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Introduction
Statistics is a computable expression of economics. It is used to design economic theories and is a basis for business planning. Moreover, statistics can help in the prognosis of a future event. The calculation of the median of given data is a vital part of statistical studies. Let us understand the definition of median and the method to calculate median.
What is Median?
When you study statistics, the median is a commonly used term. It represents the middle of a given set of values. It is different from mean and mode. The median is a positional average.
Median of ungrouped data
When given a set of random values, it is called ungrouped data. The first step in the median calculator is to arrange the data in ascending or descending order. The median value depends on the number of values in the given data.
(i) When ‘n’ is odd
If the total number of values in the given set is odd, the median is the [(n+1)/2]th term.
(ii) When ‘n’ is even
If the total number of values is even, then the median is the average of (n/2)th term and [(n/2)+1]th term.
Median of grouped data
In many cases, the data is given in groups. Here, you must know the lower limit, total number of values, cumulative frequency, frequency of median class, and class size. Cumulative frequency is the sum of frequencies obtained by adding frequencies of each previous step. We assume that the types are of equal measure.
Median Formula = l + [ (n/2 – cf)/f ]. h
- l represents the lower limit of the median class
- n is the number of observations
- cf is the cumulative frequency of the class preceding the median class
- f is the frequency of median class
- h is the class size
Difference between Mean, Median, and Mode
As explained earlier, the median is the central value of a given data set when placed in ascending or descending order. Mean is the average of the provided data of the set. It can be calculated using three methods: step deviation, direct approach, and assumed mean way. Mode represents the number that recurs the most frequently in a given set of values.
For example, if you are given a set of values such as 4,9,1,5,8,3,6,8, and you wish to find mean, median, and mode, follow the steps below:
- First of all, we present the data in ascending or descending order. The data becomes 1,3,4,5,6,8,8,9.
- The mode is the most repeated number in this set, 8. Hence, mode=8.
- The mean is the average of all values (1+3+4+5+6+8+8+9)/8. Hence, mean=5.5
- The number of values is 8, which is even. Hence the median is the average of 4th and 5th terms. The median values are 5 and 6. Hence, the median is (5+6)/2=5.5
Solved examples
Here are some solved examples:
Example 1:
Find the median of the following:
4, 19, 77, 25, 22, 24, 92, 82, 40, 24, 14, 12, 70, 23, 56
Solution:
First, we arrange the numbers in ascending order; we have:
4 ,12, 14, 19, 22, 23, 24, 24, 25, 40, 56, 70, 77, 82,92
We have fifteen numbers in total. The eighth number is the middle number.
Hence, the median value of this set of numbers is 24.
Example 2:
A survey on the heights (in cm) of 50 girls of class X was conducted at a school, and the following data were obtained:
Height (cm) | 120-130 | 130-140 | 140-150 | 150-160 | 160-170 | Total |
Number of girls | 2 | 8 | 12 | 20 | 8 | 50 |
Find the median of the above-grouped data.
Solution:
To find the median, we find cumulative frequencies.
Consider the table:
Class intervals | Number of girls (fi) | Cumulative frequency |
120-130 | 2 | 2 |
130-140 | 8 | 10 |
140-150 | 12 | 22 |
150-160 | 20 | 42 |
160-170 | 8 | 50 |
n = 50, n/2 = 25
Median class = 150 – 160
l = 150, c = 22, f = 20, h = 10
Median = l + [(n/2−c)/f] × h = 150 + [((50/2) – 22)/20] × 10 = 150 + 1.5 = 151.5
Therefore, the median = 151.5.
Example 3:
The following table shows the weekly spending of 200 families. Find the median of the weekly spending.
Weekly Expenditure ($) | 0-1000 | 1000-2000 | 2000-3000 | 3000-4000 | 4000-5000 | Total |
Number of Families | 34 | 12 | 43 | 60 | 51 | 200 |
Find the median of the above-grouped data.
Solution:
To find the median, we calculate cumulative frequencies.
Consider the table:
Weekly Expenditure | No. of families (fi) | Cumulative frequency (c) |
0 – 1000 | 34 | 34 |
1000 – 2000 | 12 | 34 + 12 = 46 |
2000 – 3000 | 43 | 46 + 43 = 89 |
3000 – 4000 | 60 | 89 + 60 = 149 |
4000 – 5000 | 51 | 159 + 51 = 200 |
n = 200, n/2 = 200/2 = 100
Median Class = 3000 – 4000
l = 3000, c = 89, f = 60, h = 1000
Median Formula= l + [(n/2−c)/f] × h = 3000 + [(200/2 – 89)/60] × 1000 = 3000 + 183 = 3183.
Therefore, the median is 3183.
Example 4:
The following set of numbers are arranged in ascending order of their values:
19, 20, 21, 22, 23, x-10, x-8, x-5, x-9, 66, 70, 85.
The median of these numbers is 44; find the value of ‘x.’
Solution:
As per the data given, the number of terms is 12 (even).
In case of even number of terms, the formula for median is-
Median Formula= [(n/2)th term +{(n/2)+1}th term]/2
Median= [(12/2)th term +{(12/2)+1}th term]/2
Median= [(6th term +7th term]/2
Median= [(x-10)+(x-8)]/2
44= [2x-18]/2
44*2=2x-18
2x-18=88
2x=88+18
2x=106
x=106/2
x=53.
Hence, the value of x is 53.
The terms are 19, 20, 21, 22, 23, 43, 45, 48, 44, 66, 70, 85.
Conclusion
Median is an important part of statistical studies. It is vital to understand the difference between mean, median, and mode and apply their formulas in the given problems. The definition, formulae, significance, and examples are explained in this article in a nutshell.
