Processing math: 100%

Mean of Grouped Data

If the values of the variable (i.e., observations or variates) be x1, x2, x3, x4, ....., xn and their corresponding frequencies are f1, f2, f3, f4, ....., fn then the mean of the data is given by

Mean = A (or ¯x) = x1f1+x2f2+x3f3+x4f4+....+xnfnf1+f2+f3+f4+.....+fn

Symbolically, A = xi.fifi; i = 1, 2, 3, 4, ...., n.

In words,

Mean = Sum of products of the Variables and their corresponding FrequenciesTotal Frequency

This is the formula for finding the mean of the grouped data by direct method. 

For Example:

The number of Mobile sold is given in the table below. Find the mean of the number of Mobile sold.

Number of Mobile Sold

2

5

6

10

12

Number of Shops

6

10

8

1

5

Solution:

Here, x1 = 2, x2 = 5, x3 = 6, x4 = 10, x5 = 12.

f1 = 6, f2 = 10, f3 = 8, f4 = 1, f5 = 5.

Therefore, mean = x1f1+x2f2+x3f3+x4f4+x5f5f1+f2+f3+f4+f5

                         = 2×6+5×10+6×8+10×1+12×56+10+8+1+5

                         = 12+50+4810+6030

                         = 18030

                         = 6.

Therefore, mean number of Mobile sold is 6.


Short-cut method for finding the mean of grouped data:

We know that the direct method of finding mean for grouped data gives

mean A =  xi.fifi

where x1, x2, x3, x4, ....., xn are variates and f1, f2, f3, f4, ....., fn  are their corresponding frequencies.

Let a = a number taken as assumed mean from which the diviation of the variate is di = xi - a.

Then, A =(a+di)fifi

            = afi+dififi

            =  afi+dififi

            = a + dififi

Therefore, A = a + dififi, where di = xi - a.


For Example: 

Find the mean of the following distribution using the short-cut method.


Variate

20

40

60

80

100

Frequency

15

22

18

30

16


Solution:

Putting the calculated values in a tabular form, we have the following.

Variate

Frequency

Deviation di from assumed mean a = 60, i.e., (xi - a)

dixi

20

15

-40

-600

40

22

-20

-440

60

18

0

0

80

30

20

600

100

16

40

640


fi = 101


difi = 200


Therefore, mean A = a + dififi

                            = 60 + 200101

                            = 6199101

                            = 61.98.


Solved Examples on Mean of Grouped Data or Mean of the Arrayed Data:

1. A class has 20 students whose ages (in years) are as follows.

14, 13, 14, 15, 12, 13, 13, 14, 15, 12, 15, 14, 12, 16, 13, 14, 14, 15, 16, 12

Find the mean ago of the students of the class.

Solution:

In the data, only five different numbers appear respectively. So, we write the frequencies of the variates as below.


Age (in years)

(xi)

12

13

14

15

16

Total

Number of Students

(fi)

4

4

6

4

2

20


Therefore, mean A = x1f1+x2f2+x3f3+x4f4+x5f5f1+f2+f3+f4+f5

= 12×4+13×4+14×6+15×4+16×24+4+6+4+2

= 48+52+84+60+3220

= 27620

= 13.8

Therefore, the mean age of the students of the class = 13.8 years.


2. The weights (in kg) of 30 boxes are as given below.

40, 41, 41, 42, 44, 47, 49, 50, 48, 41, 43, 45, 46, 47, 49, 41, 40, 43, 46, 47, 48, 48, 50, 50, 40, 44, 44, 47, 48, 50.

Find the mean weight of the boxes by preparing a frequency table of the arrayed data.

Solution:

The frequency table for the given data is 

Weight (in Kg)

(xi)

Tally Mark

Frequency

(fi)

xifi

40

///

3

120

41

////

4

164

42

/

1

42

43

//

2

86

44

///

3

132

45

/

1

45

46

//

2

92

47

////

4

188

48

////

4

192

49

//

2

98

50

////

4

200

fi = 30

xifi = 1359

By formula, mean = xififi

                           = 135930

                           = 45.3.

Therefore, the mean weight of the boxes = 45.3 kg.


3. Four variates are 2, 4, 6 and 8. The frequencies of the first three variates are 3, 2 and 1 respectively. If the mean of the variates is 4 then find the frequency of the fourth variate. 

Solution:

Let the frequency of the fourth variate (8) be f. Then, 

mean A = x1f1+x2f2+x3f3+x4f4f1+f2+f3+f4

⟹ 4 = 2×3+4×2+6×1+8×f3+2+1+f

⟹ 4 = 6+8+6+8f6+f

⟹ 24 + 4f = 20 + 8f

⟹ 4f = 4

⟹ f = 1

Therefore, the frequency of 8 is 1.

Formula for Finding the Mean of the Grouped Data

4. Find the mean of the following data.


Variate (x) 

1

2

3

4

5

Cumulative Frequency

3

5

9

12

15


Solution:

The frequency table and calculations involved in finding the mean are given below.

Variate

(xi)

Cumulative Frequency

Frequency

(fi)

xifi

1

3

3

3

2

5

2

4

3

9

4

12

4

12

3

12

5

15

3

15

fi = 15

xifi = 46

Therefore, mean = xififi

                         = 4615

                         = 3.07.


5. Find the mean mark from the following frequency table by using the short-cut method.


Marks Obtained

30

35

40

45

50

Number of Students

45

26

12

10

7


Solution:

Taking the assumed mean a = 40, the calculations will be as follows.

Marks Obtained

(xi)

Number of Students

(fi)

Deviation di = xi - a = xi - 40

difi

30

45

-10

-450

35

26

-5

-130

40

12

0

0

45

10

5

50

50

7

10

70

fi = 100

difi = -460

Therefore, mean = a + dififi

                         = 40 + 460100

                         = 40 - 4.6

                         = 35.4.

Therefore, the mean mark is 35.4.






9th Grade Math

From Mean of Grouped Data to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Area, Perimeter and Volume | Square, Rectangle, Cube,Cubo

    Jul 25, 25 12:21 PM

    In this worksheet on area perimeter and volume you will get different types of questions on find the perimeter of a rectangle, find the perimeter of a square, find the area of a rectangle, find the ar…

    Read More

  2. Worksheet on Volume of a Cube and Cuboid |The Volume of a RectangleBox

    Jul 25, 25 03:15 AM

    Volume of a Cube and Cuboid
    We will practice the questions given in the worksheet on volume of a cube and cuboid. We know the volume of an object is the amount of space occupied by the object.1. Fill in the blanks:

    Read More

  3. Volume of a Cuboid | Volume of Cuboid Formula | How to Find the Volume

    Jul 24, 25 03:46 PM

    Volume of Cuboid
    Cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid will have a length, breadth and height. Hence we can conclude that volume is 3 dimensional. To measur…

    Read More

  4. Volume of a Cube | How to Calculate the Volume of a Cube? | Examples

    Jul 23, 25 11:37 AM

    Volume of a Cube
    A cube is a solid box whose every surface is a square of same area. Take an empty box with open top in the shape of a cube whose each edge is 2 cm. Now fit cubes of edges 1 cm in it. From the figure i…

    Read More

  5. 5th Grade Volume | Units of Volume | Measurement of Volume|Cubic Units

    Jul 20, 25 10:22 AM

    Cubes in Cuboid
    Volume is the amount of space enclosed by an object or shape, how much 3-dimensional space (length, height, and width) it occupies. A flat shape like triangle, square and rectangle occupies surface on…

    Read More