At the end of this lesson, students should be able to:

1. differentiate all types of variables.
2. understand scale of measurements.
3. describe and calculate the properties of data.

• Introduction
• Types of Variables
• Scales of Measurements
• Population & Samples
• Measures of Central Tendency
• Measure of Dispersion
• Properties of Distribution

• Collection, analysis, interpretation and presentation of data to discover its underlying causes, patterns, relationships and trends.

• Two major branches of statistics:

  1. Descriptive statistics
  2. Inferential statistics

• Procedures used to summarise and organise a set of scores or observations in a meaningful way.
• Typically presented graphically, in tabular form (in tables), or as summary statistics (single values).
• Descriptive statistics do not allow us to make conclusions beyond the data we have analysed or reach conclusions regarding any hypotheses we might have made.

• Allow simpler interpretation of data.
• Used when the intent is to describe the data that they actually collected.
• Example:
  • A clinical psychologist conducted a study in which she gave some of her clients a new depression treatment and she wanted to describe the average depression score of only those clients who got the treatment.

• Procedures that allow researchers to infer or generalize observations made with samples to the larger population from which they are selected.
• Infer the value of population parameter from a sample statistics.
• Determine the probability of characteristics of population based on the characteristics of the sample.

• They help assess strength of the relationship between your independent (causal) variables, and your dependent (effect) variables.
• Examples:
  • A clinical psychologist conducted a study in which she gave some of her clients a new depression treatment and she wanted to estimate what the results would be if she were to give the same treatment to additional clients.

• Variables are measurements or observations that are typically numeric.

• Three categories of variables:

  1. Independent VS Dependent
  2. Continuous VS Discrete
  3. Quantitative VS Qualitative

1. Independent variable (IV)
   • Variable with two or more levels that are expected to have effects on another variable. (affect/change other outcome variable)
   • Sometimes called as predictor or experimental variable.
2. Dependent variable (DV)
   • The outcome variable that is used to compare the effects of the different independent variable (IV) levels.

• Example:

1. Independent variable (IV)
   • The new treatment
   • No treatment
2. Dependent variable (DV)
   • Amount of depression

• A continuous variable is measured along a continuum.
  • measured in whole units or fractional units.
  • e.g.: If we measure a height of 35cm and 36cm, an infinite number of heights is possible in the range of 35 and 36.
• A discrete variable is not measured along a continuum.
  • measured in whole units or categories.
  • e.g.: The number of your siblings and your family’s socioeconomic class (working class, middle class, upper class)

• Quantitative data is information about quantities; that is, information that can be measured and written down with numbers.
  • varies by amount
  • e.g : height, shoe size, and the length of your fingernails.
• Qualitative data is information about qualities; information that cannot be easily measured, but can be observed subjectively.
  • varies by class
  • e.g : the smells, tastes, textures, attractiveness, color.

• Measured in numeric units, so both continuous and discrete unit can be quantitative.
• e.g.: we can measure food intake in calories (a continuous variable) or we can count the number of pieces of food consumed (a discrete variable)

• Only discrete variables can fall into this category.
• e.g.: socioeconomic class (working class, middle class, upper class), mental disorders/depression (unipolar/bipolar) or drug use (none, experimental, abusive)

• Important to determine the kind of statistical procedures that can be used on that variable.

• Four different scales of measurement:

  1. Nominal
  2. Ordinal
  3. Interval
  4. Ratio

• Known as categorical data.
• Cannot be measured, because of their qualitative nature.
• Do not denote quantity and not in order.
• Categorise individuals into groups that are qualitatively different from other groups.
• Usually this categorical variables have been coded (converted into numeric)
  • e.g. person’s race (malay [1], chinese [2], indian [3]), gender (female [1] & male[2]), marital status (single[1], married[2])

• Sometimes, data that might have been expressed on different scale of measurements (ordinal, interval, ratio) may be recorded in nominal categories.
• E.g.
  • Marks of between 50.0 and 100.0 = PASS
  • Marks of between 0.0 and 49.0 = FAIL

• An ordinal scale of measurement is one that conveys order alone (no indication of how much more).
• Only indicates that some value is greater or less than another value, so differences between ranks do not have meaning.
• E.g.:
  • level of education (PhD, MSc, Bachelor’s)
  • level of satisfaction (Very Unsatisfied, Unsatisfied, Neutral, Satisfied, Very Satisfied)
  • degree of illnesses (mild, moderate, severe).

