Statistics - Arithmetic Median of Discrete Series

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When data is given alongwith 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	14	36	45	70	105	145
Frequency	2	5	1	3	12	0

Solution:

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

$M = Value\ of\ (\frac{N+1}{2})^{th}\ item. \\[7pt] \, = Value\ of\ (\frac{6+1}{2})^{th}\ item. \\[7pt] \, = Value\ of\ 3.5^{th}\ item. \\[7pt] \, = Value\ of\ (\frac{3^{rd}\ item\ +\ 4^{th}\ item}{2})\\[7pt] \, = (\frac{45\ +\ 70}{2}) \, = {57.5}$

The Arithmetic Median of the given numbers is 57.5.

In case of a group having odd 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	14	36	45	70	105
Frequency	2	5	1	3	12

Given numbers are 5, an odd number thus middle number is the Arithmetic Median.

∴ The Arithmetic Median of the given numbers is 45.