Measures of Central Tendency

Mode

The mode is the most frequent piece of data in a frequency distribution. Looking at our histogram below, which highlights the marks of our 100 students, the mode will be the tallest bar (bar highlighted in blue).



However, in examining the histogram we can see that no one bar was the tallest. In other words, there wasn’t a most frequent mark. Instead, five students scored 40 out of 100 whilst another five scored 81 out of 100. This is one of the problems of using the mode. After all, which of the scores do we choose to represent the data? Clearly, the difference between a score of 40 and 81 out of 100 is extremely large. Added to this problem, there will be times when the most frequent number is not at all representative of the data. For example, a small group of students, say 6 of the 100 in our group of data, may have scored particularly low even though the vast majority scored reasonably high. In this instance, the use of the mode as a measure of central tendency would result in a misrepresentation of the data. In other words, it would suggest that the overall performance of students was much worse than was actually the case.

Median

The median measures the middle position of a frequency distribution for a group of data. In our example of the coursework marks of 100 students, the marks achieved were:

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

In order to calculate the median we need to order our group of data from the lowest (the 1 st score in our order) to the highest (the 100 th score in our order) as shown below.

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