MATHEMATICS O LEVEL(FORM TWO) NOTES – STATISTICS

STATISTICS

Definition

Statistics is a branch of mathematics dealing with the study of methods of collecting, organizing, analyzing, presenting, and interpreting numerical data to reach conclusions.

Frequency distribution

Frequency distribution is the number of times each data point occurs.

Example 1

1. Make a frequency table from the following data of ages of 10 students:

14, 15, 16, 14, 17, 15, 16, 13,

2. In a mathematics test, the following marks were obtained:

48, 47, 42, 67, 73, 50, 76, 47, 44, 44, 57, 58, 54, 45, 58, 56, 66, 67, 45, 43, 71, 48, 64, 52, 42, 54, 62, 32, 49, 34, 35, 46, 89, 37, 47, 54, 45, 60, 64, 44.

If the class size of the class interval is 8, group the marks starting with the interval 32-39 and draw the frequency distribution table.

Solution

Example

From the example above, find the class mark of the class intervals 88–95 and 80–87.

Class mark: and

= 91.5 and 83.5

Class limits example in class interval 88 – 95

88 is the class lower limit.

95 is the class upper limit.

Class real limits

Class lower real limit is the number obtained by subtracting 0.5 from a class lower limit, e.g., 88 – 0.5 = 87.5.

Class upper real limit is obtained by adding 0.5 to the upper class limit, e.g., 95 + 0.5 = 95.5.

Class size

Class size is the value obtained by the difference between the upper real limit and the lower real limit.

Example

From class interval 88-95 and 31-35, find the class size.

Solution:

Lower class real limit = 88 – 0.5 = 87.5

Upper class real limit = 95 + 0.5 = 95.5

Class size = 95.5 – 87.5 = 8

Lower class real limit = 31 – 0.5 = 30.5

Upper class real limit = 35 + 0.5 = 35.5

Class size = 35.5 – 30.5 = 5

Exercise 1:

1. In a biology class test, the following marks were obtained:

54, 54, 40, 55, 54, 43, 73, 34, 75, 47, 35, 45, 73, 46, 31, 43, 47, 35, 35, 60, 67, 51, 44, 48, 55, 45, 50, 37, 51, 36

By grouping the marks in class intervals 20-29, 30-39, 40-49, etc., construct the frequency distribution table.

Solution:

2. The following data represent the masses of 10 people in kg. Construct the frequency distribution table for these people:

30, 25, 35, 28, 38, 40, 25, 25, 40, 24

3. The following is a set of marks on a geography examination presenting the frequency distribution table with class intervals, real limits, class marks, and interval size starting with the interval 8-15 at the bottom.

Solution:

4. Fill in the blank columns:

Distribution of 100 math examination scores:

Note: Class real limits are also known as class boundaries.

GRAPHS OF FREQUENCY DISTRIBUTIONS:

HISTOGRAMS

Histograms of frequency distribution are rectangular figures plotted with class marks against frequency. The width of the histogram equals the class size.

Example:

1. Draw a histogram of 100 mathematics examination scores in the table below:

2. Use the following distribution table below to draw a histogram:

FREQUENCY POLYGON

A frequency polygon is the line graph of class frequency plotted against class marks.

Steps:

1. Add one interval below the lowest interval and one above the highest interval and assign them zero frequency.
2. Plot points and join them by straight lines.

Example:

1. Draw a frequency polygon from the following data.

EXERCISE

1. The following table shows female deaths between 0 and 34 years to the nearest numbers. Represent this information by:

• A) Histogram
• B) Frequency polygon

Expected death of female per 100 women:

Solution:

A) Histogram

B) Frequency polygon

2. Table below shows the distribution of marks obtained by 110 students in two different monthly tests. Draw the frequency polygon on the same chart.

3. Platforms in each square metre of a lawn were counted and recorded as follows. Draw an ogive for the platforms.

REVISION EXERCISE

1. The ages of the 22 players in a football match were recorded as follows:

17, 18, 15, 16, 16, 16, 18, 15, 18, 15, 15, 18, 18, 15, 16, 17, 15, 16, 17, 15, 15, 16, 15, 18, 15

Express the data in a frequency table.

Solution:

2. The examination marks of 45 students are:

65, 58, 71, 62, 64, 35, 72, 32, 64, 46, 59, 82, 73, 76, 64, 63, 75, 71, 61, 36, 64, 80, 61, 64, 76, 64, 60, 68, 48, 35, 92, 73, 46, 24, 35, 43, 30, 50, 70, 40, 46, 64, 24, 28

A) Make a frequency distribution using class intervals 21-30, 31-40, 41-50, etc.

Solution:

B) Draw cumulative frequency curve.

Solution:

3. Two plots A and B were treated with different fertilizers. The frequency number of potatoes on samples of 100 plants on each plot are shown below:

Draw a histogram for plot B.

4. In a certain examination the results were as follows:

• 3 students got marks between 0 and 10
• 5 students got marks between 10 and 15
• 6 students got marks between 20 and 40
• 4 students got marks between 30 and 40
• 2 students got marks between 40 and 50

Construct a histogram.

5. Final scores of history examination were recorded as shown in the table below:

A) What is the size of class intervals?

Solution:

5 is the size of class intervals.

B) Draw a histogram to represent the scores.