• Written By shivani Agrawal

# Average Formula: Definition, Symbol and Examples

Average Formula: The Average of a set of numbers is equal to the sum of a given set of values divided by the total number of values in the group. In simple terms, the Average is the central value of a group of numbers. The formula of finding the Average has many applications in real life.

To understand it better, let’s take an example:

Suppose, a, b, c, d, … represents the observation ‘n’. So the average observation is given as-

Average Value = (a + b + c + d + …)/ n

‘n’ is the total number of observations.

## Average Formula

The formula to find the average of the group of the numbers is easy to remember. We have to add all the numbers of the given group and then divide the result by the number of values provided. The average is denoted by $$\overline x$$  It is also indicated by the symbol $$\mu$$. The formula of Average to calculate the group of numbers:

$$Average = {{Sum of Numbers} \over {Numbers of Unit}}$$

### How To Calculate Average?

As we have mentioned above, to calculate the average, we have to add all the given groups of values and divide the result by the number of given values:

Let us find the average of the following group of numbers 30, 31, 32, 33, 28, 34, 30, 33.

By formula of average, we know-

$$Average = {{Sum of Numbers} \over {Numbers of Unit}}$$

Average = (30 + 31 + 32 + 33 + 28 + 34 + 30 + 33)/ 8

Average = 251/ 8

Average = 31.375

#### Calculating Average of Negative Numbers

There is the same process, formula, and method to calculate the average of negative numbers if found in the group of numbers given. For Example-

Find the average of 29, 41, -32, 54, -21, 12.

Solution-

= 29 + 41 -32 + 54 -21 + 12

Total Units = 6

As we know-

$$Average = {{Sum of Numbers} \over {Numbers of Unit}}$$

So,

Average = 29 + 41 -32 + 54 -21 + 12/ 6

Average = 83/ 6

Average = 13.833

### Different Ways To Calculate Average

Learn about the three common ways of calculating the average of the groups of numbers with a detailed explanation of the following:

1. Mean
2. Median
3. Mode

There are three types of Mean-

1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean

#### Arithmetic Mean

The arithmetic mean of a group of the data set is computed by adding all the values and dividing the whole by the total number of values. The mean of any group of the data set is obtained by adding all the numbers and dividing the sum by the cumulative numbers in the given group. It provides the central value of the groups of the values. If the number ‘n’ is given, each number indicated by ai (where i= 1,2,3,4,5….,n).

The formula for Calculating Arithmetic Mean by Step-Deviation Method is-

$$\overline X = A + {{\sum {fu} } \over {\sum f }} \times h$$

Where,

Know how to calculate the arithmetic mean with the help of the following examples-

Example- Below data shows the distance covered by 100 people to perform their regular jobs.

Calculate the arithmetic mean by the step-deviation method and explain “why it is better than the direct method” in this specific case.

Solution-

The given distribution is grouped data, and the variable included is distance covered, while the number of people signifies frequencies.

Calculate the Arithmetic Mean as-

$$\overline X = A + {{\sum {fu} } \over {\sum f }} \times h$$

Where,

A = 15, $$\sum {}$$fu= 90, $$\sum {}$$f= 100, h = 10

$$\overline X$$ =15 +9010010=24 km

#### Geometric Mean

It is a method to find the central tendency of a group of numbers by finding the ‘nth’ root of the output of ‘n’ numbers.

The formula of Geometric Mean is-

$$\root n \of {{\bf{x1}},{\rm{ }}{\bf{x2}},{\rm{ }}{\bf{x3}},…..,{\bf{xn}}}$$

x1, x2, x3,…..,xn are the individual items up to ‘n’ term.

Example- Find the geometric mean (GM) of 4 and 7.

Solution- By using the formula for the geometric mean,

GM of 4 and 7 will be-

Geometric Mean will be √(4×7)

= √28

= 5.29

So, Geometric Mean (GM) = 5.29

#### Harmonic Mean

Harmonic Mean is the reciprocal of the mathematics (arithmetic) mean of the provided data values. This means if the values are less so the harmonic mean will large and vice versa.

