Data analysis Part 1: Mean and Median. When Median is recommended?

By Admin January 22, 2023

Data analysis Part 1: Mean and Median

Mean, Median are two important parameters in data analysis to calculate the spread of data

Let's calculate the two parameters with India's historical inflation data

Table 1. India's historical inflation data

Sl	Year	% inflation
1	2021	5.13
2	2020	6.62
3	2019	3.73
4	2018	3.94
5	2017	3.33
6	2016	4.95
7	2015	4.91
8	2014	6.67
9	2013	11.06
10	2012	9.31
11	2011	8.86
12	2010	11.99
13	2009	10.88
14	2008	8.35
15	2007	6.37
16	2006	5.8
17	2005	4.25
18	2004	3.77
19	2003	3.81
20	2002	4.3
21	2001	3.78
22	2000	4.01
23	1999	4.67
24	1998	13.23
25	1997	7.16
26	1996	8.98
27	1995	10.22
28	1994	10.25
29	1993	6.33
30	1992	11.79
31	1991	13.87
32	1990	8.97
33	1989	7.07
34	1988	9.38
35	1987	8.8
36	1986	8.73
37	1985	5.56
38	1984	8.32
39	1983	11.87
40	1982	7.89
41	1981	13.11
42	1980	11.35
43	1979	6.28
44	1978	2.52
45	1977	8.31
46	1976	-7.63
47	1975	5.75
48	1974	28.6
49	1973	16.94
50	1972	6.44
51	1971	3.08
52	1970	5.09
53	1969	-0.58
54	1968	3.24
55	1967	13.06
56	1966	10.8
57	1965	9.47
58	1964	13.36
59	1963	2.95
60	1962	3.63
61	1961	1.7
62	1960	1.78

1. Mean

1.1. Significance

It represents the average value of the data. It is the most commonly used parameter for estimating central tendency of the data.

1.2. Calculation

It is measured by dividing sum of the individual measurements by the number of measurements

As per Table 1

Mean is 7.39

1.3. Drawback

If the data is not uniformly distributed, it does not represent middle value of the data.

2. Median

2.1. Significance of median

To overcome the drawbacks of mean, i.e. if the data is not uniformly distributed, the term median can represent central tendency. This can be understood from an example.

2.2. Median vs Mean

Example 1

Suppose in an office, there are 6 employees. Employees are categorized in 6 groups based upon their salary.

Lowest salary is Rs 10,000, whereas highest salary is Rs 2,00,000/- and others salaries fall in between both the extremes

₹ 10,000/-
₹ 20,000/-
₹ 25,000/-
₹ 35,000/-
₹ 50,000/-
₹ 2,00,000/-

Mean: ₹ 56,667/-

Median: ₹ 30,000/-

It can be seen that 5 out of 6 people's salaries are in between ₹ 10,000-50,000/- but due to one person's salary of ₹ 2,00,000/-, the mean gets shifted to higher range and no longer represents central tendency. But median of ₹ 30,000/- still represents central tendency. That is the beauty of median. Median represents central tendency better than mean for non uniformly distributed data, whereas for uniformly distributed data, both mean and median values shall be pretty close with each other.

Example 2

Suppose in an Ph.D. course work class, there are 10 students. There ages are as following

22
22.1
22.3
22.4
22.5
22.9
23
23.1
23.4
23.9

Mean: 22.8

Median: 22.7

Here both mean (22.8) and median (22.7) represents the central tendency, since the data is uniformly distributed.

But if one student, lets say an in-service person in deputation of age 50 joins the class,

22
22.1
22.3
22.4
22.5
22.9
23
23.1
23.4
23.9
50

the new values shall be

Mean: 25.2

Median: 22.9

Here mean (25.2) no longer represents the central tendency but median (22.9) still does.

Median in comparison to mean is less susceptible to change due to extreme values and is more reliable to represent central tendency for either uniform or for non uniform data.

Uniform data means data with similar magnitude. Non uniform data means in addition to data with similar magnitude, few data with extreme values. Extreme values means either very high or very low values.

2.3. Calculation

It is measured by first arranging the data in increasing or decreasing trend and then middle value of the data is identified.

If the total no. of measurements is odd no., measurement is simple. Exactly middle value of the properly arranged data is the median.