• Interval scales are measurements where the values have no true zero and the distance between each value is equidistant.
  • True zero – values where the value 0 truly indicates nothing.
  • Equidistant – those values whose intervals are distributed in equal units.
• E.g.: temperature in Celcius or Fahrenheit. Difference between 5°F and 3°F is similar to 8°F and 6°F (equidistant) but 0°F is not the absence of heat.

• Similar to interval scales in that scores are distributed in equal units (equidistant).
• Unlike interval scales, a distribution scores on a ratio scale has a true zero (the absence of quantity being measured).
• E.g: salary amount, duration of drug abuse (in years), score (from 0 to 100%) on an exam, weight (in pounds) of an infant.

State whether the variable is continuous or discrete, and quantitative or qualitative.

Name the variable being measured, (2) state whether it is continuous or discrete, and (3) state whether the variable is quantitative or qualitative.

1. A researcher records the month of birth among patients with schizophrenia.
2. A professor records the number of students absent during final exam.
3. A researcher asks children to choose which type of cereal they prefer (one with a toy inside or one without). He records the choice of cereal.
4. A therapist measures the time (in hours) that clients continue a recommended program of counseling.

• Population is a group of all things that share a set of characteristics.
  • Population parameter – the value that would be obtained if the entire population were actually studied.
• Sample is a subset of population that is intended to represent the population.
  • Sample statistic – the value obtained from the sample. It is used to estimate the population parameter value.

• A set of data is a sample from our population. The sample is a subset of the population. Statistical inference – the process that we used to draw conclusions from a data.
• Inference involves using statistics we calculate from the sample to make and informed guess about population.
• When we do statistical inference we are interested in drawing conclusions from a set of data (sample) so that we can estimate population parameters.

1. Measures of Central Tendency
2. Measures of Dispersion

• Also known as measures of central location (locate central distribution).
• Three kinds of averages of a data set to answer “where do the data center?”

• Measures include:

  1. Mean
  2. Mode
  3. Median

• The usual “average” that is familiar to everyone.
• Adds up all the numbers (\(\sum{X}\)) and divide by how many numbers there are (N for population or n for sample).
• Formula:
  • Sample mean: \[\bar{X}=\frac{\sum_{i=1}^n X_i}{n}\]
  • Population mean: \[\mu=\frac{\sum_{i=1}^N X_i}{N}\]

• Substituting the figures from Table 1.1 into the equation for the mean, we obtain:

\[\begin{aligned} &= (20 + 25 + 21 + 34 + 41 + 37)/6 \\ &= 178/6 \\ &= 29.67 \end{aligned}\]

• Each datum point in the distribution does not contribute equally to the overall calculation of the mean.
• Data is divided into groups, each of which possesses different weighting.
• Formula:
  \[\frac{\sum_{j=1}^N w_j X_j}{N}\]

• Substituting the figures from Table 1.2 into the equation for the weighted arithmetic mean, we obtain:

\[\begin{aligned} &= (20 \times 3) + (12 \times 2) + (6 \times 1) / 20 \\ &= 36/20 \\ &= 1.8 \end{aligned}\]

- Mean \(= 2350 / 50 = 47\)

• An alternative method of describing the central nature of data.
• Relatively unaffected by the nature of the spread of data.
• Is the middle number. It is found by putting the numbers in order and taking the actual middle number if there is one, or the average of the two middle numbers if not.

• Number 192.5 is called outlier (far removed from most or all the remaining measurements).
• Usually is the result of some sort of error (but not always).
• Mean is sensitive to extreme values.
• So, a better measure of the “center” of the data can be obtained if we were to arrange the data in numerical order.

• The order
  

  18.5

  22.8

  23.7

  24.6

  24.8

  25.2

  192.5

  
  

• Then select the middle number in the list, in this case 24.6.

• In this sense, it locates the center of the data.

• If there are an even number of measurements in the data sets, there will be two middle elements -> take the mean of middle two as the median
• Example:
  

  18.5

  22.8

  23.7

  24.6

  24.8

  25.2

  28.9

  192.5

  
  

• Median: (24.6 + 24.8) / 2 = 24.7

• The easiest measure of the average.
• Defined as the item of data with the highest frequency.
• Most frequently occurring number.
• For any data set there is always exactly one mean and exactly one median.
• However, several different values could occur with the highest frequency.

• Data set 1:
  

  -1

  0

  2

  0

  
  
• The mode of this data set is 0.

