How we calculate the median depends on whether there is an even or odd number in your group of data.
- Even numbers: If the group of data contains an even number of scores; in other words, if your group of data contains 100 scores, for example, as in our example above, then we calculate the median by adding the two middle scores together and dividing them by two. As a result, we must take the middle two values, namely the 50th and 51st scores (highlighted in red in the table above), 58 and 59 out of 100 respectively, add them together (58 + 59 = 117) and divide them by 2 (117 ÷ 2 = 58.5). Hence, the median for our group of data is 58.5 out of 100.
- Odd numbers: If the group of data contains an odd number of scores; in other words, if your group of data contains 9 scores, for example, then the median will simply be the middle score, the score of the 5th student.
The median is considered to be a good measure of central tendency because it describes the middle position of a frequency distribution for a group of data. However, it has one main weakness. Like the mode, the median fails to take into consideration all the values in the group of data. This problem is illustrated below by making some crude changes to the marks achieved by our 100 students. Whilst we are unlikely to see such marks being achieved in real life, these numbers help to illustrate our point.
Order |
Score |
Order |
Score |
Order |
Score |
Order |
Score |
Order |
Score |
1st |
40 |
21st |
40 |
41st |
40 |
61st |
64 |
81st |
74 |
2nd |
40 |
22nd |
40 |
42nd |
40 |
62nd |
64 |
82nd |
74 |
3rd |
40 |
23rd |
40 |
43rd |
40 |
63rd |
65 |
83rd |
74 |
4th |
40 |
24th |
40 |
44th |
40 |
64th |
66 |
84th |
75 |
5th |
40 |
25th |
40 |
45th |
40 |
65th |
67 |
85th |
75 |
6th |
40 |
26th |
40 |
46th |
40 |
66th |
67 |
86th |
76 |
7th |
40 |
27th |
40 |
47th |
40 |
67th |
67 |
87th |
77 |
8th |
40 |
28th |
40 |
48th |
40 |
68th |
67 |
88th |
77 |
9th |
40 |
29th |
40 |
49th |
40 |
69th |
68 |
89th |
79 |
10th |
40 |
30th |
40 |
50th |
58 |
70th |
69 |
90th |
80 |
11th |
40 |
31st |
40 |
51st |
59 |
71st |
69 |
91st |
81 |
12th |
40 |
32nd |
40 |
52nd |
60 |
72nd |
69 |
92nd |
81 |
13th |
40 |
33rd |
40 |
53rd |
61 |
73rd |
70 |
93rd |
81 |
14th |
40 |
34th |
40 |
54th |
62 |
74th |
70 |
94th |
81 |
15th |
40 |
35th |
40 |
55th |
62 |
75th |
71 |
95th |
81 |
16th |
40 |
36th |
40 |
56th |
62 |
76th |
71 |
96th |
81 |
17th |
40 |
37th |
40 |
57th |
63 |
77th |
71 |
97th |
83 |
18th |
40 |
38th |
40 |
58th |
63 |
78th |
72 |
98th |
84 |
19th |
40 |
39th |
40 |
59th |
64 |
79th |
74 |
99th |
84 |
20th |
42 |
40th |
40 |
60th |
64 |
80th |
74 |
100th |
85 |
In the above example, the marks above the median have been kept the same but all the marks below the median have been changed to 40 (highlighted in blue). Despite this, the median score of 58.5 remains (highlighted in red). Clearly, the median of 58.5 is no longer as accurate a representation of the performance of our students as it was previously because of the lower marks introduced. Nonetheless, without looking at the ‘raw’ data (in other words, every single mark) we would not know this. That said, the median does have benefits over the mean (discussed next), being a particularly useful statistic when the group of data is not normally distributed. This is because the median is not influenced by outliers or large deviations from the mean score (see the statistical guide on
frequency distributions).
Mean
The mean, otherwise called the arithmetic mean or average, is the most commonly used measure of central tendency. In order to calculate the mean, we simply add up all the values in the group of data and divide them by the number of values. Therefore, in our example we added up the scores of each of the 100 students and divided it by number of students (in other words, divide it by 100), which gives us a mean score of
58.75 out of 100 (see below).
