Measures of Central Tendency – JSS3 Mathematics Lesson Note (Third Term)

A measure of central tendency is a single value that describes a set of data by identifying its central position. The three types are the mean, median and mode.

Behavioural objectives

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

1. explain what a measure of central tendency is;
2. calculate the mean of a set of data;
3. find the median of a set of data with an odd or an even number of values;
4. identify the mode of a set of data, including data with more than one mode or no mode;
5. calculate the range of a set of data.

What is a measure of central tendency?

A measure of central tendency is a single value that represents the centre of a data set and is typical of the whole set.

• Mean: the sum of the values divided by the number of values.
• Median: the middle value of data arranged in order.
• Mode: the value that occurs most often.
• Use: each measure suits different data, so choose the one that best represents the set.

What is the mean?

The mean (or average) is the sum of all the values in a data set divided by the number of values.

Formula: Mean = (Sum of all values) ÷ (Number of values)

• Symbol: the formula is also written with the Greek letter ∑ (sigma), which means “sum of”.
• Popularity: the mean is the best-known measure of central tendency.

How do you calculate the mean?

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

Example: Find the mean of 5, 8, 12, 7 and 3.

• Sum: 5 + 8 + 12 + 7 + 3 = 35.
• Number of values: 5.
• Mean: 35 ÷ 5 = 7.

How do outliers affect the mean?

An outlier is a value far from the rest of the data, and it pulls the mean towards itself.

• Example data: 10, 12, 11, 13 and 64.
• Mean: 110 ÷ 5 = 22, which is higher than four of the five values.
• Median: 12, which stays close to most of the data.
• Choice: when a data set has outliers, the median is usually the better measure.

What is the median?

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

To find it, first arrange the data in order of magnitude (smallest to largest). Consider the data below:

We first rearrange the data in order of magnitude (smallest to largest).

The median mark in this example is 56, since 56 (highlighted) is in the middle rank of the data set.

How do you find the median of an odd number of values?

With an odd number of values, the median is the single middle value after ordering.

1. Arrange the values from smallest to largest.
2. Find the value with the same number of values on each side.

Example: Find the median of 12, 5, 8, 20 and 9.

• Ordered: 5, 8, 9, 12, 20.
• Median: 9, the third value.

How do you find the median of an even number of values?

With an even number of values, the median is the average of the two middle values.

Consider the example below:

We arrange the data in ascending order:

The median lies between the two middle values. We take the 5th and 6th scores from the ordered list and average them to get a median of 55.5.

Another example: Find the median of 6, 2, 9, 4, 7 and 10.

• Ordered: 2, 4, 6, 7, 9, 10.
• Middle values: 6 and 7.
• Median: (6 + 7) ÷ 2 = 6.5.

What is the mode?

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

1. Collect and organise the data.
2. Count how often each value occurs.
3. Pick the value with the highest count.

In the list below, 10 is the mode because it appears more often than any other number.

2, 2, 5, 4, 10, 10, 10, 21, 25, 39, 42

Can a data set have more than one mode, or no mode?

Yes, a data set can have one mode, two modes, several modes or no mode.

• Bimodal: two values occur with equal highest frequency.
• Trimodal: three values occur with equal highest frequency.
• No mode: no value occurs more than once.

4, 4, 4, 9, 15, 15, 15, 27, 37, 48

In this example, both 4 and 15 are modes, so the set is bimodal.

What is the range?

The range of a set of data is the difference between the highest and lowest values.

Formula: Range = Highest value – Lowest value

• Note: the range shows how spread out the data are. It is not a measure of central tendency.

Example: Find the range of these test scores: 77, 75, 94, 85, 91, 90, 94.

• Highest value: 94.
• Lowest value: 75.
• Range: 94 – 75 = 19 points.

How is the range used in daily life?

The range tells you the gap between the highest and lowest prices or scores.

Example: A student recorded the price of a sachet of water in five shops: ₦200, ₦250, ₦180, ₦300 and ₦220.

• Range: 300 – 180 = ₦120.
• Mean: 1150 ÷ 5 = ₦230.
• Median: ordered values are 180, 200, 220, 250, 300, so the median is ₦220.

Which measure should you use?

Choose the measure that best represents the data you have.

Evaluation

1. Find the mean and the range of the following percentages: 45%, 52%, 57%, 30%, 63%, 70%.
2. A researcher recorded the percentage of time participants spent in different houses: 57%, 31%, 42%, 39%, 64%, 47%, 51%, 42%. Find the range.
3. Find the median of 14, 3, 9, 21, 6 and 11.
4. State the mode of 7, 8, 8, 9, 9, 10. Say what kind of mode this is.

Assignment

Answer all four parts in your exercise book, showing your working.

1. Find the mean of 4, 6, 8, 10, 12.
2. Find the median of 3, 7, 5, 9, 1.
3. Find the mode of 2, 3, 3, 5, 5, 5, 8.
4. Find the range of 12, 7, 19, 4, 15.