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

Measures of Central Tendency

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that dataset. This value is considered typical of how data are distributed. Central tendency can be useful when data is skewed or unevenly distributed. The three main types of central tendency are the mean, median, and mode.

The mean, median, and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In this section, we’ll look at when it is appropriate to use the mean, median, or mode, and learn how to calculate them and what situations they are most applicable to.

Mean (Arithmetic)

The mean (or average) is the most popular and well-known measure of central tendency. It is calculated by taking the sum of all the values in a data set and dividing that number by the number of values in the data set. The mean is often referred to as the “average.”

Formula: 

Mean = (Sum of all values in the data set) / (Number of values)

The formula is usually written in a slightly different manner using the Greek symbol ∑ (Sigma), which means “sum of.”

You may have noticed that the mean formula works with all types of data. That’s why it’s called a simple mean. The issue is that complex, skewed, and grouped data are very different from simple data. The difference is so large that using a single mean formula may give misleading results for some types. So statisticians developed methods to calculate means and further explain averages, so we can take different types of data into account.

The mean is commonly misused if your data is full of outliers. This will distort the mean, making it less relevant. However, the mean is still one of the best calculations that you can have when all outliers are low. Moreover, you can make sure that the outliers themselves go in the same direction as any other value in your dataset. It is best to use the one that produces the lowest amount of errors when applied to your data.

Median

The median is the middle score in a set of data that has been arranged in order of magnitude. The median is less affected by outliers or skewed data, and to calculate the median, the data has to be arranged in order. Consider the data below: 

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

Our median mark in this example is 56 since 56 (highlighted) is in the middle rank of the dataset. For an even number of data points, the median is found by taking the average of the two middle scores. Consider the example below;

We arrange in ascending order;

In this case, the median lies between the two middle values. So we take the 5th and 6th scores from the ordered list and average them to get a median of 55.5.

Definition of Mode

A statistical term that refers to the most frequently occurring number found in a set of numbers. The model is found by collecting and organizing the data and counting how often each value occurs. The mode is the value that appears most frequently in a data set.

For example, in the following list of numbers, 10 is the mode since it appears more than any other number in the set.

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

In a set of numbers, we can have more than one mode. This is known as bimodal if multiple numbers occur with equal frequency, and trimodal if three values occur with equal frequency.

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

In this example, both 4 and 15 are the modes. If no number in a set of numbers occurs more than once, then that set has no mode.

Range

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

Formula

Range = Highest value – Lowest value

Example:  

Consider the following test scores:  

77, 75, 94, 85, 91, 90, 94

Range = 94 (highest) – 75 (lowest) = 19

The range of these test scores is 19 points.

Example of Range in Practical Use

A family went on a vacation and kept track of the prices they paid for different gas stations during their trip. What was the range of prices they paid?

The range between the highest and lowest prices gives an idea of the price difference.

Evaluation

1. Find the average range of the following percentages:

   – 45%, 52%, 57%, 30%, 63%, 70%

2. A researcher has compiled data from participants. What is the range of time participants spent in different houses:  

   57%, 31%, 42%, 39%, 64%, 47%, 51%, 42%?