If the total no. of measurements is even no., two middle values are added and are divided by 2.

Table 1a: India's historical inflation values arranged in increasing trend (1960-2021)

New Sl	Old Sl	Year	% Inflation
1	46	1976	-7.63
2	53	1969	-0.58
3	61	1961	1.7
4	62	1960	1.78
5	44	1978	2.52
6	59	1963	2.95
7	51	1971	3.08
8	54	1968	3.24
9	5	2017	3.33
10	60	1962	3.63
11	3	2019	3.73
12	18	2004	3.77
13	21	2001	3.78
14	19	2003	3.81
15	4	2018	3.94
16	22	2000	4.01
17	17	2005	4.25
18	20	2002	4.3
19	23	1999	4.67
20	7	2015	4.91
21	6	2016	4.95
22	52	1970	5.09
23	1	2021	5.13
24	37	1985	5.56
25	47	1975	5.75
26	16	2006	5.8
27	43	1979	6.28
28	29	1993	6.33
29	15	2007	6.37
30	50	1972	6.44
31	2	2020	6.62
32	8	2014	6.67
33	33	1989	7.07
34	25	1997	7.16
35	40	1982	7.89
36	45	1977	8.31
37	38	1984	8.32
38	14	2008	8.35
39	36	1986	8.73
40	35	1987	8.8
41	11	2011	8.86
42	32	1990	8.97
43	26	1996	8.98
44	10	2012	9.31
45	34	1988	9.38
46	57	1965	9.47
47	27	1995	10.22
48	28	1994	10.25
49	56	1966	10.8
50	13	2009	10.88
51	9	2013	11.06
52	42	1980	11.35
53	30	1992	11.79
54	39	1983	11.87
55	12	2010	11.99
56	55	1967	13.06
57	41	1981	13.11
58	24	1998	13.23
59	58	1964	13.36
60	31	1991	13.87
61	49	1973	16.94
62	48	1974	28.6

It can be seen as per Table 1a that, total no. of measurements is even no., so median is (6.62+6.67)/ 2 = 6.645.

Table 1b: India's historical inflation values arranged in increasing trend with India's projected inflation of 2022 (1960-2022)

New Sl.	Old Sl.	Year	% Inflation
1	47	1976	-7.63
2	54	1969	-0.58
3	62	1961	1.7
4	63	1960	1.78
5	45	1978	2.52
6	60	1963	2.95
7	52	1971	3.08
8	55	1968	3.24
9	6	2017	3.33
10	61	1962	3.63
11	4	2019	3.73
12	19	2004	3.77
13	22	2001	3.78
14	20	2003	3.81
15	5	2018	3.94
16	23	2000	4.01
17	18	2005	4.25
18	21	2002	4.3
19	24	1999	4.67
20	8	2015	4.91
21	7	2016	4.95
22	53	1970	5.09
23	2	2021	5.13
24	38	1985	5.56
25	48	1975	5.75
26	17	2006	5.8
27	44	1979	6.28
28	30	1993	6.33
29	16	2007	6.37
30	51	1972	6.44
31	3	2020	6.62
32	9	2014	6.67
33	1	2022	6.7
34	34	1989	7.07
35	26	1997	7.16
36	41	1982	7.89
37	46	1977	8.31
38	39	1984	8.32
39	15	2008	8.35
40	37	1986	8.73
41	36	1987	8.8
42	12	2011	8.86
43	33	1990	8.97
44	27	1996	8.98
45	11	2012	9.31
46	35	1988	9.38
47	58	1965	9.47
48	28	1995	10.22
49	29	1994	10.25
50	57	1966	10.8
51	14	2009	10.88
52	10	2013	11.06
53	43	1980	11.35
54	31	1992	11.79
55	40	1983	11.87
56	13	2010	11.99
57	56	1967	13.06
58	42	1981	13.11
59	25	1998	13.23
60	59	1964	13.36
61	32	1991	13.87
62	50	1973	16.94
63	49	1974	28.6

It can be seen as per Table 1b that, total no. of measurements is odd no., so median is 6.67.

Conclusion

Median in comparison to mean is less susceptible to change due to extreme values and is more reliable to represent central tendency for either uniform or for non uniform data.

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