The mean is what we usually call the average of a set of numbers. To find the mean, we add up all the numbers in the data set and then divide that total by how many numbers there are. It is often used in statistics to show the center or typical value of the data.

There are two types of means — population mean and sample mean.

• The population mean is used when we have data for the entire group or population.
• Sometimes, it's not possible to collect data from the whole population because it’s too large. In such cases, we take a sample from the population. The average of that sample is called the sample mean, and it gives us an idea of the average for the whole population.

It's important to note that the mean value will fall between the maximum and minimum values in the data set. The mean may not always be a number in the data set, but occasionally it could be equal to a value from the set.

How to Calculate Mean for Ungrouped Data

The formula for calculating the mean of ungrouped data is as follows:

Let's say we have n observations of a data set, denoted as x 1 , x 2 , x 3 ,….., x n . The mean of these values would be:

\(\begin{array}{l}\overline{x}=\frac{\sum x_i}{n}\end{array} \)

Where,

x i = ith observation, 1 ≤ i ≤ n

∑x i = Sum of observations

n = Number of observations

How to Calculate Mean for Grouped Data

When dealing with grouped data, there are three methods to calculate the mean, depending on the size of the data. These methods include:

• Direct Method
• Assumed Mean Method
• Step-deviation Method

Below, we'll go through the formulas for these three methods:

Direct Method

Suppose we have n observations denoted as x 1 , x 2 , x 3 ,…., x n , each with respective frequencies f 1 , f 2 , f 3 ,…., f n . This implies that the observation x 1 occurs f 1 times, x 2 occurs f 2 times, x 3 occurs f 3 times and so on. The formula to calculate the mean using the direct method is:

\(\begin{array}{l}\overline{x}=\frac{f_1x_1+f_2x_2+f_3x_3+….+f_nx_n}{f_1+f_2+f_3+….+f_n}\end{array} \)

Or

\(\begin{array}{l}\overline{x}=\frac{\sum_{i=1}^{n}f_ix_i}{\sum_{i=1}^{n}f_i}\end{array} \)

Where,

∑f i x i = Sum of all the observations

∑f i = Sum of frequencies or observations

This method is typically used when the number of observations is small.

Assumed Mean Method

In this method, we typically assume a value as the mean (let's call it a ). This value is used to calculate the deviations, upon which the formula is based. Additionally, the data will be presented in the form of a frequency distribution table with classes. The formula to find the mean in the assumed mean method is:

\(\begin{array}{l}Mean ,\ (\overline{x})=a+\frac{\sum f_id_i}{\sum f_i}\end{array} \)

Where,

a = assumed mean

f i = frequency of ith class

d i = x i – a = deviation of ith class

Σf i = N = Total number of observations

x i = class mark = (upper class limit + lower class limit)/2

Learn more about the assumed mean method here.

Step-deviation Method

The step-deviation method is used to calculate the mean when the data values are large. The formula is given by:

\(\begin{array}{l}Mean,\ (\overline{x})=a+h\frac{\sum f_iu_i}{\sum f_i}\end{array} \)

Where,

a = assumed mean

f i = frequency of ith class

x i – a = deviation of ith class

u i = (x i – a)/h

Σf i = N = Total number of observations

x i = class mark = (upper class limit + lower class limit)/2.

Types of Mean in Math 

In mathematics, the mean is a type of average used to represent the center or typical value in a set of numbers. There are different types of means, and each one is used in different situations. The main types of mean are:

1. Arithmetic Mean

This is the most common type and is usually what we mean when we say “mean” in everyday use. To find the arithmetic mean, you add up all the numbers in your data set and then divide by how many numbers there are.
Formula:

• For simple data: Mean = (Sum of all values) ÷ (Number of values)
   
• For grouped data: Mean = Σfixi / Σfi,
  where:
  • xi = value
  • fi = frequency

2. Weighted Mean

A weighted mean is used when some values are more important or carry more weight than others. Each value has a weight (wi) assigned to it.
Formula:
Weighted Mean = Σwixi / Σwi,
where:

• xi = data value
• wi = weight of each value

3. Geometric Mean

The geometric mean is used when dealing with growth rates, percentages, or ratios. You multiply all the numbers together and then take the nth root (where n is the total number of values).
Formula:
G.M. = √(x1 × x2 × x3 × ... × xn)

4. Harmonic Mean

The harmonic mean is used when the values are rates or speeds. It is found by dividing the number of values by the sum of the reciprocals (1/value).
Formula:
H.M. = n / [Σ(1/xi)],
where n is the total number of values.

Arithmetic Mean is the average found by adding all values and dividing by the number of values. Geometric Mean is the average rate of growth, found by multiplying all values and taking the nth root. While AM is best for regular data, GM is ideal for comparing ratios or growth over time.

Mean Examples 

Example 1: Calculate the mean of the following data set.

15, 25, 30, 10, 45, 50, 35, 20, 30, 50

Solution:

Given,

x i = 15, 25, 30, 10, 45, 50, 35, 20, 30, 50

n = 10

Mean = ∑x i /n

= (15 + 25 + 30 + 10 + 45 + 50 + 35 + 20 + 30 + 50)/10

= 310/10

= 31

Therefore, the mean of the given data set is 31.

Example 2: Calculate the mean of the following distribution, which represents the scores obtained by students in a quiz.

Scores: 30, 50, 40, 45, 35, 30, 32, 25

Number of students: 22, 2, 5, 3, 18, 27, 30, 8

Solution: We'll create a table to find the sum:

Mean = (∑fᵢ × xᵢ) / ∑fᵢ
= 3695 / 115
= 32.13

Therefore, the mean of the given distribution is 32.13.

Example 3:Calculate the mean of the following distribution, which shows the number of books read by students in a month.

Books Read (xᵢ): 1, 2, 3, 4, 5

Number of Students (fᵢ): 4, 6, 10, 5, 5

Solution: We will create a table to find the sum of fᵢ and fᵢ × xᵢ.

Mean = (∑fᵢ × xᵢ) / ∑fᵢ
= 91 / 30
= 3.03

Therefore, the mean number of books read by students is 3.03.