Statistics - Discrete Series Arithmetic Median


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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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