STEPS IN CALCULATING MEDIAN

Dr. Anshul Singh Thapa

MEDIAN

• The arithmetic mean is affected by the presence of extreme values in the data. If you take a measure of central tendency which is based on middle position of the data, it is not affected by extreme items. Median is that positional value of the variable which divides the distribution into two equal parts, one part comprises all values greater than or equal to the median value and the other comprises all values less than or equal to it. The Median is the “middle” element when the data set is arranged in order of the magnitude.

DEFINITION

• According to Connor “The median is that value of the variable which divides the group into two equal parts, one part comprising all values greater than the median value and the other part comprising all the values smaller than the median value”.

MEDIAN

MEDIAN

FOR UNGROUPED DATA

MEDIAN

FOR GROUPED DATA

EVEN NUMBER

ODD NUMBER

DISCRETE SERIES

CONTINUOUS SERIES

Position of median = (N+1)^th item

2

Median = size of (N+1)^th item

2

(N+1)/2

L + (N/2 - c.f.) × i

f

COMPUTATION OF MEDIAN FOR UNGROUPED DATA

• The median can be easily computed by sorting the data from smallest to largest and counting the middle value.

Example

• Suppose we have the following observation in a data set: 5, 7, 6, 1, 8,10, 12, 4, and 3.
• Arranging the data, in ascending order you have: 1, 3, 4, 5, 6, 7, 8, 10, 12.
• The “middle score” is 6, so the median is 6. Half of the scores are larger than 6 and half of the scores are smaller.
• If there are even numbers in the data, there will be two observations which fall in the middle.

Example

• The following data provides marks of 20 students. You are required to calculate the median marks. 25, 72, 28, 65, 29, 60, 30, 54, 32, 53, 33, 52, 35, 51, 42, 48, 45, 47, 46, 33.
• Arranging the data in an ascending order, you get 25, 28, 29, 30, 32, 33, 33, 35, 42, 45, 46, 47, 48, 51, 52, 53, 54, 60,65, 72.
• You can see that there are two observations in the middle, 45 and 46. The median can be obtained by taking the mean of the two observations:
• Median = 45+ 46 = 45.5 marks

2

• In order to calculate median it is important to know the position of the median i.e. item/items at which the median lies. The position of the median can be calculated by the following formula:
• Position of median = (N+1)^th item

2

• Where N = number of items.
• We may note that the above formula gives us the position of the median in an ordered array, not the median itself. Median is computed by the formula:
• Median = size of (N+1)^th item = (20+1) = 10.5^th item

2 2

Size of 10.5^th item = 10^th item + 11^th item = 45 + 46 = 45.5

2 2

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Discrete Series

• In case of discrete series the position of median i.e. (N+1)/2th item can be located through cumulative frequency. The corresponding value at this position is the value of median.

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Example

• The frequency distribution of the number of persons and their respective incomes (in Rs) are given below. Calculate the median income.

The median is located in the (N+1)/2 = (20+1)/2 = 10.5th observation. This can be easily located through cumulative frequency. The 10.5^th observation lies in the c.f. of 16. The income corresponding to this is Rs 30, so the median income is Rs 30.

The median is located in the (N+1)/2 = (20+1)/2 = 10.5th observation. This can be easily located through cumulative frequency. The 10.5^th observation lies in the c.f. of 16. The income corresponding to this is Rs 30, so the median income is Rs 30.

Continuous Series

• In case of continuous series you have to locate the median class where N/2th item [not (N+1)/2th item] lies. The median can then be obtained as follows:
• Median = L + (N/2 - c.f.) × i

f

• Where, L = lower limit of the median class, c.f. = cumulative frequency of the class preceding the median class, f = frequency of the median class, i = class interval of the median class.
• No adjustment is required if frequency is of unequal size or magnitude.

Example

• Following data relates to daily wages of persons working in a factory. Compute the median daily wage.

Position of the median = N/2 = 160/2 = 80 ^th item

​

Median = L + (N/2 - c.f.) × i

f

= 35 + (80 - 75) x 5 = 35.83

30

• Thus, the median daily wage is Rs 35.83. This means that 50% of the workers are getting less than or equal to Rs 35.83 and 50% of the workers are getting more than or equal to this wage.