- Two most frequently observed values in this data set are 1 and 2. Therefore mode is a set of two values : {1,2}

Weight of luggage presented by airline passengers at check-in (measured to the nearest kg).

• Mode: 20

• Median: 20
  • put the numbers in order first and take the actual middle number (odd count) or the average middle number (even count).
    

- Mean = (15+18+19+20+20+20+21+23+23+24+24) / 11 = 20.64

• Mean is good for dataset that is evenly spread.
• For normally distributed data, the numerical values of the mean and median should be identical and either term may successfully be used to describe the central point.
• The use of median is preferable for distributions that possess extreme values (mean is unacceptably distorted).
  

- The mean salary is 30.7k

• Also known as measures of variation
• How spread out are the data?
• Describing quantitative data will not be complete without knowing how observed values are spread out from the average.
• E.g: two classes who sat the same exam might have the same mean mark but the marks may vary in a different pattern around this.

• Measures include:

  1. Range
  2. Variance
  3. Standard Deviation (SD)

• Example:
  
  

  Table 1.3: Individual values associated with two sets of data possessing identical means.

  Set A	Set B
  10	28
  20	29
  30	30
  20	29
  10	28
  Mean: 30	Mean: 30

• Very simple measure of dispersion.
• Difference between the maximum value and the minimum value (max-min).
• Example
  1. Marks on test A

     

  2. Marks on test B

     

1. Marks on test A

   

2. Marks on test B

   

   On test A, the range of marks is 70-45=25.
   On test B, the range of marks is 65-45=20.

1. Marks on test C

   

   
   

• On test C, the range of marks is 75-40=32.

• Squared deviations from the mean.
• The sample variance of a set of n sample data is the number (\(s^2\)) defined by the formula: \[s^2=\frac{\sum(X-\bar{X})^2}{n-1}\]
• The population variance (\(\sigma^2\)) formula : \[\sigma^2=\frac{\sum(X-\mu)^2}{N}\]

• Measure of variation (deviation) of all values from the mean.
• Positive square root of the variance.
• Properties include:
  • The value is usually positive.
  • 0 indicates no variation.
  • Larger values indicate greater variation.
  • The value can increase dramatically with the inclusion of one or more outliers.
  • The units are the same as the units of the original data values.

• Sample standard deviation formula \[s=\sqrt\frac{\sum(X-\bar{X})^2}{n-1}\]
• The population standard deviation formula \[\sigma=\sqrt{\sigma^2}=\sqrt\frac{\sum(X-\mu)^2}{N}\]

1. Compute the mean (\(\bar{X}\))
2. Subtract the mean from each individual value to get a list of deviations of the form: (\(X – \bar{X}\))
3. Square each of the differences obtained from step 2: \((X – \bar{X})^2\)
4. Add all of the squares obtained from step 3: \(\sum(X – \bar{X})^2\)
5. Divide the total from step 4 by the number (n – 1), which is 1 less than the total number of values present: \(\frac{\sum(X – \bar{X})^2}{n-1}\)
6. Find the square root of the result of step 5: \(\sqrt{\frac{\sum(X – \bar{X})^2}{n-1}}\)

• Multiple waiting line: 1, 3, 4
• Compute the mean (\(\bar{X}\)) = 18 / 3 = 6 min
  

• Variance: 98 / 2 = 49
• Standard deviation: \(\sqrt{49}\) = 7.0 min

• For interpreting a known value of the standard deviation

  1. Minimum “usual” value = (mean) – 2 x standard deviation
  2. Maximum “usual” value = (mean) + 2 x standard deviation

• With a mean of 40.05 cm and a standard deviation of 1.64 cm, we use the range rule of thumb to find the minimum and maximum usual circumferences as follows:
  1. Minimum usual value = (mean) – 2 x (standard deviation) = 40.05 – 2 x 1.64 = 36.77 cm
  2. Maximum usual value = (mean) + 2 x (standard deviation) = 40.05 + 2 x 1.64 = 43.33 cm
• Based on these results, we expect that typical two-month-old girls have head circumferences between 36.77 cm and 43.33 cm.
• Because 42.6 cm falls within those limits, it would be considered usual or typical, not unusual.

• Skewness
  • Refer to shape of distribution, either symmetry or assymetry
• Kurtosis
  • Refer to the flatness or peakness of a distribution

• Tail pointing to the larger value
• Mean > Median > Mode

  

  
  

• Tail pointing to the smaller value
• Mode > Median > Mean