What is the Formula of Mean?

The mean of a data set is derived by dividing the total of all data points by the number of data points in the data set. It's a data point in a collection that represents the average of all the data points in the set. The mean is the most popular and widely used approach in statistics for determining the centre of a data set. It's a basic, but crucial, aspect of statistical data analysis. The population mean is what we get when we calculate the average value of a population set. Sometimes, population data is so large that we can't do any analysis on it. As a result, in that scenario, we take a sample and average it. That sample represents the population, and the sample mean is the average of this subset of the data.

Now we will understand what is the formula of mean. It is given below:

Mean = \[\frac{\text{Sum of observations}}{\text{Number of observations}}\]

In this article, we will learn what is the formula of mean and Mean Formula For Ungrouped Data along with solved examples.

It's worth noting that the mean value is the average value in the data set, which will fall between the maximum and minimum values. The mean value will not be the same as the data set's value, but its values are sometimes the same.


Mean Formula for Ungrouped Data

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

Consider x\[_{1}\], x\[_{2}\], x\[_{3}\]...x\[_{n}\] be n observations of a data set, then the mean of these values can be calculated as:

\[\bar{x}\] = \[\frac{\sum x_{i}}{n}\]

Here,

x\[_{i}\] = i\[^{th}\]observation, 1 ≤ i ≤ n

\[\sum\] x\[_{i}\] = Sum of observations

n = Number of observations


Mean Formula for Grouped Data

Depending on the amount of the data, there are three approaches for finding the mean for grouped data. They are as follows:

  • Direct Method

  • Assumed Mean Method

  • Step-deviation Method

Let's look at the formulas for the three approaches listed below:


Step Deviation Method Formula

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

Mean \[\bar{x}\] = a + h\[\frac{\sum f_{i} u_{i}}{\sum f_{i}}\]

Here,

a = assumed mean

f\[_{i}\] = frequency of ith class

x\[_{i}\] - a = deviation of ith class

u\[_{i}\] = \[\frac{(x_{i}-a)}{h}\]

Σf\[_{i}\] = N = Total number of observations

x\[_{i}\] = class mark = \[\frac{\text{(upper class limit + lower class limit)}}{2}\]


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. As a result, the direct method's mean calculation formula is:

\[\bar{x}\] = \[\frac{f_{1}x_{1} + f_{2}x_{2} + f_{3}x_{3}+...+f_{n}x_{n}}{f_{1}+f_{2}+f_{3}+...+f_{n}}\]

\[\sum\]f\[_{i}\]x\[_{i}\] = Sum of all the observations

\[\sum\]f\[_{i}\] = Sum of frequencies or observations

When the number of observations is small, this strategy is utilised.


Assumed Mean Method Formula

We use this method to assume a value as the mean (namely a). This value is used to calculate the deviations that the formula is based on. In addition, the information will be presented as a frequency distribution table with classifications. As a result, the formula for calculating the mean in the assumed mean technique is:

Mean (\[\bar{x}\]) = a + \[\frac{\sum f_{i} d_{i}}{\sum f_{i}}\]

Here,

a = assumed mean

f\[_{i}\] = frequency of ith class

d\[_{i}\] = x\[_{i}\] - a = deviation of i\[^{th}\] class

\[\sum\]f\[_{i}\] = N =  Total number of observations

x\[_{i}\] = class mark = \[\frac{\text{(upper class limit + lower class limit)}}{\text{2}}\]


Solved Examples:

Question 1: Find the mean of the following distribution, which contains the quiz results of the students.


Marks

25

43

38

42

33

28

29

20

Number of students

20

1

4

2

15

24

28

6


Solution:

We will 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

As a result, the distribution's mean is 28.78.


Question 2: The table below shows the results of an examination taken by 110 students.


Class

0-10

10-20

20-30

30-40

40-50

Frequency

12

28

32

25

13


Find the mean marks of the students using the assumed mean method.

Answer: We will create a table to calculate x\[_{i}\], d\[_{i}\] and f\[_{i}\], d\[_{i}\].


Class (CI)

Frequency (f\[_{i}\])

Class mark (x\[_{i}\])

d\[_{i}\] = x\[_{i}\] - a

f\[_{i}\]d\[_{i}\]

0-10

12

5

5 – 25 = – 20

-240

10-20

28

15

15 – 25 = – 10

-280

20-30

32

25 = a

25-25 = 0

0

30-40

25

35

35-25 = 10

250

40-50

13

45

45-25 = 20

260

Total

Σf\[_{i}\] =110



Σf\[_{i}\]d\[_{i}\] = -10


Assumed mean = a = 25

Mean of the data:

\[\bar{x}\] = a + \[\frac{\sum f_{i} d_{i}}{f_{i}}\]

= 25 + (-10/ 110)

= 25 -( 1/11)

= (275-1)/11

= 274/11

=24.9

Hence, the mean marks of the students = 24.9.


Conclusion

To conclude the above discussion, we can say that the mean can be used to get an overall idea or picture of the data set. For a data collection containing numbers that are close together, the mean is the best option. When the form of the sample is adequate, the mean should be utilised. The mean is a reasonable summary of the average when the data is regularly distributed. We have learned about the step deviation method formula and terms and different mean method formulas.

FAQs (Frequently Asked Questions)

Q1: Which Method is Best for Finding the Mean?

Answer: When the number of observations is decreasing, the direct mean approach is employed to get the arithmetic mean. It is difficult to determine the mean using the direct technique when the number of observations in the data is enormous.

Q2: When Should We Use the Step Deviation Method?

Answer: When data is presented in the form of huge numbers, calculating the mean using the assumed mean approach can be complex and time-consuming. In such instances, we use a different method known as the step deviation method. We may minimise the calculation even more by using this strategy.

Q3: What are the Factors on Which the Assumed Mean Method Depends?

Answer: The assumed mean approach is based on estimating the mean and rounding to a convenient number. This value is deducted from all sample values once more. The computations are performed once the data are transformed into equal size ranges or class intervals.