Mean of Grouped Data (With Class-Interval)

When the data is grouped in the form of class interval then the mean can be calculated by three methods.

1. Direct Method

In this method, we use a midpoint which represents the whole class. It is called the class mark. It is the average of the upper limit and the lower limit.

\mathrm{Class\ Mark=\dfrac{Upper\ Class\ Limit-Lower\ Class\ Mark}{2}}

\bar x=\dfrac{\sum f_ix_i}{\sum f_i}

Example

A teacher marks the test result of the class of 55 students for mathematics. Find the mean for the given group. 

Marks of Students0 – 1010 – 2020 – 3030 – 4040 – 5050 – 60
Frequency27107542

To find the mean we need to find the mid-point or class mark for each class interval which will be the x and then by multiplying frequency and midpoint we get fx.

Marks of studentsFrequency(f)Midpoint(x)fx
0 – 10275135
10 – 201015150
20 – 30725175
30 – 40535175
40 – 50445180
50 – 60255110
  \sum f = 55\sum fx = 925

\bar x=\dfrac{f_1x_1+f_2x_2+....+f_nx_n}{f_1+f_2+...+f_n}

\bar x=\dfrac{925}{55}=16.8 marks

2. Deviation or Assumed Mean Method

If we have to calculate the large numbers then we can use this method to make our calculations easy. In this method, we choose one of the x’s as assumed mean and let it as “a”. Then we find the deviation which is the difference of assumed mean and each of the x. The rest of the method is the same as the direct method.

\bar x=a+\dfrac{\sum f_id_i}{\sum f_i}

Example

If we have the table of the expenditure of the company’s workers in the household, then what will be the mean of their expenses?

Expense(Rs.)100 – 150150 – 200200 – 250250 – 300300 – 350350 – 400
Frequency244033283022

Solution

Here we take 275 as the assumed mean.

Expenses(Rs.)Frequency(f)Mid value(x)d = x – 275fd
100 – 15024125– 150– 3600
150 – 20040175– 100– 4000
200 – 25036225– 50-1650
250 – 3002827500
300 – 35030325501500
350 – 400223751002200
 \sum f = 180  \sum fd = – 5550

\bar x=275+\dfrac{-5550}{180}

=275-30.83

=244.17

3. Step Deviation Method

In this method, we divide the values of d with a number “h” to make our calculations easier.

\bar x=a+\left(\dfrac{\sum f_iu_i}{\sum f_i}\right)\times h

Example

The wages of the workers are given in the table. Find the mean by step deviation method.

Wages 20 – 3020 – 3030 – 4040 – 5050 – 60
No. of workers8912116

Solution

WagesNo. of workers (f) Mid-point(x)Assume mean (a) = 35, d = x – ah = 10, u = (x – a)/hfu
10 – 20815-20-2-16
20 – 30925-10-1-9
30 – 401235000
40 – 50114510111
50 – 6065520212
 \sum f = 46    \sumfu = -2

\bar x=a+\left(\dfrac{\sum f_iu_i}{\sum f_i}\right)\times h

35+\left(\dfrac{-2}{46}\right)\times10=34.57

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