Calculation of Mean in Continuous Series | Formula of Mean

The mean, also known as the average, is a measure of central tendency that summarizes a set of data by identifying the central point. In a continuous series, data is grouped into class intervals, and the mean is calculated differently than in a discrete series. The mean provides a comprehensive overview of the dataset, making it useful for comparing different datasets and understanding the overall distribution

What is Mean?

Mean is the sum of a set of numbers divided by the total number of values. It is also referred to as the average. For instance, if there are four items in a series, i.e. 2, 5, 8, 3, and 9. The simple arithmetic mean is (2 + 5 + 8 + 3 + 9) / 5 = 5.4.

What is a Continuous Series?

In continuous series (grouped frequency distribution), the value of a variable is grouped into several class intervals (such as 0-5,5-10,10-15) along with the corresponding frequencies. The method used to determine the arithmetic average in a continuous series is the same as that used in a discrete series. The midpoints of several class intervals replace the class interval in a continuous series. When it is done, a continuous series and a discrete series are the same.

Example of Continuous Series

If 15 students of a class score marks between 50-60, 10 students score marks between 60-70, and 20 students score marks between 70-80, then this information will be shown as:

Mean in Continuous Series

The arithmetic mean in continuous series can be calculated by using:

1. Direct Method;
2. Shortcut Method; and
3. Step Deviation Method

Solved Examples on Calculation of Mean in Continuous Series

Example 1

Calculate the mean of the following data using Direct Method and Short-Cut Method:

Solution:

Direct Method:

\bar{X}=\frac{\sum{fm}}{\sum{f}}

\bar{X}=\frac{2,150}{50}

Mean (\bar{X})  = 43

Short-Cut Method:

\bar{X}=A+\frac{\sum{fd}}{\sum{f}}

\bar{X}=45+\frac{(-100)}{50}

Mean (\bar{X})  = 43

Example 2

Find the missing frequency of the following series if the average marks is 30.5:

Solution:

Let us assume that the missing frequency is f.

\bar{X}=\frac{\sum{fm}}{\sum{f}}

30.5=\frac{920+25f}{28+f}

854 + 30.5f = 920 + 25f

5.5f = 66

f = 12

Missing Frequency (f) = 12

Example 3:

Calculate average profit earned by 50 companies from the following data using Step Deviation Method:

Solution:

\bar{X}=A+\frac{\sum{fd'}}{\sum{f}}\times{C}

\bar{X}=50+\frac{(-15)}{50}\times{20}

\bar{X}=50-6

Average Profit (\bar{X})  = ₹44 Crores

Practice Problems on Calculation of Mean in Continuous Series

1. Given the following continuous series, find the mean:

2. Calculate the mean for the following data:

3. Find the mean for this continuous series:

4. Determine the mean for the following data set: