we use frequency density for each class calculated as \(fd = \frac{f}{i}\).

Class limits are given hence to obtain class boundaries (true class limits), we adjust the limits by using the correction factor.

unequal_hist <- data.frame(
  lower = c(14.5, 19.5, 29.5, 34.5, 39.5, 54.5),
  upper = c(19.5, 29.5, 34.5, 39.5, 54.5, 59.5),
  freq  = c(5, 8, 22, 35, 20, 10),
  width = c(5, 10, 5, 5, 15, 5)
)
unequal_hist$fd <- unequal_hist$freq / unequal_hist$width

ggplot(unequal_hist,
       aes(xmin = lower, xmax = upper, ymin = 0, ymax = fd)) +
  geom_rect(fill = "#117a65", color = "white", linewidth = 0.5) +
  scale_x_continuous(breaks = c(14.5, 19.5, 29.5, 34.5, 39.5, 54.5, 59.5)) +
  labs(title = "Histogram using frequency density (unequal class widths)",
       x     = "Class boundaries",
       y     = "Frequency density (f/i)") +
  theme_classic(base_size = 13) +
  theme(plot.title    = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        axis.text.x   = element_text(angle = 45, hjust = 1))

Figure 1.3: Histogram of unequal class widths (frequency density)

Exercise: Suppose the classes were of equal widths, then construct a histogram (DIY).

Class limits	15–19	20–24	25–29	30–34	35–39	40–44
Frequency	1	4	22	35	20	8

1.5.2 Frequency Polygon

A frequency polygon is a graphical form of representation of data. It is used to depict the shape of the data and to depict trends. It is usually drawn with the help of a histogram but can be drawn without it as well. If a histogram is already drawn and the midpoint of adjacent rectangles joined by straight lines we will obtain frequency polygons.

Steps to Draw a Frequency Polygon

1. Mark the class intervals for each class on the horizontal axis. We will plot the frequency on the vertical axis.
2. Calculate the class mark for each class interval. The formula for class mark is:

\[\text{Class mark} = \frac{\text{Upper limit} + \text{Lower limit}}{2}\]

3. Mark all the class marks on the horizontal axis. It is also known as the mid-value of every class.
4. Corresponding to each class mark, plot the frequency as given to you. The height always depicts the frequency. Make sure that the frequency is plotted against the class mark and not the upper or lower limit of any class.
5. Join all the plotted points using a line segment. The curve obtained will be kinked.
6. This resulting curve is called the frequency polygon.

N/B: It can be drawn without rectangles.

Example: Frequency Polygon of Student Marks

Plot the frequency polygon of the marks of students given in (a) above.

Solution:

Note: It is customary to add the extensions PQ and RS to the next lower and next higher midpoints which have corresponding class frequencies of zero.

fp_data <- data.frame(
  midpoint  = c(0, 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 105),
  frequency = c(0, 5, 11, 19, 21, 16, 10,  8,  6,  3,  1,   0)
)

# Underlying histogram bars
hist_bars <- data.frame(
  lower = seq(0, 90, by = 10),
  upper = seq(10, 100, by = 10),
  freq  = c(5, 11, 19, 21, 16, 10, 8, 6, 3, 1)
)

ggplot() +
  geom_rect(data = hist_bars,
            aes(xmin = lower, xmax = upper, ymin = 0, ymax = freq),
            fill = "#aed6f1", color = "white", linewidth = 0.4, alpha = 0.6) +
  geom_line(data = fp_data,
            aes(x = midpoint, y = frequency),
            color = "#1a5276", linewidth = 1.2) +
  geom_point(data = fp_data,
             aes(x = midpoint, y = frequency),
             color = "#e74c3c", size = 2.5) +
  scale_x_continuous(breaks = seq(0, 105, by = 10)) +
  labs(title = "Frequency polygon of student marks",
       x     = "Marks (midpoints)",
       y     = "Frequency") +
  theme_classic(base_size = 13) +
  theme(plot.title = element_text(hjust = 0.5, color = "#1a5276", face = "bold"))

Figure 1.4: Frequency polygon of student marks

Plot the Frequency Polygon – Given Data Set

fp2_data <- data.frame(
  midpoint  = c(12, 17, 22, 27, 32, 37, 42, 47),
  frequency = c( 0,  1,  4, 22, 35, 20,  8,  0)
)

hist2_bars <- data.frame(
  lower = c(14.5, 19.5, 24.5, 29.5, 34.5, 39.5),
  upper = c(19.5, 24.5, 29.5, 34.5, 39.5, 44.5),
  freq  = c(1, 4, 22, 35, 20, 8)
)

ggplot() +
  geom_rect(data = hist2_bars,
            aes(xmin = lower, xmax = upper, ymin = 0, ymax = freq),
            fill = "#a9dfbf", color = "white", linewidth = 0.4, alpha = 0.6) +
  geom_line(data = fp2_data,
            aes(x = midpoint, y = frequency),
            color = "#117a65", linewidth = 1.2) +
  geom_point(data = fp2_data,
             aes(x = midpoint, y = frequency),
             color = "#e74c3c", size = 2.5) +
  scale_x_continuous(breaks = c(12, 17, 22, 27, 32, 37, 42, 47)) +
  labs(title = "Frequency polygon",
       x     = "Class mid-points",
       y     = "Frequency") +
  theme_classic(base_size = 13) +
  theme(plot.title = element_text(hjust = 0.5, color = "#1a5276", face = "bold"))

Figure 1.5: Frequency polygon – second data set

1.5.3 Bar Graph (Chart)

The height of the bar is proportional to the frequency of the variate but the thickness of the bar is insignificant. A bar chart comprises a number of spaced rectangles and thus do not suggest continuity and which generally have their major axes vertical. They can be used to represent a large variety of statistical data. The bar chart is appropriate for displaying discrete data with only a few categories.

(a) Simple Bar Chart

Example 1.5 (i)– Birth Rates by Country

The following table gives the birth rate per thousand of different countries over a certain period of time.

Represent the above data by a suitable diagram.

