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The World of Statistics – 6

By Dr. Magdi Abadir, PhD

Article – 6: The Central Tendency (2)

3. The median of ungrouped data

In the last article, the mean value was introduced as a single value that more or less summarizes the data present in the distribution. However, one main problem with using the mean as a measure for central tendency is the presence of odd values (outliers) as was explained in the said article.

If this is the case, then the median can be used as a better measure for central tendency. The median is the central value of the distribution if all data are grouped in ascending (or descending) order. This is simply calculated for ungrouped data as follows:

• Group the data in ascending (or descending) order. Let their number = n.
If n is odd, then the central value of rank (n+1)/2 is the median.
If n is even, then there are two medians of rank n/2 and n/2 + 1.

For example, consider these 25 age values (refer to Article 3):

32	45	48	23	24	56	36	39	42	47	59	26	30
35	35	23	49	62	48	55	34	38	25	58	40

These are first grouped in ascending order:

23	23	24	25	26	30	32	34	35	35	36	38	39
40	42	45	47	48	48	49	55	56	58	59	62

The rank of the median is (25+1)/2 = 13. This is 39, the number highlighted in yellow.
Now, consider the following case where only 10 values are involved:

Since the number of values is even, there will be two median values of ranks 10/2

and 10/2 + 1, that is, the 5th and 6th numbers (highlighted in yellow): 32 and 35. It is customary, however, to consider the median value of the set as the mean value of these two values: (32+35)/2 = 33.5

One interesting property of the median is that if any of the values lower than the median is changed to a lower value the median remains unchanged. It also remains unchanged if a value greater than the median is changed to a higher value. For example, in the last set of data, if the second value in the ordered set is decreased from 25 to 19 and the 8th value is changed from 47 to 55, the ordered data become:

The median values are seen to remain unchanged.

Reference:
(1) Hoel P.G. (1976). “Elementary Statistics” 4th Ed., Reading, Mass.: Wiley Int. Edition, Chapter 2

Dr. Magdi Fouad Abadir, Ph. D.: Dr. M. F. Abadir is currently a professor with the Chemical Engineering Department at the Faculty of Engineering, University of Cairo, Egypt. His major interests are in the fields of high temperature science and technology. During his career, he has supervised more than 110 MSc and PhD theses and published more than a hundred papers mostly in international peer review journals. He currently teaches courses in High Temperature Technology and Industrial Statistics. He is also a consultant for several industrial businesses.

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The World of Statistics – 6

By Dr. Magdi Abadir, PhD

Article – 6: The Central Tendency (2)

3. The median of ungrouped data

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