# 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.**Computation of median*The median can be easily computed by sorting the data from smallest to largest and counting the middle value.

**Example 5**

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. The median in this case is computed as the arithmetic mean of the two middle values.

**Example 6**

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, namely 45 and 46. The median can be obtained by taking the mean of the two observations:

Median =

= 45.5 marks

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 = item

Where N = number of items.

You may note that the above formula gives you the position of the median in an ordered array, not the median itself. Median is computed by the formula:

Median = size of item

# 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.**Example 7**

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

**TABLE 5.4**

**Computation of Median for Discrete Series**

^{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/2)th item [not [(N+1)/2]th item] lies. The median can then be obtained as follows:

Median = L +

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, h = magnitude of the median class interval. No adjustment is required if frequency is of unequal size or magnitude.

**Example 8**

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

Number of workers: 7 13 15 20 30 33 28 14

**TABLE 5.5**

**Computation of Median for Continuous Series**

Median =L +

= 35 +

=Rs 35.83

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. You should remember that median, as a measure of central tendency, is not sensitive to all the values in the series. It concentrates on the values of the central items of the data.

**Activities**

- Find mean and median for all four values of the series. What do you observe?

**TABLE 5.6**

**Mean and Median of different series**

- Is median a better method than mean?