Statistics - Discrete Series Arithmetic Median

When data is given along with their frequencies. Following is an example of discrete series −

Items	5	10	20	30	40	50	60	70
Frequency	2	5	1	3	12	0	5	7

In case of a group having even number of distribution, Arithmetic Median is found out by taking out the Arithmetic Mean of two middle values after arranging the numbers in ascending order.

Formula

Median = Value of ($\frac{N+1}{2})^{th}\ item$.

Where −

${N}$ = Number of observations

Example

Problem Statement −

Let's calculate Arithmetic Median for the following discrete data −

Items, ${X}$	14	36	45	70	105	145
Frequency, ${f}$	2	5	2	3	12	4
Comulative Frequency, ${C_f}$	2	7	9	12	24	28
Terms	1-2	3-7	8-9	10-12	13-24	25-28

Solution −

Based on the above mentioned formula, Arithmetic Median M will be −

$M = Value\ of\ (\frac{N+1}{2})^{th}\ item. \\[7pt] \, = Value\ of\ (\frac{28+1}{2})^{th}\ item. \\[7pt] \, = Value\ of\ 14.5^{th}\ item. \\[7pt] \, = Value\ of\ (\frac{14^{th}\ item\ +\ 15^{th}\ item}{2})\\[7pt] \, = (\frac{105\ +\ 105}{2}) \, = {105}$

The Arithmetic Median of the given numbers is 2.

In case of a group having even number of distribution, Arithmetic Median is the middle number after arranging the numbers in ascending order.

Example

Let's calculate Arithmetic Median for the following discrete data −

Items, ${X}$	14	36	45	70	105
Frequency, ${f}$	2	5	1	4	13
Comulative Frequency, ${C_f}$	2	7	8	12	25
Terms	1-2	3-7	8-8	9-12	13-25

Given numbers are 25, an odd number thus middle number, 12th term is the Arithmetic Median.

∴ The Arithmetic Median of the given numbers is 70.

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