Mean is an arithmetic average of the data set and it can be calculated by dividing a sum of all the data points with the number of data points in the data set. It is a point in a data set that is the average of all the data points we have in a set. In statistics, mean is the most common and frequently used method to measure the center of a data set. It’s a fundamental yet essential part of the statistical analysis of data. If we calculate the average value of the population set, then it is called the population mean. Sometimes, population data is vast, and we cannot perform analysis on that data set. Hence, in that case, we take a sample out of it and take an average. That sample represents the population set and the mean of this part of the data is called a sample mean.

\(Mean= \frac{Sum of observations}{Number of observations}\) |

An important note is that the mean value is the average value, which will fall between the maximum and minimum value in the data set. The mean value will not be the number in the data set, but its values are sometimes equal to the data set’s value.

## Mean Formula For Ungrouped Data

The formula to find the mean of an ungrouped data is given below:

Suppose x_{1}, x_{2}, x_{3},….., x_{n} be n observations of a data set, then the mean of these values is:

Here,

x_{i} = ith observation, 1 ≤ i ≤ n

∑x_{i} = Sum of observations

n = Number of observations

## Mean Formula For Grouped Data

There are three methods to find the mean for grouped data, depending on the size of the data. They are:

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

Let us go through the formulas in these three methods given below:

### Direct Method

Suppose x_{1}, x_{2}, x_{3},…., x_{n} be n observations with respective frequencies f_{1}, f_{2}, f_{3},…., f_{n}. This means, the observation x_{1} occurs f_{1} times, x_{2} occurs f_{2} times, x_{3} occurs f_{3} times and so on. Hence, the formula to calculate the mean in the direct method is:

Or

\(\overline{x}=\frac{\sum_{i=1}^{n}f_ix_i}{\sum_{i=1}^{n}f_i}\)Here,

∑f_{i}x_{i} = Sum of all the observations

∑f_{i} = Sum of frequencies or observations

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

### Assumed Mean Method

In this method, we generally assume a value as the mean (namely **a**). This value is taken for calculating the deviations based on which the formula is defined. Also, the data will be in the form of a frequency distribution table with classes. Thus, the formula to find the mean in assumed mean method is:

Mean \((\overline{x})=a+\frac{\sum f_id_i}{\sum f_i}\)

Here,

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

Click here to learn more about the assumed mean method in detail.

### Step-deviation Method

When the data values are large, the step-deviation method is used to find the mean. The formula is given by:

Mean \((\overline{x})=a+h\frac{\sum f_iu_i}{\sum f_i}\)

Here,

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

### Examples

**Question 1:** Find the mean of the following data set.

10, 20, 36, 12, 35, 40, 36, 30, 36, 40

**Solution:**

Given,

x_{i} = 10, 20, 36, 12, 35, 40, 36, 30, 36, 40

n = 10

Mean = ∑x_{i}/n

= (10 + 20 + 36 + 12 + 35 + 40 + 36 + 30 + 36 + 40)/10

= 295/10

= 29.5

Therefore, the mean of the given data set is 29.5.

**Question 2:** Find the mean of the following distribution, which gives the scores obtained by the students in a quiz.

Marks | 25 | 43 | 38 | 42 | 33 | 28 | 29 | 20 |

Number of students | 20 | 1 | 4 | 2 | 15 | 24 | 28 | 6 |

**Solution:**

Let us create a table to find the sum:

Marks (x_{i}) |
Number of students (f_{i}) |
f_{i}x_{i} |

25 | 20 | 500 |

43 | 1 | 43 |

38 | 4 | 152 |

42 | 2 | 84 |

33 | 15 | 495 |

28 | 24 | 672 |

29 | 28 | 812 |

20 | 6 | 120 |

Sum | 100 | 2878 |

Mean = (∑f_{i}x_{i})/ ∑f_{i}

= 2878/100

= 28.78

Thus, the mean of the given distribution is 28.78.