Measures of Central Tendency

Overview

Measures of central tendency are ways of describing the central position of a frequency distribution for a group of data. During this guide we will be introducing the different methods to describe central tendency including the mode, median and mean. What does this mean in simple terms? Let’s use an example.

Setting the scene:

A tutor sets a piece of coursework for 100 students. The students could achieve a score between 0 and 100. The following marks were achieved.

62	81	49	47	69	52	77	45	57	64
71	60	51	58	40	39	44	40	67	64
63	81	74	74	49	83	81	49	71	76
84	37	39	48	66	42	79	45	74	61
85	51	52	38	67	42	69	69	55	62
80	42	68	74	53	64	75	40	75	64
51	35	41	49	57	81	77	67	54	70
74	39	42	56	51	45	39	84	67	81
41	63	59	39	62	53	58	81	55	45
42	37	72	65	71	70	40	40	44	40

Even with 100 pieces of data (i.e. the marks of 100 students) it is difficult for the tutor at first sight to gauge the overall performance of the students.

Frequency Distribution

To get a better idea of the overall performance of the students, the tutor could simply decide to calculate the frequency of each mark. In other words, how many students scored the same marks? For example, how many students scored 35 out of 100 or 64 out of 100? Looking again at the group of data below, one student scored 35 (numbers in red) whilst four scored 64 (numbers in blue).

62	81	49	47	69	52	77	45	57	64
71	60	51	58	40	39	44	40	67	64
63	81	74	74	49	83	81	49	71	76
84	37	39	48	66	42	79	45	74	61
85	51	52	38	67	42	69	69	55	62
80	42	68	74	53	64	75	40	75	64
51	35	41	49	57	81	77	67	54	70
74	39	42	56	51	45	39	84	67	81
41	63	59	39	62	53	58	81	55	45
42	37	72	65	71	70	40	40	44	40

Whilst this data can be presented in a tabulated format, we can also illustrate it graphically in a histogram (see below).



The histogram is useful because it shows the range of marks achieved, or what we call the frequency distribution. In other words, it shows the distribution and pattern of marks from the lowest to the highest (see the statistical guide on frequency distributions).

Measures of Central Tendency

Whilst knowing the frequency distribution for a group of data (in this case, the coursework marks for a group of 100 students) is useful, it does not summarise the findings in the most effective way. When we talk about summarising the findings, we want to know if there is a single mark that is most effective in representing the data. In other words, we are trying to find the central position of our frequency distribution. There are a number of ways of doing this, including calculating the mode, median and mean.