# Measures of Central Tendency

A central tendency refers to an average or a central value of a statistical series. Measures of central tendency refer to all those methods of statistical analysis by which averages of statistical series are calculated. Let’s start with our first post on the central tendency. Today, we will learn only about mean but to understand that we will focus on the types of series.

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## Arithmetic Mean

It is the number which is obtained by adding the values of all the items of a series and dividing the total by the number of items.

Arithmetic Mean = Sum of Observation/ Number of observation

X refers to items

N number of observation

For example, pocket allowances of 5 students is 15, 20, 30, 22, and 25. Find the arithmetic mean.

Arithmetic mean = 15 + 20 + 30 + 22 + 25 / 5

Arithmetic mean = 112/ 5 = 22.4

What is frequency?

Frequency in statistics, refers to the number of times an item is repeated. For example, 5 students got 20 marks in maths. Here, 5 is the frequency.

## Types of Series

### 1. Individual Series

These are those series when the only X that is, items are given. Frequency is not given, or you can say each item in the question repeats once even if the items are mentioned twice.

Under Individual series there are 2 methods:

#### a. Direct Method

This is the most commonly used method. Under this individual series arithmetic mean is calculated using the below formula:

Arithmetic Mean = Sum of Observation/ Number of observation

#### b. Short-cut method

Short cut method is also known as the assumed mean method. Under this,individual series arithmetic mean is calculated using the below formula:

Arithmetic mean = Assumed mean + Summation of deviation/ Number of observation

A – Assumed mean taken from X, generally we take the middle value

D – Deviation & D = X-A

N- Number of items

### 2. Discreet series

These are those series where X is given as a discreet number with frequency. Under discreet series there are 3 methods:

#### a. Direct Method

As I mentioned in discreet series we have X (items) and F (frequency). So, in the formula:

Arithmetic mean = Summation of (F multiply with X) / Summation F

#### b. Short-cut method/ Assumed mean method

As the name suggest, we will have a assumed mean from X. Look at this formula:

The arithmetic mean = Assumed mean + Summation of (F multiply with X)/ Summation of F

#### c. Step-deviation method

Since, the name of the method says step deviation, we try to do the calculation faster by taking out a common factor from X. Look at the formula:

Arithmetic mean = Assumed mean + Summation of (F multiply with D’)/ Summation F multiply with Common factor

### 3. Continuous Series

Continuous series are those when the X is mentioned as an interval, like 0-10, 10-20, with frequency F. Remember, one thing, we always find mid-points in case of continuous series mean calculation. This series also have 3 methods:

#### a. Direct method

It is a simple method, you find the mid-point from X, multiply them with F, them add them up to find summation FM, where m refers to the mid-point from X, finally the sum is divided by the summation F.

#### b. Short-cut or Assumed mean method

As the name suggest, take the assumed mean A from X and use the following formula:

#### c. Step-deviation method

Step-deviation method, required the common factor, but here common factor will be the common gap or difference between the range of X. For example, 10-20 has 10 as common gap (20 minus 10). Have a look at the below formula:

I know, these formulas seems confusing, we will be discussing the questions on each method in the coming posts. To help you learn the formulas better, you can download all the formulas from the below table:

I hope it was helpful, you can refer more posts related to the statistics.

Thank You!

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