Formula of Harmonic Mean-

HM = n / [(1/x1) + (1/x2) + (1/x3) + (1/x4)…+ (1/xn)]

Where,

x1, x2, x3, x4…, xn are the individual values up to n terms.

### Mean

Mean is calculated as a measure of central tendencies in Statistics. The calculation of the mean includes all values in the data. If we change any value, the mean changes. But, the mean doesn’t always locate the center of the data accurately.

The formula for Calculating Mean is:

Mean = Sum of Given Data/ Total Number of Data

In terms of sigma ($$\sum$$) notation, the formula of the mean is-

$$\overline X = {{\sum\limits_i^n { = 1Xi} } \over N}$$

Where,

$$\sum$$xi = Sum of Given Data Values

N = Number of Given Data

i = Index of summation

ai = data value of the given index

### Median

To identify the median from the groups of numbers, we need to place all the values from descending order to ascending order and finds the value one in the middle.

The formula of Median based on different data set:

If the group of given numbers of observations/ data is odd. The formula will be used to find the median is:

Median = {(n+1)/2}th term

If the group of given numbers of observations/ data is even. The formula will be used to find the median is:

Median = [(n/2)th term + {(n/2)+1}th term]/2

Where,

‘n’ is the number of observations.

Example-

Find the median from the given data set- 5, 7, 4, 8, 6.

Solution-

Given- 4,5,6,7,8

As we can see in the given data set the number of the observations/ data is odd i.e., 5.

So, n = 5

Arrange the data set in ascending order,

1,3,4,6,7

Now we will put the formula of finding the median from odd observations-

Median = {(n+1)/2}th term

Median = {(5+1)/2}th term

Median = 6th term

Here the sixth (6th) term is ‘6’.

Therefore, the median of the given data set is ‘6’.

Example-

Find the median from the given data set- 4,7,3,8,6,2

Solution-

Given- 4,7,3,8,6,2

As we can see in the given data set the number of the observations/ data is even i.e., 6.

So, n = 6

Now we will put the formula of finding the median from even observations-

Median = [(n/2)th term + {(n/2)+1}th term]/ 2

Median = [(6/2)th term + {(6/2)+1}th term]/ 2

Median = (3rd term + 4th term)/ 2

Here, the 3rd term is ‘4’ and the 4th term is ‘6’

Median = (4+6)/ 2

= 10/ 2 = 5

Therefore, the median of the given data set is ‘5’.

### Mode

Mode is identified by the highest frequency of the value mentioned in the given groups of values. It means the value that appears the most number of times in the data set. It can be said that the value in a data set with a high frequency or rises more frequently is known as the mode or modal value. It is also come under the measures of central tendency, aside from mean and median.

Suppose grouped frequency distribution appears in the data set, so the calculation of mode as per the frequency is not possible. To determine the mode of data of such cases, we have to calculate the modal class. Mode lies inside the modal class.

The formula for the mode of data is given below:

$$Mode = l + ({{f1 – f0} \over {2f1 – f0 – f2}}) \times h$$

Where,

l = lower limit

h = size of the class interval

f1 = frequency

f0 = frequency of the class preceding

f2 = frequency of the class succeeding

Example-

Find the mode of 5, 5, 5, 10, 14, 14, 14, 28, 47, 58 data set.

Solution-

Given, 5, 5, 5, 10, 14, 14, 14, 28, 47, 58

As per the definition, the mode of the data set is both ‘5’ and ‘14’.

Example-

In a class of 30 students, marks secured by students in maths out of 50 are tabulated below. Calculate the mode of data given in the table:

Solution-

The maximum frequency of the class is 13 and the interval of the class corresponding to this frequency is 15-25. So the modal of the class is 15-25.

l = 15

h = 10

f1 = 13

f0 = 8

f2= 3

Put the value in the formula-

$$Mode = l + ({{f1 – f0} \over {2f1 – f0 – f2}}) \times h$$

$$Mode = 15 + ({{13 – 8} \over {2(13) – 8 – 3}}) \times 10$$

Mode = 15 + (5/15) x 10

Mode = 15 + 50/15

Mode = 18.33…