Solution: The appropriate diagram for this data is a simple bar diagram.

birth_rate <- data.frame(
  Country    = c("Kenya","India","China","Uganda","U.K.","Sweden"),
  Birth_Rate = c(30, 33, 40, 29, 20, 15)
)
birth_rate$Country <- factor(birth_rate$Country,
                             levels = birth_rate$Country[order(birth_rate$Birth_Rate)])

ggplot(birth_rate, aes(x = Country, y = Birth_Rate)) +
  geom_bar(stat = "identity", fill = "#2e86c1", width = 0.6) +
  geom_text(aes(label = Birth_Rate), vjust = -0.4, size = 4, color = "#1a5276") +
  labs(title = "Birth rate per thousand by country",
       x     = "Country",
       y     = "Birth Rate (per thousand)") +
  theme_classic(base_size = 13) +
  theme(plot.title = element_text(hjust = 0.5, color = "#1a5276", face = "bold"))

Figure 1.6: Simple bar chart of birth rates by country

Comparing the size of the bars, you can easily see that China has the highest birth rate while Sweden has the lowest.

Example 1.5 (ii): Bacterial Meningitis Cases

Consider data relating to the number of patients diagnosed with Bacterial meningitis in a hospital each year.

This data can be represented by the bar chart as shown below.

The number of patients diagnosed with Bacterial meningitis in a hospital during the period 2001 – 2005.

mening <- data.frame(
  Year     = factor(2001:2005),
  Patients = c(141, 225, 205, 108, 192)
)

ggplot(mening, aes(x = Year, y = Patients)) +
  geom_bar(stat = "identity", fill = "#117a65", width = 0.6) +
  geom_text(aes(label = Patients), vjust = -0.4, size = 4, color = "#117a65") +
  labs(title = "Bacterial meningitis patients (2001–2005)",
       x     = "Year",
       y     = "Number of patients") +
  theme_classic(base_size = 13) +
  theme(plot.title = element_text(hjust = 0.5, color = "#1a5276", face = "bold"))

Figure 1.7: Bacterial meningitis cases 2001–2005

Notice that it is now easy to see that there are variations in the number of cases over this period of time.

(b) Multiple Bar Chart Bar charts often prove most useful if we have two (or more) sets of comparable data, and wish to compare and contrast them.

Example 1.6

Suppose that apart from the data relating to the number of patients diagnosed with Bacterial meningitis in a hospital each year, we also have the corresponding numbers for Malaria cases.

multi_long <- data.frame(
  Year     = rep(factor(2001:2005), 2),
  Disease  = c(rep("Meningitis", 5), rep("Malaria", 5)),
  Patients = c(141, 225, 205, 108, 192, 321, 251, 123, 547, 148)
)

ggplot(multi_long, aes(x = Year, y = Patients, fill = Disease)) +
  geom_bar(stat = "identity", position = "dodge", width = 0.7) +
  geom_text(aes(label = Patients),
            position = position_dodge(width = 0.7),
            vjust = -0.4, size = 3.5) +
  scale_fill_manual(values = c("Meningitis" = "#2e86c1",
                               "Malaria"    = "#e67e22")) +
  labs(title = "Meningitis vs malaria cases (2001–2005)",
       x     = "Year",
       y     = "Number of patients",
       fill  = "Disease") +
  theme_classic(base_size = 13) +
  theme(plot.title   = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        legend.position = "top")

Figure 1.8: Multiple bar chart: meningitis vs malaria cases 2001–2005

(c) Component Bar Charts (Sub-divided Bar Diagrams)

In this type of bar chart each bar is subdivided into two or more components.

Example 1.7

Suppose further that the data in the example above is grouped according to sex as follows:

This data can be represented in a component bar chart as shown in the figure below. Looking at this presentation, it is possible to discern two main features; firstly, we can see how the meningitis cases vary from year to year and secondly we can get a good idea of the make up of this total in terms of proportions of patients who are male or female.

comp_long <- data.frame(
  Year     = rep(factor(2001:2005), 2),
  Sex      = c(rep("Male", 5), rep("Female", 5)),
  Patients = c(100, 125, 90, 20, 102, 41, 100, 115, 88, 90)
)

ggplot(comp_long, aes(x = Year, y = Patients, fill = Sex)) +
  geom_bar(stat = "identity", position = "stack", width = 0.6) +
  geom_text(aes(label = Patients),
            position = position_stack(vjust = 0.5),
            color = "white", size = 4, fontface = "bold") +
  scale_fill_manual(values = c("Male" = "#2e86c1", "Female" = "#e74c3c")) +
  labs(title = "Component bar chart: meningitis patients by sex (2001–2005)",
       x     = "Year",
       y     = "Number of patients",
       fill  = "Sex") +
  theme_classic(base_size = 13) +
  theme(plot.title      = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        legend.position = "top")

Figure 1.9: Component bar chart: meningitis cases by sex 2001–2005

1.5.4 Pie Chart

A pie chart presents data in the form of a circle. The slices represent absolute or relative proportions. A pie chart is formed by making a portion of the pie corresponding to each characteristic being displayed.

Example 1.8

A researcher studying the distribution of manufacturing costs in ABC Ltd found that 20% of the firm’s unit cost is due to labour, 40% raw materials, 25% maintenance costs and 15% debt servicing. Present this information in a pie chart.

Fig 2: A pie chart representing the distribution of ABC Ltd per unit manufacturing cost during the year.

pie_data <- data.frame(
  Component  = c("Labour","Raw Materials","Maintenance costs","Debt servicing"),
  Percentage = c(20, 40, 25, 15)
)
pie_data$Component <- factor(pie_data$Component,
                             levels = pie_data$Component)
pie_data$label     <- paste0(pie_data$Component, "\n", pie_data$Percentage, "%")

ggplot(pie_data, aes(x = "", y = Percentage, fill = Component)) +
  geom_bar(stat = "identity", width = 1, color = "white", linewidth = 0.7) +
  coord_polar(theta = "y", start = 0) +
  geom_text(aes(label = paste0(Percentage, "%")),
            position = position_stack(vjust = 0.5),
            color = "white", size = 5, fontface = "bold") +
  scale_fill_manual(values = c("Labour"           = "#2e86c1",
                               "Raw Materials"     = "#e67e22",
                               "Maintenance costs" = "#117a65",
                               "Debt servicing"    = "#8e44ad")) +
  labs(title = "ABC Ltd: per unit manufacturing cost distribution",
       fill  = "Component") +
  theme_void(base_size = 13) +
  theme(plot.title      = element_text(hjust = 0.5, color = "#1a5276",
                                       face = "bold", size = 13),
        legend.position = "right")

