Mean Deviation about Mean for Grouped Data

Data can be grouped in two ways - discrete frequency distribution and continuous frequency distribution.
 
A collection of data consisting of distinct values occurring with certain frequencies is called discrete frequency distribution.
 
Example of discrete frequency distribution:
The table displays the data of the performance of 50 students in a math test.
 


Classify the data into different class-intervals along with their respective frequencies.

When data are arranged in different class intervals without any gaps along with their frequencies, it is called continuous frequency distribution.
 
An example of continuous frequency distribution:
 
The table displays the marks scored by 50 students.

Marks Scored	Number of Students
90 - 100	4
75 - 90	9
60 - 75	14
40 - 60	18
0 - 40	5

 
Mean deviation about the mean for discrete frequency distribution:

Observations	Frequencies
x1	f1
x2	f2
x3	f3
....	....
xn	fn

 
$\stackrel{\_}{\text{x}}\text{=}\frac{\sum _{\text{i = 1}}^{\text{n}}{{\text{x}}_{\text{i}}\text{f}}_{\text{i}}}{\sum _{\text{i = 1}}^{\text{n}}{\text{f}}_{\text{i}}}\text{=}\frac{\text{1}}{\text{N}}\sum _{\text{i = 1}}^{\text{n}}{{\text{x}}_{\text{i}}\text{f}}_{\text{i}}$

N = Sum of the frequencies (Number of observations)
Deviation from the mean, $\stackrel{\_}{\text{x}}$ = $$${\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}$$$, where i = 1, 2, 3, ... , n
Absolute values of the deviations = | $$${\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}$$$ |, where i = 1, 2, 3, ... , n
Mean deviation about the mean = MD ($\stackrel{\_}{\text{x}}$)

$$$\text{=}\frac{\sum _{\text{i = 1}}^{\text{n}}{\text{f}}_{\text{i}}\text{|}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{|}}{\sum _{\text{i = 1}}^{\text{n}}{\text{f}}_{\text{i}}}$

$\text{=}\frac{\text{1}}{\text{N}}\sum _{\text{i = 1}}^{\text{n}}{\text{f}}_{\text{i}}\text{|}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{|}$

Ex: Consider the weights of 20 students.

Weight in kg (xi)	Number of Students (fi)
48	3
49	4
50	6
51	4
52	3

 
To find the mean deviation of the given data, mean of the data is found.

Weight in kg (xi)	Number of Students (fi)	fixi
48	3	144
49	4	196
50	6	300
51	4	204
52	3	156
	$\sum _{\text{i = 1}}^{\text{5}}{\text{f}}_{\text{i}}\text{= 20}$	$\sum _{\text{i = 1}}^{\text{5}}{{\text{f}}_{\text{i}}\text{x}}_{\text{i}}$  = 1000

 
Mean ($\stackrel{\_}{\text{x}}$) = $$$\frac{\text{1000}}{\text{20}}\text{= 50 kg}$

Modulus of deviation of each observation from the mean and the sum of the products of the deviations of the observations and the frequency concerned:

Weight in kg (xi)	Number of Students (fi)	fixi	$\text{|}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{| = |}{\text{x}}_{\text{i}}\text{- 50|}$	${\text{f}}_{\text{i}}\text{|}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{|}$
48	3	144	|48 - 50| = 2	6
49	4	196	|49 - 50| = 1	4
50	6	300	|50 - 50| = 0	0
51	4	204	|51 - 50| = 1	4
52	3	156	|52 - 50| = 2	6
	$\sum _{\text{i = 1}}^{\text{5}}{\text{f}}_{\text{i}}\text{= 20}$	$\sum _{\text{i = 1}}^{\text{5}}{{\text{f}}_{\text{i}}\text{x}}_{\text{i}}$  = 1000		$\sum _{\text{i = 1}}^{\text{5}}{{\text{f}}_{\text{i}}\text{| x}}_{\text{i}}$  - $\stackrel{\_}{\text{x}}$ |= 20

 
MD ($\stackrel{\_}{\text{x}}$) = $\frac{\text{1}}{\text{n}}$ $\sum _{\text{i = 1}}^{\text{n}}{{\text{f}}_{\text{i}}\text{| x}}_{\text{i}}$  - $\stackrel{\_}{\text{x}}$ | = $\frac{\text{20}}{\text{20}}$ = 1
 
Mean deviation about the mean for Continuous frequency distribution:

Marks Scored	Number of Students (fi)
80 - 100	3
60 - 80	15
40 - 60	19
20 - 40	10
0 - 20	3

 
Mean of the data:

Marks Scored	Number of Students (fi)	Mid-values (xi)	fixi
80 - 100	3	90	270
60 - 80	15	70	1050
40 - 60	19	50	950
20 - 40	10	30	300
0 - 20	3	10	30
	∑ fi = 50		$\sum _{\text{i = 1}}^{\text{5}}{{\text{f}}_{\text{i}}\text{x}}_{\text{i}}$  = 2600

 
Mean ($\stackrel{\_}{\text{x}}$) = $\frac{\sum {{\text{x}}_{\text{i}}\text{f}}_{\text{i}}}{\sum {\text{f}}_{\text{i}}}$  = $\frac{\text{2600}}{\text{50}}$   = 52 
 
If the number of observations is large, then the mathematical calculation is tedious.
 
For data with a large number of observations, the assumed mean method or the step-deviation method can be used to find the mean.

Find the modulus of the deviations of the mid-values from the mean and find the product of the deviations with their respective frequencies.

Marks Scored	Number of Students (fi)	Mid-values (xi)	fixi	$\text{|}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{| = |}{\text{x}}_{\text{i}}\text{- 52|}$	${\text{f}}_{\text{i}}\text{|}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{|}$
80 - 100	3	90	270	|90 - 52| = 38	114
60 - 80	15	70	1050	|70 - 52| = 18	270
40 - 60	19	50	950	|50 - 52| = 2	38
20 - 40	10	30	300	|30 - 52| = 22	220
0 - 20	3	10	30	|10 - 52| = 42	126
	∑ fi = 50		$\sum _{\text{i = 1}}^{\text{5}}{{\text{f}}_{\text{i}}\text{x}}_{\text{i}}$  = 2600		$\sum _{\text{i = 1}}^{\text{5}}{{\text{f}}_{\text{i}}\text{| x}}_{\text{i}}$  - $\stackrel{\_}{\text{x}}$ |= 768

 
 
 Mean deviation about mean = MD ($\stackrel{\_}{\text{x}}$) = $\frac{\text{1}}{\text{n}}$ $\sum _{\text{i = 1}}^{\text{n}}{{\text{f}}_{\text{i}}\text{| x}}_{\text{i}}$  - $\stackrel{\_}{\text{x}}$ | = $\frac{\text{768}}{\text{50}}$ = 15.36
 
Therefore, the mean deviation about the mean is equal to 768 divided by 50, which is equal to 15.36.