Statistics: Mean, Median & Mode – JSS1 Mathematics Lesson Note (Third Term)

An average is a single value used to represent a set of numbers.

• Mean: the sum of the values divided by the number of values.
• Median: the middle value after the data is arranged in order.
• Mode: the value that occurs most often.

Behavioural objectives

By the end of the lesson, students should be able to:

1. explain what an average is;
2. calculate the mean of a set of data;
3. find the median of a set of data, including when there are two middle values;
4. identify the mode, including when there are two modes (bimodal);
5. prepare a frequency table from raw data and draw a bar chart from it.

What is an average?

An average is a single value used to represent a set of numbers (that is, all the values in a set of data).

• Data: a collection of facts or numbers, such as the ages of pupils in a class.
• Statistics: the collection, arrangement and study of data.
• Use: an average lets us describe a whole set of data with one number.
• Types: the three averages in this lesson are the mean, the median and the mode.
• The average is the most commonly used statistic.

How do you find the mean, median and mode?

There are three averages: the mean, the median and the mode, and each has its own rule.

How do you find the mean?

The mean is the sum of all the values divided by the number of values.

1. Add all the values together.
2. Count how many values there are.
3. Divide the sum by the number of values.

• Formula: Mean = (∑x) ÷ n, where ∑x is the sum of the values and n is the number of values.
• Example: for 4, 6, 8, 10, 12, the sum is 40 and n is 5, so the mean is 40 ÷ 5 = 8.

How do you find the median?

The median is the middle value of data arranged in order.

1. Arrange the values from smallest to largest (or largest to smallest).
2. If the number of values is odd, pick the single middle value.
3. If the number of values is even, pick the two middle values.
4. For two middle values, add them and divide by 2.

• Odd example: 7, 3, 9, 5, 11 becomes 3, 5, 7, 9, 11. The median is 7.
• Even example: 2, 4, 6, 8 has middle values 4 and 6. The median is (4 + 6) ÷ 2 = 5.

How do you find the mode?

The mode is the value that occurs most often in a set of data.

• One mode: one value occurs most often, as 3 in 1, 3, 3, 4.
• Two modes (bimodal): two values tie for most often, as 2 and 5 in 2, 2, 3, 5, 5.
• No mode: every value occurs the same number of times, as in 1, 2, 3.

Worked example

Calculate the mean, median and mode of the following data.

1. 45, 50, 55, 54, 48, 53, 50, 55
2. 38, 35, 36, 30.8, 34.7, 37.9, 33.1
3. 3, 0, 4, 7, 0, 5, 3, 4, 0, 3, 6, 5, 5, 4, 6, 5

Solution for the first set of data:

Mean

Mean = (∑x) ÷ n, that is, the addition of all the given items divided by the number of items.

1. Mean = (45 + 50 + 55 + 54 + 48 + 53 + 50 + 55) ÷ 8
2. Mean = 410 ÷ 8
3. Mean = 51.25

Median

1. Arrange the data in order: 45, 48, 50, 50, 53, 54, 55, 55.
2. There are eight values, so there are two middle values, 50 and 53.
3. The median is their mean: (50 + 53) ÷ 2 = 103 ÷ 2 = 51.5.

Mode

• 50 occurs twice and 55 occurs twice. No other value occurs twice.
• The data is bimodal. The modes are 50 and 55.

How do you prepare a frequency table and bar chart?

A frequency table shows each value in a set of data and the number of times it occurs (its frequency).

The figures below show the number of children per family in a sample of 40 houses in a street.

1, 2, 4, 3, 5, 3, 8, 3, 2, 2, 3, 4, 5, 6
5, 4, 2, 1, 3, 2, 4, 5, 3, 8, 7, 6, 5, 4
5, 7, 6, 3, 8, 6, 3, 5, 7, 5, 4, 3

Prepare a frequency table to represent the information. Then draw a bar chart to represent the information.

Steps for the frequency table

1. List every possible value (1 to 8) in the first column.
2. Go through the data and count how many times each value occurs.
3. Write each count in the frequency column.
4. Add the frequencies. The total must equal the number of values, here 40.

• Check: 2 + 5 + 9 + 6 + 8 + 4 + 3 + 3 = 40.
• Mode from the table: 3 children per family, because it has the highest frequency (9).

Bar chart showing the number of children per family in a sample of 40 houses

A bar chart uses bars of equal width, with the height of each bar showing the frequency.

• Vertical axis: frequency, from 0 to 10.
• Horizontal axis: number of children per family, from 1 to 8.

1. Draw the two axes and mark equal spaces on each.
2. Label the axes and give the chart a title.
3. Draw one bar for each value, with its height equal to its frequency.
4. Leave equal gaps between the bars.

Evaluation

1. What is an average?
2. State how to find the mean, the median and the mode of a set of data.
3. How do you find the median when there are two middle values?
4. Calculate the mean, median and mode of the second and third sets of data in the worked example.

Assignment

Do these exercises:

• Exercise 24.1, page 263, numbers 2, 4, 5 and 6
• Exercise 24.3, page 266, numbers 3 and 4