Figure 1.10: Pie chart: ABC Ltd manufacturing cost distribution

1.6.1 Frequency Curve

Consider for example the sales data for some company over a period of six years as shown in the table below:

sales_df2 <- data.frame(
  Year  = c(2000, 2001, 2002, 2003, 2004),
  Sales = c(420000, 370000, 360000, 380000, 540000)
)

ggplot(sales_df2, aes(x = Year, y = Sales)) +
  geom_line(color = "#1a237e", linewidth = 1.2) +
  geom_point(color = "#1a237e", size = 2) +
  scale_x_continuous(breaks = c(2000, 2001, 2002, 2003, 2004),
                     labels = c("2000","2001","2002","2003","2004")) +
  scale_y_continuous(breaks = seq(0, 600000, by = 100000),
                     labels = c("0","100,000","200,000","300,000",
                                "400,000","500,000","600,000"),
                     limits = c(0, 620000)) +
  labs(title = NULL, x = NULL, y = NULL) +
  theme_gray(base_size = 12) +
  theme(
    panel.background = element_rect(fill = "#c0c0c0", color = NA),
    plot.background  = element_rect(fill = "white", color = "black", linewidth = 0.8),
    panel.grid.major = element_line(color = "white", linewidth = 0.4),
    panel.grid.minor = element_blank(),
    axis.text        = element_text(color = "black", size = 10)
  )

Figure 1.11: Sales data for a company (2000–2004)

This original data can be presented in a graphical form as follows.

1.6.2 Cumulative Frequency Curve (Ogive)

(i) “Less Than” Ogive

The cumulative frequency curve is obtained by first plotting the points with the upper class boundaries of each class interval on the X-axis and their corresponding cumulative frequencies on the Y-axis. The points are joined by means of a freehand smooth curve. The cumulative frequency curve is specifically called the “Less than” Ogive curve.

Example 1.9

Plot the “Less than” ogive curve of the marks of students given in example 2 above.

Solution:

lt_data <- data.frame(
  upper_boundary = c(0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100),
  cum_freq       = c(0,  5, 16, 35, 56, 72, 82, 90, 96, 99, 100)
)

ggplot(lt_data, aes(x = upper_boundary, y = cum_freq)) +
  geom_line(color = "#1f618d", linewidth = 1.2) +
  geom_point(color = "#e74c3c", size = 3) +
  scale_x_continuous(breaks = seq(0, 100, by = 10)) +
  scale_y_continuous(breaks = seq(0, 100, by = 10)) +
  labs(title = '"Less than" Ogive Curve of Student Marks',
       x     = "Upper Class Boundary (Marks)",
       y     = "Cumulative Frequency") +
  theme_classic(base_size = 13) +
  theme(plot.title = element_text(hjust = 0.5, color = "#1a5276", face = "bold"))

Figure 1.12: ‘Less than’ ogive curve of student marks

From the graph there are “y” students who scored less than “x” marks.

(ii) “More Than” Ogive

If we plot the “more than” cumulative frequencies against the corresponding lower class boundaries and join the points by a smooth curve, we get a “more than” ogive.

Example 1.10

Plot the “More than” ogive curve of the marks of students given in example 2 above.

Solution:

mt_data <- data.frame(
  lower_boundary = c(0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100),
  cum_freq       = c(100, 95, 84, 65, 44, 28, 18, 10,  4,  1,   0)
)

ggplot(mt_data, aes(x = lower_boundary, y = cum_freq)) +
  geom_line(color = "#117a65", linewidth = 1.2) +
  geom_point(color = "#e74c3c", size = 3) +
  scale_x_continuous(breaks = seq(0, 100, by = 10)) +
  scale_y_continuous(breaks = seq(0, 100, by = 10)) +
  labs(title = '"More than" Ogive Curve of Student Marks',
       x     = "Lower Class Boundary (Marks)",
       y     = "More Than Cumulative Frequency") +
  theme_classic(base_size = 13) +
  theme(plot.title = element_text(hjust = 0.5, color = "#1a5276", face = "bold"))

Figure 1.13: ‘More than’ ogive curve of student marks

From the graph there are “y” students who scored more than “x” marks.

The value of x at the intersection of the two graphs is the median value.

Both Ogive Curves on the Same Graph

The intersection of the “less than” and “more than” ogive curves gives the median.

lt_data$type <- "Less than"
mt_data2 <- data.frame(
  upper_boundary = c(0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100),
  cum_freq       = c(100, 95, 84, 65, 44, 28, 18, 10,  4,  1,   0),
  type           = "More than"
)
names(lt_data)[1] <- "boundary"
names(mt_data2)[1] <- "boundary"

both_ogive <- rbind(lt_data, mt_data2)

ggplot(both_ogive, aes(x = boundary, y = cum_freq, color = type)) +
  geom_line(linewidth = 1.2) +
  geom_point(size = 2.5) +
  scale_color_manual(values = c("Less than" = "#1f618d", "More than" = "#117a65")) +
  scale_x_continuous(breaks = seq(0, 100, by = 10)) +
  scale_y_continuous(breaks = seq(0, 100, by = 10)) +
  labs(title = '"Less than" and "More than" Ogive Curves',
       x     = "Class Boundary (Marks)",
       y     = "Cumulative Frequency",
       color = "Ogive type") +
  theme_classic(base_size = 13) +
  theme(plot.title      = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        legend.position = "top")

Figure 1.14: Both ogive curves – median at intersection

This is a graph of upper class boundaries and cumulative frequencies (\(c_f\)).

Exercise 1.2

1. Consider the following data:

Arrange the data in a frequency distribution with the first class interval 10 – 19.

2. The highway patrol set up a radar checkpoint and recorded the speed in miles per hour of a random sample of 50 cars that passed the checkpoint in one hour. The speed of the cars was recorded as follows:

Make a frequency distribution table using 5 as the class width.

3. Given the data below:

Form a frequency distribution using the classes 2.7–2.9, 3.0–3.2, 3.3–3.5, …

4. Using Sturges’ rule,

\[K = 1 + 3.322 \log_{10} N\]

where \(K\) = number of class-intervals and \(N\) = total number of observations; classify, in equal intervals, the following hours worked by 20 workers in a factory for one month:

Find the percentage frequency in each class-interval.

5. Represent the following data by a histogram.

6. Using the data classified in questions 1, 2 and 3, draw:

   1. A Histogram
   2. A Frequency polygon
   3. “Less than” and “more than” Ogive curves

7. A nutritionist is interested in knowing the percent of calories from fat which Kenyans intake on a daily basis. To study this, the nutritionist randomly selects 25 Kenyans and evaluates the percent of calories from fat consumed in a typical day. The results of the study are as follows:

Construct a frequency distribution and the corresponding histogram.

