Mean is an average of the given numbers: a calculated central value of a set of numbers. In simple words, it is the average of the set of values. In statistics, the mean is one of the Measures of central tendency, apart from the mode and median. But altogether, all three measures (mean, median, mode) define the central value of given data or observations.

Example: What is the mean of 2, 4, 6, 8 and 10?

First add all the numbers, 

2+4+6+8+10 = 30

Now divide by 5 (total number of observations)

Mean = 30/5 = 6

Definition of Mean in Statistics

Mean is nothing but the average of the given set of values. It denotes the equal distribution of values for a given data set. Central tendency is the statistical measure that recognizes a single value as representative of the entire distribution. It strives to provide an exact description of the whole data. It is the unique value that is the which represents collected data. The mean, median and mode are the three commonly used measures of central tendency.

In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x) and then adding all these products together.

Mean Symbol (X Bar)

The symbol of mean is usually given by the symbol ‘x̄’. The bar above the letter x, represents the mean of x number of values. 

X̄ = (Sum of values ÷ Number of values)

X̄ = (x₁ + x₂ + x₃ +….+x_n)/n

What is Mean in Maths?

Mean, in Maths, is nothing but the average value of the given numbers or data. To calculate the mean, we need to add the total values given in a datasheet and then divide the sum by the total number of values. Suppose, in a data table, the price values of 10 clothing materials are mentioned. If we have to find the mean of the prices, then add the prices of each clothing material and divide the total sum by 10. It will result in an average value.

Another example is, if we have to find the average age of students of a class, then we have to add the age of individual students present in the class and then divide the sum by the total number of students present in the class.

Mean is a method that is commonly used in Statistics. In primary schools, we have learned this concept with the term ‘average’. But in higher classes, when we are introduced to the topic mean, it refers to an advanced version of sequence or series of a number. In the real-world, when there is huge data available, we use statistics to deal with it. Along with mean, we also learn about median and mode. Median is the middle value of a given data when all the values are arranged in ascending order. Whereas mode is the number in the list, which is repeated a maximum number of times.

Mean Formula

The basic formula to calculate the mean is calculated based on the given data set. Each term in the data set is considered while evaluating the mean. The general formula for mean is given by the ratio of sum of all the terms and total number of terms. Hence, we can say;

Mean = Sum of the Given Data/Total number of Data

To calculate the arithmetic mean of a set of data we must first add up (sum) all of the data values (x) and then divide the result by the number of values (n). Since ∑ is the symbol used to indicate that values are to be summed (see Sigma Notation) we obtain the following formula for the mean (x̄):

x̄=∑ x/n

How to Find Mean?

To find the mean of any given data set, we have to use the average. The example given below will help you in understanding how to find the mean of given data.

For example, in a class there are 20 students and they have secured a percentage of: 88,82,88,85,84,80,81,82,83,85,84,74,75,76,89,90,89,80,82,83. Find the average of the percentage obtained by the class.

Solution: Average = Total of percentage obtained by 20 students in class/Total number of students

Avg = [88+82+88+85+84+80+81+82+83+85+84+74+75+76+89+90+89+80+82+83]/20

Avg.=1660/20 = 83

Hence, the average percentage of each student in class is 83%.

In the same way, we find the mean of the given data set, in statistics.

Mean of Negative Numbers

We have seen examples of finding the mean of positive numbers till now. But what if the numbers in the observation list include negative numbers. Let us understand with an instance,

Example: Find the mean of 9, 6, -3, 2, -7, 1.

Add all the numbers first:

Total: 9+6+(-3)+2+(-7)+1 = 9+6-3+2-7+1 = 8

Now divide the total from 6, to get the mean.

Mean = 8/6 = 1.33

Types of Mean

There are majorly three different types of mean value that you will be studying in statistics.

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

Arithmetic Mean

When you add up all the values and divide by the number of values it is called Arithmetic Mean. To calculate, just add up all the given numbers then divide by how many numbers are given.

Example: What is the mean of 3, 5, 9, 5, 7, 2?

Now add up all the given numbers:

3 + 5 + 9 + 5 + 7 + 2 = 31

Now divide by how many numbers provided in the sequence:

316= 5.16

5.16 is the answer.

Geometric Mean

The geometric mean of two numbers x and y is xy. If you have three numbers x, y, and z, their geometric mean is 3xyz.

\(\large Geometric\;Mean=\sqrt[n]{x_{1}x_{2}x_{3}…..x_{n}}\)

Example: Find the geometric mean of 4 and 3 ?

Geometric Mean =\( \sqrt{4 \times 3} = 2 \sqrt{3} = 3.46\)

Harmonic Mean

The harmonic mean is used to average ratios. For two numbers x and y, the harmonic mean is 2xy(x+y). For, three numbers x, y, and z, the harmonic mean is 3xyz(xy+xz+yz)

\(\large Harmonic\;Mean (H) = \frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+……\frac{1}{x_{n}}}\)

Root Mean Square (Quadratic)

The root mean square is used in many engineering and statistical applications, especially when there are data points that can be negative.

\(\large X_{rms}=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}….x_{n}^{2}}{n}}\)

Contraharmonic Mean

The contraharmonic mean of x and y is (x2 + y2)/(x + y). For n values,

\(\large \frac{(x_{1}^{2}+x_{2}^{2}+….+x_{n}^{2})}{(x_{1}+x_{2}+…..x_{n})}\)

Practice Problems

Q.1: Find the mean of 5,10,15,20,25.

Q.2: Find the mean of the given data set: 10,20,30,40,50,60,70,80,90.

Q.3: Find the mean of the first 10 even numbers.

Q.4: Find the mean of first 10 odd numbers.