Chapter 8: Statistics

8.1 Measures of Central Tendency for Grouped Data

In previous grades, we focused on ungrouped data. In Form Three, we analyze data organized into frequency distributions (grouped data).

1. The Mean (\(\bar{x}\))

The mean for grouped data is calculated by finding the midpoint (\(x\)) of each class, multiplying it by the frequency (\(f\)), and dividing the sum by the total frequency.

Formula: \[\bar{x} = \frac{\sum fx}{\sum f}\]

2. The Median

The median is the value that separates the higher half from the lower half of the data.

Formula: \[\text{Median} = L_m + \left( \frac{\frac{n}{2} - c}{f} \right) \cdot h\] Where: * \(L_m\): Lower boundary of the median class. * \(n\): Total frequency. * \(c\): Cumulative frequency of the class preceding the median class. * \(f\): Frequency of the median class. * \(h\): Class width.

3. The Mode

The mode is the value with the highest frequency.

Formula: \[\text{Mode} = L_m + \left( \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} \right) \cdot h\] Where \(f_m\) is the modal frequency, \(f_1\) is the frequency before it, and \(f_2\) is the frequency after it.

8.2 Correlation and Scatter Diagrams

Correlation measures the strength and direction of the relationship between two variables.

Real-Life Example: Study Hours vs. Test Scores

Figure 1: Positive Correlation between Study Hours and Scores

Figure 1: Positive Correlation between Study Hours and Scores

Types of Correlation: 1. Strong Positive: Points lie close to a line rising from left to right. 2. Strong Negative: Points lie close to a line falling from left to right. 3. No Correlation: Points are scattered randomly.

Chapter 8 Exercises

Exercise 8A: Grouped Mean The table below shows the marks of 50 students in a math test. Calculate the mean mark.

Marks Frequency (\(f\)) Midpoint (\(x\)) \(fx\)
21-30 2 25.5 51
31-40 5 35.5 177.5
41-50 7 45.5 318.5
51-60 9 55.5 499.5
61-70 11 65.5 720.5

Exercise 8B: Correlation A farmer records the amount of fertilizer used (kg) and the crop yield (tonnes). 1. Plot a scatter diagram for the data: (10, 2), (20, 4), (30, 5), (40, 7), (50, 8). 2. Describe the relationship between fertilizer and yield.

Chapter 9: Probability

9.1 Probability Terminology

  • Experiment: An activity that generates results (e.g., tossing a coin).
  • Outcome: A specific result of an experiment.
  • Sample Space (\(S\)): The set of all possible outcomes.
  • Event (\(E\)): A subset of the sample space.

9.2 Calculating Probability

The probability of an event \(E\) is: \[P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}\]

Properties: * \(0 \le P(E) \le 1\) * \(P(E) = 0\) (Impossible event) * \(P(E) = 1\) (Certain event)

9.3 Rules of Probability

1. Addition Rule (OR)

For two events \(A\) and \(B\): \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] If \(A\) and \(B\) are mutually exclusive (cannot happen together), then \(P(A \cap B) = 0\).

2. Multiplication Rule (AND)

For independent events: \[P(A \cap B) = P(A) \cdot P(B)\]

3. Conditional Probability

The probability of \(B\) occurring given that \(A\) has already occurred: \[P(B|A) = \frac{P(A \cap B)}{P(A)}\]

9.4 Visualizing Sample Spaces

When rolling two dice, the sample space consists of \(6 \times 6 = 36\) outcomes.

##    Die1 Die2 Sum
## 1     1    1   2
## 2     2    1   3
## 3     3    1   4
## 4     4    1   5
## 5     5    1   6
## 6     6    1   7
## 7     1    2   3
## 8     2    2   4
## 9     3    2   5
## 10    4    2   6

Chapter 9 Exercises

Exercise 9A: Basic Probability A bag contains 5 blue, 5 green, 6 white, and 4 red discs. If one disc is picked at random, find the probability that it is: 1. Green 2. Blue or Red 3. Not White

Exercise 9B: Independent Events A coin is tossed and a six-sided die is rolled. What is the probability of getting a “Head” and a “6”?

Exercise 9C: Dependent Events A bag contains 3 red and 5 white balls. Two balls are picked one after the other without replacement. 1. What is the probability that both are red? 2. What is the probability of picking a red ball then a white ball?

Exercise 9D: Real-Life Application In a survey of 200 men, 142 like camel milk, while 58 dislike it. 1. What is the probability that a man chosen at random likes camel milk? 2. If two men are chosen, what is the probability they both dislike camel milk?

Summary Table: Statistics vs Probability

Concept Statistics (Ch 8) Probability (Ch 9)
Focus Analyzing past/collected data Predicting future outcomes
Key Tool Mean, Median, Mode Sample Space, Rules of Addition
Visual Scatter Plots, Histograms Tree Diagrams, Venn Diagrams

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Key Features of this Draft:

  1. Mathematical Accuracy: All formulas for grouped data (Mean, Median, Mode) and Probability (Addition/Multiplication rules) are formatted using LaTeX for professional rendering.
  2. R Integration:
    • Included a Scatter Plot using R code to demonstrate positive correlation.
    • Included a Dice Grid generation to show how R handles sample spaces.
  3. Curriculum Alignment: The exercises are directly adapted from the provided textbook pages (e.g., the “Marks of 50 students” table and the “Camel milk survey”).
  4. Real-Life Context: Examples like “Study hours vs. Test scores” and “Fertilizer vs. Crop yield” make the statistics section relatable.
  5. Clean Formatting: Uses an R Markdown structure with a Table of Contents and themed output for easy reading.