Measures of Central Tendency

What is central tendency?

There are three measures of central tendency we need to know about:

• the mean (also referred to as the average),
• the median and
• the mode

Mean as a Measure of Central Tendency

The process to find the mean is to sum all the values of the data set, and then divide that by the number of data points. The mean is usually represented by the Greek letter \(\mu\), pronounced /mi/:

\[\frac{value_1 + value_2 + value_3 +...+ value_n}{n}\]

This can also be expressed with the following equation:

\[\frac{\sum^{n}_{i=1} x_i}{n}\]

This means that all values (represented with x) starting at value number 1 (i = 1) will be added until we reach value n, and then this number will be divided by n (the total number of values).

The mean is one of the most commonly used measures of central tendency in data analysis and is often used to summarize data sets. However, the mean can be influenced by extreme values in the data set, known as outliers, which can skew the results. In such cases, other measures of central tendency, such as the median or mode, may be more appropriate.

Median as a Measure of Central Tendency

When we have a set of data that is able to be ordered in some way, we can find the median. The process to find the median is as follows:

1. Order the data, from smallest to largest.

1. If the number of data points is odd, the middle number is the median, meaning we take the \((\frac{n+1}{2})^{th}\) value.
2. If the number of data points is even, then we take the mean value of the middle two values. This means we take mean of the \((\frac{n}{2})^{th}\) and \((\frac{n+2}{2})^{th}\) value.

Mode as a Measure of Central Tendency

The mode of a set of data is the most common value in the dataset. If there are two or more values which are most common, both of these values are the mode.

Choosing suitable measures of central tendency

Each measure of central tendency has its own advantages and disadvantages.

For the mean, the advantages are that it uses all of the data, and is, therefore, representative of all the data. However, there are disadvantages to using the mean. It is disproportionately influenced by extreme values, which can throw the mean. The mean also cannot be used if our data isn't numerical, and takes the most computation out of all our measures of central tendency.

For the mode, the advantages are that we can find the mode of a set of data, be it numeric or otherwise. There is also limited computation, as we only need to tally the data, meaning if our data comes pre-tallied then this aids the mode. However, a downside is that the mode doesn't necessarily exist. In addition, we can have multiple modes, which doesn't help us describe a lot about the data set. As well as this, the mode doesn't take into account the full data set.

Our final measure of central tendency is the median. The advantages are that the median isn't affected by any outliers or extreme values, and we have very little calculation to do. On the flip side, it does require us to order the set of data, which for large sets of data, is lengthy and time-consuming. It also doesn't take into account the full set of data, which means this could bring in weak results.