Measures of Spread

Overview

Measures of spread are ways of summarizing a group of data by describing how spread out the scores are. The following example helps to explain this in simple terms (this is the same example as the one used in the statistical guide, Measures of Central Tendency).

Setting the Scene

A tutor set 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

Again, we are interested in gauging the overall performance of the students. The tutorial on measures of central tendency showed how we could do this by using the mode, median, and in particular, the mean, to help us to describe the central position of a frequency distribution for a group of data.

The mean score for the above marks, for example, was 58.75 out of 100. However, the mean does not provide us with all the information we need. Rather, measures of spread help us to understand the variability in the scores around the mean. This is very important in order to understand how representative the mean is of our group of data. In other words, does a mean score of 58.75 really inform us about the performance of our 100 students?

To know the answer to this question, we must examine the degree of spread within our group of data. A large spread would suggest that the mean is not so representative because there are large differences between the individual scores, whilst a small spread would suggest that the mean is quite representative. Nonetheless, there are a number of ways of finding out how spread out the scores in a group of data are, some of which do not take the mean as their starting point. These measures include calculating the range, quartiles, variation (both absolute deviation, mean absolute deviation and variance) and standard deviation. Each of these is discussed in turn.

Range

The range is the difference between the highest and lowest scores in a distribution. If we examine the marks of the 100 students above, then we can see that the highest score was 85 and the lowest was 35. Therefore, the range is 50 (85 – 35 = 50). Whilst the range is limited as a means of telling us about the general spread of a group of data, it does set the boundaries of the scores.

Quartiles

Quartiles tell us about the spread within a frequency distribution by breaking the group of data into quarters, just like the median breaks it in half. If we examine the marks of the 100 students, which have been ordered below from the lowest to the highest scores, the quartiles are clearly visible (highlighted in red).

Order	Score	Order	Score	Order	Score	Order	Score	Order	Score
1st	35	21st	42	41st	53	61st	64	81st	74
2nd	37	22nd	42	42nd	53	62nd	64	82nd	74
3rd	37	23rd	44	43rd	54	63rd	65	83rd	74
4th	38	24th	44	44th	55	64th	66	84th	75
5th	39	25th	45	45th	55	65th	67	85th	75
6th	39	26th	45	46th	56	66th	67	86th	76
7th	39	27th	45	47th	57	67th	67	87th	77
8th	39	28th	45	48th	57	68th	67	88th	77
9th	39	29th	47	49th	58	69th	68	89th	79
10th	40	30th	48	50th	58	70th	69	90th	80
11th	40	31st	49	51st	59	71st	69	91st	81
12th	40	32nd	49	52nd	60	72nd	69	92nd	81
13th	40	33rd	49	53rd	61	73rd	70	93rd	81
14th	40	34th	49	54th	62	74th	70	94th	81
15th	40	35th	51	55th	62	75th	71	95th	81
16th	41	36th	51	56th	62	76th	71	96th	81
17th	41	37th	51	57th	63	77th	71	97th	83
18th	42	38th	51	58th	63	78th	72	98th	84
19th	42	39th	52	59th	64	79th	74	99th	84
20th	42	40th	52	60th	64	80th	74	100th	85