8. In Kenya, approximately 45% of the population has blood type O; 40% type A; 11% type B; and 4% type AB. Illustrate this distribution of blood types with a pie chart.

blood_df <- data.frame(
  Type       = c("Type O", "Type A", "Type B", "Type AB"),
  Percentage = c(45, 40, 11, 4)
)

ggplot(blood_df, aes(x = "", y = Percentage, fill = Type)) +
  geom_bar(stat = "identity", width = 1, color = "white", linewidth = 0.7) +
  coord_polar(theta = "y", start = 0) +
  geom_text(aes(label = paste0(Percentage, "%")),
            position = position_stack(vjust = 0.5),
            color = "white", size = 5, fontface = "bold") +
  scale_fill_manual(values = c("Type O"  = "#2e86c1",
                               "Type A"  = "#e67e22",
                               "Type B"  = "#117a65",
                               "Type AB" = "#8e44ad")) +
  labs(title = "Distribution of Blood Types in Kenya",
       fill  = "Blood Type") +
  theme_void(base_size = 13) +
  theme(plot.title      = element_text(hjust = 0.5, color = "#1a5276",
                                       face = "bold", size = 13),
        legend.position = "right")

Figure 1.15: Distribution of blood types in Kenya

9. In the academic years 1982 to 1985, the number of students in College ABC were as follows:

Represent the data by an appropriate diagram (Component bar chart).

ex9_long <- data.frame(
  Year    = rep(c("1982–83", "1983–84", "1984–85"), 3),
  Faculty = c(rep("Science", 3), rep("Arts", 3), rep("Law", 3)),
  Count   = c(1000, 1600, 2100, 1500, 2000, 4000, 200, 350, 420)
)
ex9_long$Year    <- factor(ex9_long$Year, levels = c("1982–83","1983–84","1984–85"))
ex9_long$Faculty <- factor(ex9_long$Faculty, levels = c("Law","Science","Arts"))

ggplot(ex9_long, aes(x = Year, y = Count, fill = Faculty)) +
  geom_bar(stat = "identity", position = "stack", width = 0.6) +
  geom_text(aes(label = Count),
            position = position_stack(vjust = 0.5),
            color = "white", size = 4, fontface = "bold") +
  scale_fill_manual(values = c("Science" = "#2e86c1",
                               "Arts"    = "#e67e22",
                               "Law"     = "#117a65")) +
  labs(title = "Students in College ABC by Faculty (1982–1985)",
       x     = "Academic Year",
       y     = "Number of Students",
       fill  = "Faculty") +
  theme_classic(base_size = 13) +
  theme(plot.title      = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        legend.position = "top")

Figure 1.16: Component bar chart: students in College ABC by faculty (1982–1985)

10. The table below gives data relating to the Kenyan exports and imports (in millions of Ksh) during the four years ending 1999–2004:

Represent this information using a suitable diagram (multiple bar chart).

ex10_long <- data.frame(
  Year  = rep(c("1999–2000","2000–2001","2001–2002","2002–2003","2003–2004"), 2),
  Type  = c(rep("Export", 5), rep("Import", 5)),
  Value = c(160000,170000,180000,200000,200000,
            200000,300000,350000,300000,380000)
)
ex10_long$Year <- factor(ex10_long$Year,
                         levels = c("1999–2000","2000–2001","2001–2002",
                                    "2002–2003","2003–2004"))

ggplot(ex10_long, aes(x = Year, y = Value / 1000, fill = Type)) +
  geom_bar(stat = "identity", position = "dodge", width = 0.7) +
  geom_text(aes(label = paste0(Value/1000, "K")),
            position = position_dodge(width = 0.7),
            vjust = -0.4, size = 3.5) +
  scale_fill_manual(values = c("Export" = "#2e86c1", "Import" = "#e67e22")) +
  labs(title = "Kenyan Exports and Imports (1999–2004)",
       x     = "Year",
       y     = "Value (thousands of millions Ksh)",
       fill  = "Trade type",
       caption = "Source: KNBS") +
  theme_classic(base_size = 13) +
  theme(plot.title      = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        axis.text.x     = element_text(angle = 30, hjust = 1),
        legend.position = "top")

Figure 1.17: Multiple bar chart: Kenyan exports and imports (1999–2004)

11. The following table shows the Kenyan population age structure as per the 2009 census:

How best would you represent this data diagrammatically?

ex11_long <- data.frame(
  Age = rep(c("0–14","15–24","25–54","55–64","65+"), 2),
  Sex = c(rep("Male", 5), rep("Female", 5)),
  Population = c(9557274, 4552448, 8170264, 856092, 614751,
                 9497870, 4567894, 7976751, 1009075, 813320)
)
ex11_long$Age <- factor(ex11_long$Age,
                        levels = c("0–14","15–24","25–54","55–64","65+"))

ggplot(ex11_long, aes(x = Age, y = Population / 1e6, fill = Sex)) +
  geom_bar(stat = "identity", position = "dodge", width = 0.7) +
  scale_fill_manual(values = c("Male" = "#2e86c1", "Female" = "#e74c3c")) +
  labs(title = "Kenyan Population Age Structure by Sex (2009 Census)",
       x     = "Age Group",
       y     = "Population (millions)",
       fill  = "Sex",
       caption = "Source: CIA World Factbook 2017") +
  theme_classic(base_size = 13) +
  theme(plot.title      = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        legend.position = "top")

Figure 1.18: Component bar chart: Kenyan population by age group and sex (2009 Census)

12. The following data represents the maximum temperatures in degrees centigrade predicted for some 55 major cities on the 24th September 1993.

