Statistics

Grade 12 Math - Statistics

Lesson Objectives

Understand various methods of data collection.
Organize data using frequency tables and charts.
Calculate and interpret mean, median, and mode.
Compare datasets using measures of central tendency.
Apply statistics to real-world problems and past exam formats.

Lesson Introduction

Statistics involves collecting, organizing, analyzing, and interpreting numerical data. From business forecasting to government policy, statistics provides tools for making informed decisions. This lesson will cover how to collect and present data and how to summarize it using measures of central tendency—mean, median, and mode.

Core Lesson Content

Data Collection Methods

Surveys
Interviews
Observation
Experiments
Questionnaires

Presenting Data

Frequency tables
Bar charts
Histograms
Pictograms
Pie charts

Measures of Central Tendency

Mean: \( \bar{x} = \frac{\sum x}{n} \)
Median: Middle value when data is arranged in order
Mode: Most frequently occurring value

Worked Example

Example 1: Find the mean of 4, 8, 6, 5, 3.
\( \bar{x} = \frac{4 + 8 + 6 + 5 + 3}{5} = \frac{26}{5} = 5.2 \)

Example 2: Find the median of 9, 2, 7, 5, 3.
Ordered: 2, 3, 5, 7, 9 → Middle value is \( 5 \)

Example 3: Find the mode of 3, 4, 4, 6, 7.
Mode = \( 4 \) (appears most)

Example 4: The ages of students are: 14, 15, 16, 16, 17. Find the mean.
\( \bar{x} = \frac{14 + 15 + 16 + 16 + 17}{5} = \frac{78}{5} = 15.6 \)

Example 5: Find the median of 11, 13, 15, 17.
Even number: Median = \( \frac{13 + 15}{2} = 14 \)

Example 6: Frequency table:

Value:      2  4  6
Frequency:  1  3  2

Mean = \( \frac{(2 \times 1) + (4 \times 3) + (6 \times 2)}{1 + 3 + 2} = \frac{2 + 12 + 12}{6} = \frac{26}{6} = 4.33 \)

Example 7: Frequency table:

Scores:     5  6  7  8
Frequency:  2  4  1  3

Mode = \( 6 \) (highest frequency = 4)

Example 8: Frequency table:

Marks:      2  3  4  5
Frequency:  3  2  1  4

Median position: \( \frac{n+1}{2} = \frac{10+1}{2} = 5.5 \) → 5th and 6th data fall at value 3 and 5 → Median = \( 4 \)

Example 9: Find the mean from this data:

Class Marks: 10 20 30
Frequency:   2  3  1

Mean = \( \frac{(10 \times 2) + (20 \times 3) + (30 \times 1)}{2 + 3 + 1} = \frac{20 + 60 + 30}{6} = \frac{110}{6} = 18.33 \)

Example 10: In a class, these test scores were recorded: 12, 15, 17, 15, 18, 20. Find the mode.
Mode = \( 15 \)

Exercises

Calculate the mean of: 5, 8, 12, 10, 15.
Find the median of: 6, 11, 7, 9, 5.
Determine the mode of: 2, 4, 2, 6, 3, 2.
[WAEC] The marks obtained are: 10, 15, 15, 20, 25. Find the mean. (Past Question)
[NECO] Given: 3, 5, 8, 8, 8, 9. What is the mode? (Past Question)

In a frequency table:

Value:      1 2 3
Frequency:  2 3 5

Find the mean.

Using the same data, find the mode.
[JAMB] A student scored: 60, 65, 70, 75, 80. Find the median. (Past Question)

Find the median from the following:

Score:      10 20 30
Frequency:  1  2  2

[WAEC] Determine the mean of this dataset:
```
Score:      5 10 15
Frequency:  4  2  4
      
```
(Past Question)

Conclusion/Recap

In this lesson, we learned how data is collected and presented in various formats and how to compute the measures of central tendency—mean, median, and mode. These tools help in summarizing and interpreting data meaningfully. In the next lesson, we’ll cover Measures of Dispersion such as range, variance, and standard deviation.