Arithmetic Mean (Average) 

The arithmetic mean, often referred to as the average, is one of the easiest and most frequently used statistical tools to obtain the centre value or "typical" value within a collection of data. It aids the representation of a set of numbers by a single number, hence providing an instantaneous summarisation of the information. The arithmetic mean is applied widely in statistics, economics, education, and many more disciplines.

1.0What is the Arithmetic mean? 

The arithmetic mean is obtained by adding up all the numbers in a data set and dividing that total by the number of values found within the data set. It is a simple measure of central tendency and, therefore, extensively used to summarise data.

The general formula for arithmetic mean is:

$\mathrm{Mean}(\text{ Average })=\frac{\text{ sum of all values of a data set }}{\text{ No. of values }}$

2.0How to calculate the Arithmetic mean? 

To calculate the arithmetic mean, follow these simple steps: 

1. Add up all the values of a given dataset. 
2. Count how many values are there in the dataset. 
3. Divide step 1 from step 2 to get the arithmetic mean of a given data set. 

Let us elaborate on this with the help of an example: 

Example: Find the arithmetic mean of 24, 25, 35, 65, 46. 

Solution: Step 1: 24 + 25 + 35 + 65 + 46 = 195

Step 2: No. of values are = 5. 

Mean = 195/5 = 39. 

3.0Arithmetic Mean in Statistics 

In statistics, the arithmetic mean is a way of summarising a collection of data as the "central" or "typical" value; it may be used to represent the general trend of a data set. However, it is affected by outliers or extreme values much higher or lower than the rest of the data.

In statistics, the mean data is categorised into two groups: grouped data and ungrouped data. 

Ungrouped Data 

Ungrouped data refers to raw, individual values that are not organised into groups. In the case of handling ungrouped data, one uses the direct summation of the values and then the division by the number of data points to compute the arithmetic mean. Let us understand with an example how to solve ungrouped mean. 

Example: Find the arithmetic mean of 13, 14, 15, 16, 17. 

Solution: Sum of all the values of data = 75

No. of values = 5 

Mean = 75/5 = 15

Grouped Data 

Grouped data refers to arranged data in intervals or classes. It is usually used when the amount of data is very large and cannot be managed with individual data points. Grouped data arithmetic mean involves midpoints of the classes, also known as class marks and their frequencies. The formula for grouped data is 

$\text{ Mean }=\frac{\sum f_i{x}_i}{\sum f_i}$

$\text{ Class Mark }=\frac{\text{ Upper Limit }+\text{ Lower Limit }}{2}$

Example: The following data shows the height of students in the 10th class. Find the mean of the following data. 

Solution: 

$\text{ Mean }=\frac{\Sigma f_i{x}_i}{\Sigma f_i}$

$\text{ Mean }=\frac{4740}{32}=148.125$

4.0Weighted Arithmetic Mean

The weighted arithmetic mean is a version of the arithmetic mean to be used in cases where different values have different weights to be taken on their importance. Each value is multiplied by a "weight" representing its significance. The formula for the weighted mean is:

$\text{ Weighted Mean }=\frac{\sum x_i{w}_i}{\sum w_i}$

Where,

$x_i$ represents each value. 

${\mathrm{W}}_{\mathrm{i}}$ represents the weight associated with each value. 

Example: Consider you have two subjects with marks: 

Maths = 80 weight 2. 

English = 90 weight 2. 

Solution: Multiply each score by its weight:

80 · 2 = 160, and 90 · 2 = 180 

Adding both the values, 

160 +180 = 340 

$\text{ Weighted Mean }=\frac{340}{4}=85$