Construct a frequency distribution table for these temperatures starting with the classes 11–17, 18–24, …

Solution:

(a) Histogram of Maximum Temperatures

Represent the data using a histogram.

temp_hist <- data.frame(
  lower = c(10.5, 17.5, 24.5, 31.5, 38.5),
  upper = c(17.5, 24.5, 31.5, 38.5, 45.5),
  freq  = c(15, 15, 16, 7, 2)
)

ggplot(temp_hist, aes(xmin = lower, xmax = upper, ymin = 0, ymax = freq)) +
  geom_rect(fill = "#2e86c1", color = "white", linewidth = 0.5) +
  scale_x_continuous(breaks = c(10.5, 17.5, 24.5, 31.5, 38.5, 45.5)) +
  labs(title = "Histogram of Maximum Temperatures (55 Cities, Sept 1993)",
       x     = "Temperature (°C) – Class Boundaries",
       y     = "Frequency") +
  theme_classic(base_size = 13) +
  theme(plot.title  = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        axis.text.x = element_text(angle = 30, hjust = 1))

Figure 1.19: Histogram of maximum temperatures for 55 cities

(b) Frequency Polygon of Maximum Temperatures

temp_poly <- data.frame(
  midpoint  = c(3.5, 14, 21, 28, 35, 42, 49),
  frequency = c(  0, 15, 15, 16,  7,  2,  0)
)

temp_bars <- data.frame(
  lower = c(10.5, 17.5, 24.5, 31.5, 38.5),
  upper = c(17.5, 24.5, 31.5, 38.5, 45.5),
  freq  = c(15, 15, 16, 7, 2)
)

ggplot() +
  geom_rect(data = temp_bars,
            aes(xmin = lower, xmax = upper, ymin = 0, ymax = freq),
            fill = "#aed6f1", color = "white", linewidth = 0.4, alpha = 0.6) +
  geom_line(data = temp_poly,
            aes(x = midpoint, y = frequency),
            color = "#1a5276", linewidth = 1.2) +
  geom_point(data = temp_poly,
             aes(x = midpoint, y = frequency),
             color = "#e74c3c", size = 3) +
  scale_x_continuous(breaks = c(3.5, 14, 21, 28, 35, 42, 49)) +
  labs(title = "Frequency Polygon of Maximum Temperatures (55 Cities)",
       x     = "Class Mid-points (°C)",
       y     = "Frequency") +
  theme_classic(base_size = 13) +
  theme(plot.title  = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        axis.text.x = element_text(angle = 30, hjust = 1))

Figure 1.20: Frequency polygon of maximum temperatures for 55 cities

(c) Ogive Curves of Maximum Temperatures

temp_lt <- data.frame(
  boundary = c(10.5, 17.5, 24.5, 31.5, 38.5, 45.5),
  cum_freq = c(0, 15, 30, 46, 53, 55),
  type     = "Less than"
)
temp_mt <- data.frame(
  boundary = c(10.5, 17.5, 24.5, 31.5, 38.5, 45.5),
  cum_freq = c(55, 40, 25, 9, 2, 0),
  type     = "More than"
)
temp_both <- rbind(temp_lt, temp_mt)

ggplot(temp_both, aes(x = boundary, y = cum_freq, color = type)) +
  geom_line(linewidth = 1.2) +
  geom_point(size = 3) +
  scale_color_manual(values = c("Less than" = "#1f618d", "More than" = "#117a65")) +
  scale_x_continuous(breaks = c(10.5, 17.5, 24.5, 31.5, 38.5, 45.5)) +
  scale_y_continuous(breaks = seq(0, 55, by = 5)) +
  labs(title = "Ogive Curves for Maximum Temperatures (55 Cities)",
       x     = "Class Boundary (°C)",
       y     = "Cumulative Frequency",
       color = "Ogive type") +
  theme_classic(base_size = 13) +
  theme(plot.title      = element_text(hjust = 0.5, color = "#1a5276", face = "bold"),
        legend.position = "top",
        axis.text.x     = element_text(angle = 30, hjust = 1))

Figure 1.21: Less than and more than ogive curves for maximum temperatures

(d) Using the ogive curves, estimate:

i. The modal temperature

The modal class is 25 – 31°C (highest frequency = 16). The modal temperature is approximately 28°C.

ii. The median temperature

The median is at cumulative frequency = 55/2 = 27.5. From the “less than” ogive, reading off at \(cf = 27.5\) gives a median of approximately \(\approx 24.5°C\).

iii. The lower and upper class boundaries of the temperature range within which the middle 50% of all cities lie.

The middle 50% lies between \(Q_1\) (at \(cf = 13.75\)) and \(Q_3\) (at \(cf = 41.25\)). From the ogive: \(Q_1 \approx 17.5°C\) and \(Q_3 \approx 31.5°C\).

iv. The minimum and maximum temperature of the middle 80% of the cities.

The middle 80% lies between the 10th percentile (\(cf = 5.5\)) and the 90th percentile (\(cf = 49.5\)). From the ogive: \(P_{10} \approx 14°C\) and \(P_{90} \approx 35°C\).

v. On this particular day, a researcher was collecting data and required data from cities whose temperatures were above \(29.5°C\). How many of these cities did he include in his study?

From the “less than” ogive, at \(x = 29.5°C\), cumulative frequency \(\approx 43\). Therefore, cities with temperature \(> 29.5°C\) = \(55 - 43 = \mathbf{12}\) cities.

2.5.1 Correcting Incorrect Values

It sometimes happens that due to an oversight or mistake in copying, certain wrong values are taken while calculating the mean. The process of correction is simple: from \(\sum x\) deduct the wrong observations and add the correct observations, then divide the correct \(\sum x\) by the number of observations.

Example 2.4

i. The average weekly wage for a group of 25 persons working in a factory was calculated to be $378.40. It was later discovered that one figure was misread as $160 instead of the correct value $200. Calculate the correct average wage.

Solution:

\[\text{Incorrect } \sum x = 378.40 \times 25 = 9{,}460\]

\[\text{Correct } \sum x = 9{,}460 - 160 + 200 = 9{,}500\]

\[\text{Correct mean} = \frac{9{,}500}{25} = \mathbf{\$380}\]

ii. The mean of 200 observations was 50. Later on, it was discovered that two observations were wrongly read as 92 and 8 instead of 192 and 88. Find out the correct mean.

Solution:

\[\text{Incorrect } \sum x = 50 \times 200 = 10{,}000\]

\[\text{Correct } \sum x = 10{,}000 - 92 - 8 + 192 + 88 = 10{,}180\]

\[\text{Correct mean} = \frac{10{,}180}{200} = \mathbf{50.9}\]

2.5.2 Exercise

1. The mean of seven numbers is seven. One number is removed and the mean increases to 10. Find the number which was removed.

2. The average weight of a group of 30 friends increases by 1 kg when the weight of their football coach was added. If the average weight of the group after including the weight of the football coach is 31 kg, what is the weight of their football coach?

3. The average wages of a worker during a fortnight comprising 15 consecutive working days was $90 per day. During the first 7 days, his average wages was $87 per day and the average wages during the last 7 days was $92 per day. What was his wage on the 8th day?

