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THE MEDIAN


When a data set is at ordinal level, we can use median as an alternative to arithmetic mean. We use the alternative measure especially when we cannot calculate the arithmetic mean. For example, we want to compare the clothes size of two groups of two students.


Group A







Group B









To compare which group have larger clothes size, we cannot calculate the mean. Then, how to determine the values ​​that represent the groups’ clothes size? This is the use of the median! To determine the median of a data, the first step is to sort the data from the smallest to the largest. The data located in the middle is the median of the data. If Me is the median of a data set, then the number of data whose value is ≤ Me equals the number of data whose value is ≥ Me. Group A’s sorted clothes size data are: S - M - L - L - XL. The third data from the left, that is L, is the median of Group A’s clothes size. We denote it as: Me = L. Group B’s sorted clothes size data are: S - S - S - M - L - L - XL. The 4th data from the left, that is M, is the median of Group B’s clothes size. We denote it as: Me = M.

Determining the median
Let X1, X2, X3, ..., Xn be the data under consideration, sorted from the smallest to the largest. [So, X1 is the smallest and Xn is the largest]. The median of the data is $Me = X_{\frac{n+1}{2}}$ if n is odd and $Me = \frac{X_{n/2}+X_{1+n/2}}{2}$ if n is even.

Example 1
The following are shoe sizes of a group of men: 40, 38, 42, 44, 38, 36, 40. What is the median?

Answer:
Firstly, sort the data from the smallest to the largest. So, we have:
36   38   38   40   40   42   44
Let X1 = 36, X2 = 38, X3 = 38, X4 = 40, X5 = 40, X6 = 42, and X7 = 44. In this case, n = 7 (odd), so we use $Me = X_{\frac{n+1}{2}}$.
$\frac{n+1}{2} = \frac{7+1}{2} = 4$
Me = X4 = 40.
So, the median of the data is 40.

Example 2
The following are shoe sizes of a group of men: 40, 38, 42, 44, 38, 36, 40, 40. What is the median?

Answer:
Firstly, sort the data from the smallest to the largest. So, we have:
36   38   38   40   40   40   42   44
Let X1 = 36, X2 = 38, X3 = 38, X4 = 40, X5 = 40, X6 = 40, X7 = 42, and X8 = 44. In this case, n = 8 (even), so we use $Me = \frac{X_{n/2}+X_{1+n/2}}{2}$.
$ Me = \frac{X_{8/2}+X_{1+8/2}}{2} = \frac{X_{4}+X_{5}}{2} = \frac{40+40}{2} = 40$
So, the median is 40.


PROBLEMS
  1. The following are scores of a Social Statistics test: 28, 40, 87, 47, 59, 72, 84, 82, 19, 55, 64, 87. Calculate the median.
  2. The following is the length of the conversation using 14 mobile phones, in seconds: 16, 62, 45, 90, 55, 49, 100, 89, 72, 115, 45, 59, 128, 219. Find the median.
  3. The following are 11 data regarding the number of students attending Social Statistics class, from the first lecture to the 11th: 37, 40, 35, 36, 40, 40, 42, 37, 39, 40, 41. What is the mean of this data?

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