4. The average age of a group of 10 students was 20. The average age increased by 2 years when two new students joined the group. What is the average age of the two new students who joined the group?

2.5.3 Combined Mean

If we have the arithmetic mean and number of observations of two or more related groups, we can compute the combined average using the following formula:

\[\bar{X}_{12} = \frac{N_1 \bar{X}_1 + N_2 \bar{X}_2}{N_1 + N_2}\]

where:

• \(\bar{X}_{12}\) = Combined mean of the two groups
• \(\bar{X}_1\) = Arithmetic mean of the first group
• \(\bar{X}_2\) = Arithmetic mean of the second group
• \(N_1\) = Number of observations in the first group
• \(N_2\) = Number of observations in the second group

Example 2.5

i. There are two branches of a company employing 100 and 80 employees respectively. If the arithmetic means of the monthly salaries paid by two branches are $4,570 and $6,750 respectively, find the arithmetic mean of the salaries of the employees of the company as a whole.

Solution:

\[\bar{X}_{12} = \frac{N_1 \bar{X}_1 + N_2 \bar{X}_2}{N_1 + N_2} = \frac{100 \times 4570 + 80 \times 6750}{100 + 80} = \frac{457000 + 540000}{180} = \frac{997000}{180} = \mathbf{\$5538.89}\]

If we have to find out the combined mean of three related groups, the above formula can be extended as: \[\bar{X}_{123} = \frac{N_1\bar{X}_1 + N_2\bar{X}_2 + N_3\bar{X}_3}{N_1 + N_2 + N_3}\]

ii. The mean of marks in Statistics of 100 students of a class was 72. The mean of marks of boys was 75, while their number was 70. Find out the mean marks of girls in the class.

Solution:

We are given \(N_1 + N_2 = 100\), \(\bar{X}_{12} = 72\), mean of boys \(\bar{X}_1 = 75\), number of boys \(N_1 = 70\). We have to find out the mean marks of girls, i.e., \(\bar{X}_2\).

\[72 = \frac{70 \times 75 + 30 \times \bar{X}_2}{100}\]

\[7200 = 5250 + 30\bar{X}_2 \implies \bar{X}_2 = \frac{7200 - 5250}{30} = \frac{1950}{30} = \mathbf{65}\]

Hence the mean marks of girls in the class = 65.

iii. The mean age of a combined group of men and women is 30 years. If the mean age of the group of men is 32 and that of the group of women is 25, find out the percentage of men and women in the group.

Solution:

Let \(N_1\) represent the percentage of men and \(N_2\) represent the percentage of women so that \(N_1 + N_2 = 100\). We are given \(\bar{X}_{12} = 30\), \(\bar{X}_1 = 32\), \(\bar{X}_2 = 25\).

\[30 = \frac{32N_1 + 25N_2}{N_1 + N_2} = \frac{32N_1 + 25(100 - N_1)}{100}\]

\[3000 = 32N_1 + 2500 - 25N_1 = 7N_1 + 2500\]

\[N_1 = \frac{500}{7} \approx 71.43\% \quad \text{(men)}, \qquad N_2 \approx 28.57\% \quad \text{(women)}\]

iv. A shopkeeper has 50 cold drink bottles. Some of the bottles are 1-litre and some are 2-litre bottles. The average cold drink of the bottles is 1200 ml. Find the number of 2-litre bottles. (1 litre = 1000 ml)

Solution:

Let the number of 1-litre bottles be \(N_1\) and the number of 2-litre bottles be \(N_2\). We know that \(N_1 + N_2 = 50\). The average of group 1 (\(\bar{X}_1\)) is 1000 ml and the average of group 2 (\(\bar{X}_2\)) is 2000 ml. The weighted average is 1200 ml.

\[1200 = \frac{1000 N_1 + 2000 N_2}{N_1 + N_2} = \frac{1000 N_1 + 2000 N_2}{50}\]

\[60000 = 1000 N_1 + 2000 N_2\]

Since \(N_1 + N_2 = 50 \implies N_1 = 50 - N_2\):

\[60000 = 1000(50 - N_2) + 2000 N_2 = 50000 + 1000 N_2\]

\[N_2 = 10 \quad \text{and} \quad N_1 = 40\]

Thus, the shopkeeper has 10 bottles of 2-litre.

2.5.4 Weighted Arithmetic Mean

The arithmetic mean discussed above gives equal importance to all observations. But there are cases where the relative importance of the different observations is not the same. When this is so, we compute weighted arithmetic mean. The term “weight” stands for the relative importance of the different observations. The formula for computing weighted arithmetic mean is:

\[\bar{X}_w = \frac{\sum WX}{\sum W}\]

where \(W\) represents the respective weights.

Example 2.6

A student’s final marks in Mathematics, Physics, English and Accounting are respectively 82, 86, 90, and 70. If the respective credits received for these courses are 3, 5, 3, and 1; determine the approximate average mark.

Solution:

\[\bar{X}_w = \frac{\sum WX}{\sum W} = \frac{1016}{12} = \mathbf{84.67}\]

2.5.5 Merits and Demerits of the Arithmetic Mean

Merits: Satisfies properties (i), (ii), (iii), (iv), (v), and (vi).

Demerits: Does not satisfy property (vii), i.e., it is affected by extreme observations.

3.7.1 Merits and Limitations of Average Deviation

Merits

• It is simple to understand and easy to compute.
• It is based on each and every observation of the data.
• It is less affected by the values of extreme observations.
• Since deviations are taken from a central value, comparison about formation of different distributions can easily be made.

Limitations

• The greatest drawback of this method is that algebraic signs are ignored while taking deviations of the items.
• This method may not give us very accurate results.
• It is not capable of further algebraic treatment.
• It is rarely used in sociological and business studies.

Because of these limitations its use is limited and it is overshadowed as a measure of variation by the superior standard deviation.

3.8.1 Merits and Limitations of Quartile Deviation

Merits

• In certain respects it is superior than range as a measure of variation.
• It has a special utility in measuring variation in case of open-end distributions or one in which the data may be ranked but measured quantitatively.
• It is also useful in erratic or highly skewed distributions, where the other measures of variation would be warped by extreme values.
• It is not affected by the presence of extreme values.

Limitations

• Quartile deviation ignores 50% of the items, i.e., the first 25% and the last 25%. As the value of quartile deviation does not depend upon every observation it cannot be regarded as a good method of measuring variation.
• It is not capable of mathematical manipulation.
• Its value is very much affected by sampling fluctuations.

3.9.1 Objectives

By the end of the topic, the learner should be able to:

• Calculate and interpret standard deviation.
• Calculate and interpret coefficient of variation for given data sets.
• Calculate combined standard deviation for two or more data sets.

3.9.2 Introduction

The standard deviation is by far the most important and widely used measure of studying variation. Its significance lies in the fact that it is free from those defects from which earlier methods suffer and satisfies most of the properties of a good measure of variation. It is a measure of how much “spread” or “variability” is present in the sample. If all the numbers in the sample are very close to each other, the standard deviation is close to zero. If the numbers are well dispersed, the standard deviation will tend to be large. Standard deviation is known as the root mean square deviation for the reason that it is the square root of the means of square deviations from the arithmetic mean. Standard deviation is denoted by the small Greek letter \(\sigma\) (read as sigma) and is defined as:

\[\sigma = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n}}\]

If we square standard deviation, we get what is called Variance.

\[\text{Variance} = \sigma^2 = \frac{\sum(x_i - \bar{x})^2}{n}\]

The standard deviation measures the absolute variation of a distribution; the greater the amount of variation, the greater the standard deviation. A small standard deviation means a high degree of uniformity of the observations as well as homogeneity of a series; a large standard deviation means a low degree of uniformity of the observations as well as homogeneity of a series. Thus if we have two or more comparable series with identical means, it is the distribution with the smallest standard deviation that has the most representative mean.

3.9.3 Calculation Using Direct Method

\[S = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n}}\]

\[= \sqrt{\frac{\sum f(x_i - \bar{x})^2}{\sum f}}\]

The square of standard deviation is variance:

\[S^2 = \frac{\sum(x_i - \bar{x})^2}{n}\]

\[S^2 = \frac{\sum f(x_i - \bar{x})^2}{\sum f}\]

Proof that \(\dfrac{\sum(x_i - \bar{x})^2}{n} = \dfrac{\sum x_i^2}{n} - \left(\dfrac{\sum x_i}{n}\right)^2\)

\[\frac{\sum(x_i - \bar{x})^2}{n} = \frac{\sum(x_i^2 - 2x_i\bar{x} + \bar{x}^2)}{n}\]

\[= \frac{\sum x_i^2}{n} - 2\bar{x}\frac{\sum x}{n} + \frac{n\bar{x}^2}{n}\]

\[= \frac{\sum x_i^2}{n} - 2\bar{x}^2 + \bar{x}^2\]

\[= \frac{\sum x_i^2}{n} - \bar{x}^2\]

\[= \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2\]

Therefore:

\[S^2 = \frac{\sum f x^2}{\sum f} - \left(\frac{\sum f x}{\sum f}\right)^2\]

3.9.4 Calculation Using Assumed Mean Method

If \(d_i = x_i - a\) where \(a\) is a constant, then:

\[S^2 = \frac{\sum f d^2}{\sum f} - \left(\frac{\sum f d}{\sum f}\right)^2\]

Proof

\[S^2 = \frac{\sum f x^2}{\sum f} - \left(\frac{\sum f x}{\sum f}\right)^2 \quad \text{where } d_i = x_i - a,\ x_i = d_i + a\]

\[= \frac{\sum f(d_i + a)^2}{\sum f} - \left(\frac{\sum f(d_i + a)}{\sum f}\right)^2\]

\[= \frac{\sum f(d_i^2 + 2d_ia + a^2)}{\sum f} - \left(\frac{\sum fd_i + \sum fa}{\sum f}\right)^2\]

\[= \frac{\sum fd_i^2}{\sum f} + \frac{2a\sum fd_i}{\sum f} + \frac{\sum fa^2}{\sum f} - \left(\frac{\sum fd_i}{\sum f} + \frac{\sum fa}{\sum f}\right)^2\]

\[= \frac{\sum fd_i^2}{\sum f} + \frac{2a\sum fd_i}{\sum f} + a^2 - \left(\frac{\sum fd_i}{\sum f} + a\right)^2\]

\[= \frac{\sum fd_i^2}{\sum f} + 2a\frac{\sum fd_i}{\sum f} + a^2 - \left(\frac{\sum fd_i}{\sum f}\right)^2 - 2a\frac{\sum fd_i}{\sum f} - a^2\]

\[= \frac{\sum fd_i^2}{\sum f} - \left(\frac{\sum fd_i}{\sum f}\right)^2 = S^2\]

3.9.5 Calculation Using Coding Method

If \(d_i = x_i - a = cu_i\), then:

\[S^2 = c^2\left\{\frac{\sum fu_i^2}{\sum f} - \left(\frac{\sum fu_i}{\sum f}\right)^2\right\}\]

Proof

\[S^2 = \frac{\sum fd_i^2}{\sum f} - \left(\frac{\sum fd_i}{\sum f}\right)^2\]

\[= \frac{\sum f(cu_i)^2}{\sum f} - \left(\frac{\sum f\,cu_i}{\sum f}\right)^2\]

\[= c^2\frac{\sum fu_i^2}{\sum f} - c^2\left(\frac{\sum fu_i}{\sum f}\right)^2\]

\[S^2 = c^2\left[\frac{\sum fu_i^2}{\sum f} - \left(\frac{\sum fu_i}{\sum f}\right)^2\right]\]

Example

Using Direct Method:

\[S^2 = \frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2 = \frac{519550}{99} - \left(\frac{7130}{99}\right)^2 = 5247.9798 - 5186.9095 = 61.0703\]

Using Assumed Mean Method:

\[S^2 = \frac{\sum fd^2}{\sum f} - \left(\frac{\sum fd}{\sum f}\right)^2 = \frac{6925}{99} - \left(\frac{-295}{99}\right)^2 = 69.9495 - 8.8792 = 61.0703\]

Using Coding Method (\(c = 5\)):

\[S^2 = c^2\left[\frac{\sum fu_i^2}{\sum f} - \left(\frac{\sum fu_i}{\sum f}\right)^2\right] = 25\left[\frac{277}{99} - \left(\frac{-59}{99}\right)^2\right] = 25(2.7980 - 0.3552) = 61.0703\]

3.9.6 Pooled Mean and Variance (When \(n\) is Equal)

The following is a summary of statistics of 2 samples:

Determine mean and variance of combined sample.

Combined mean:

\[\bar{X} = \frac{\text{total of sample 1} + \text{total of sample 2}}{\text{total size}} = \frac{(100 \times 55) + (100 \times 50)}{100 + 100} = 52\]

For combined samples of sizes \(n_1\) and \(n_2\) with means \(\bar{X}_1\) and \(\bar{X}_2\) and variances \(s_1^2\) and \(s_2^2\):

\[\bar{X} = \frac{n_1 x_1 + n_2 x_2}{n_1 + n_2}\]

Combined Variance:

For first sample, \(s_1^2 = 100\):

\[s_1^2 = \frac{\sum fx^2}{n} - \bar{X}^2 \implies 100 = \frac{\sum fx^2}{100} - 55^2 \implies \sum fx^2 = 312500\]

For second sample, \(s_2^2 = 64\):

\[s_2^2 = \frac{\sum fy^2}{n} - \bar{Y}^2 \implies 64 = \frac{\sum fy^2}{150} - 50^2 \implies \sum fy^2 = 384600\]

\[\text{Combined variance} = \frac{\sum fx^2 + \sum fy^2}{n_1 + n_2} - (\text{combined mean})^2\]

\[= \frac{312500 + 384600}{100 + 100} - (52)^2 = 84.4\]

In general:

If the two samples have equal means, then:

\[\textbf{Pooled variance} = \frac{n_1 s_1^2 + n_2 s_2^2}{n_1 + n_2}\]

If means are not equal, then:

\[\textbf{Pooled variance} = \frac{(s_1^2 + \bar{X}_1^2)n_1 + (s_2^2 + \bar{X}_2^2)n_2}{n_1 + n_2} - m^2\]

3.9.7 Merits and Limitations of Standard Deviation

Merits

• It is based on each and every observation of the data.
• It is amenable to further algebraic treatment.
• It is less affected by fluctuations of sampling than most other measures of variation.
• It is possible to calculate the combined standard deviation of two or more groups. This is not possible with any other measure.
• For comparing the variability of two or more distributions, coefficient of variation is considered to be most appropriate and this measure is based on mean and standard deviation.
• Standard deviation is most prominently used in further statistical work.

Limitations

• As compared to other measures it is difficult to compute.
• It gives more weight to extreme values and less to those which are near the mean.

3.9.8 Coefficient of Variation

The standard deviation discussed so far is an absolute measure of variation. The corresponding relative measure is known as the coefficient of variation denoted by C.V. This measure developed by Karl Pearson is the most commonly used measure of relative variation. It is used in such problems where we want to compare the variability of two or more than two series. That series (or group) for which the coefficient of variation is greater is said to be more variable or conversely less consistent, less uniform, less stable or less homogeneous. On the other hand, the series (or group) for which the coefficient of variation is less is said to be less variable or conversely more consistent, more uniform, more stable or more homogeneous.

Note:

• Standard deviation of an exponential distribution = its mean, so its C.V = 1.
• Distributions with C.V \(< 1\) tend to have low variance.
• Distributions with C.V \(> 1\) tend to have high variance.

Coefficient of Variation is obtained as follows (This is the principle measure of relative dispersion. It expresses standard deviation as a % of mean. Co-efficient of variation = ratio of standard deviation \(\sigma\) to the mean \(\mu\)):

\[C.V = \frac{S}{\bar{X}} \times 100\%\]

Coefficient of Variation is more useful when the two distributions are entirely different and the units of measurement are also different.

Advantages

• The co-efficient of variation is useful because the standard deviation of data must always be understood in the context of the mean of the data.
• The co-efficient of variation is a dimensionless number. So when comparing between data sets with different units with widely different means one should use co-efficient of variation for comparison.

Disadvantages

• When the mean value is near zero, the co-efficient of variation is sensitive to small changes in the mean, limiting its usefulness.
• Unlike standard deviation it cannot be used to construct confidence intervals for the mean.

Example

Calculate co-efficient of variation of the following populations of the values obtained:

Solution:

Population A:

\[S = \sqrt{\frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2} = \sqrt{\frac{67}{5} - \left(\frac{15}{5}\right)^2} = \sqrt{4.4}\]

\[C.V = \frac{S}{\bar{X}} = \frac{\sqrt{4.4}}{3} = 0.6992 \cong 69.92\%\]

Population B:

\[S = \sqrt{\frac{\sum y^2}{n} - \left(\frac{\sum y}{n}\right)^2} = \sqrt{\frac{254}{5} - \left(\frac{30}{5}\right)^2} = \sqrt{14.8}\]

\[C.V = \frac{S}{\bar{X}} = \frac{\sqrt{14.8}}{6} = 0.6412 \cong 64.12\%\]

Since C.V \(< 1\), it has low variance.

Combined Standard Deviation (Pooled Mean and Variance)

Example 3.5

The following is a summary of statistics of 2 samples:

Determine mean and variance of combined sample.

Combined mean:

\[\bar{X} = \frac{\text{total of sample 1} + \text{total of sample 2}}{\text{total size}} = \frac{(100 \times 55) + (50 \times 150)}{100 + 150} = 52\]

For combined samples of sizes \(n_1\) and \(n_2\) with means \(\bar{X}_1\) and \(\bar{X}_2\) and variances \(s_1^2\) and \(s_2^2\):

\[\bar{X} = \frac{n_1 x_1 + n_2 x_2}{n_1 + n_2}\]

Combined Variance:

For first sample, \(s_1^2 = 100\):

\[s_1^2 = \frac{\sum fx^2}{n} - \bar{X}^2 \implies 100 = \frac{\sum fx^2}{100} - 55^2 \implies \sum fx^2 = 312500\]

For second sample, \(s_2^2 = 64\):

\[s_2^2 = \frac{\sum fy^2}{n} - \bar{Y}^2 \implies 64 = \frac{\sum fy^2}{150} - 50^2 \implies \sum fy^2 = 384600\]

\[\text{Combined variance} = \frac{\sum fx^2 + \sum fy^2}{n_1 + n_2} - (\text{combined mean})^2\]

\[= \frac{312500 + 384600}{100 + 100} - (52)^2 = 84.4\]

In general:

If the two samples have equal means, then:

\[\textbf{Pooled variance} = \frac{n_1 s_1^2 + n_2 s_2^2}{n_1 + n_2}\]

If means are not equal, then:

\[\textbf{Pooled variance} = \frac{(s_1^2 + \bar{X}_1^2)n_1 + (s_2^2 + \bar{X}_2^2)n_2}{n_1 + n_2} - m^2\]

Note: - Distributions with C.V \(< 1\) tend to have low variance. - Distributions with C.V \(> 1\) tend to have high variance.