Formula for Mean Deviation For Ungrouped Data

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What Does “Mean” Means?

Mean is the other name for average. Finding out the mean is very easy, we just have to find the sum of all the numbers and then divide them by the total number of numbers that we have. Mean deviation formula is also a measure of central tendency which can be calculated using Arithmetic Mean, Median, or Mode. It lets us know on average how far all the observations can be from the middle. Each deviation being an absolute value ignores all the negative signs therefore it can rightfully be called an absolute deviation. Also, the deviations must be equal on both sides of the Mean. The mean deviation formula of the observations or values is actually the mean of the absolute deviations from a suitable average. This suitable average can be the mean, median, or mode.

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Formula of Mean Deviation From Mean

We will find the formula of mean deviation from mean for individual series, discrete series, and continuous series. 

1) Individual Series: The formula to find the mean deviation for an individual series is:  

M.D = \[\frac{\sum|X-\bar{X}|}{N}\]

\[\sum\] = Summation

X = Observation / Values 

\[\bar{X}\] = Mean

N = Number of observations


2) Discrete Series: The formula of mean deviation from mean for a discrete series is:  

M.D = \[\frac{\sum f|X-\bar{X}|}{\sum f}\]

\[\sum\] = Summation

X = Observation / Values 

\[\bar{X}\] = Mean

f = frequency of observations


3) Continuous Series: The formula to find the mean deviation for a discrete series is:  

M.D = \[\frac{\sum f|X-\bar{X}|}{\sum f}\]

\[\sum\] = Summation

X = Mid-Value of the class 

\[\bar{X}\] = Mean

f = frequency of observations


Mean Deviation From Median

1) Individual Series: The formula to find the mean deviation for an individual series is:  

M.D = \[\frac{\sum|X-M|}{N}\]

\[\sum\] = Summation

X = Mid-Value of the class 

M = Median

N = Number of observations


2) Discrete Series: The formula to find the mean deviation for a discrete series is:  

M.D = \[\frac{\sum f|X-M|}{\sum f}\]

\[\sum\] = Summation

X = Observation / Values 

M = Median

f = frequency of observations


3) Continuous Series: The formula to find the mean deviation for a continuous series is:  

M.D = \[\frac{\sum f|X-M|}{\sum f}\]

\[\sum\] = Summation

X =  Mid-Value of the class 

M = Mean

f = frequency of observations


Mean Deviation From Mode

1) Individual Series: The formula to find the mean deviation from mode for an individual series is:  

M.D = \[\frac{\sum|X-Mode|}{N}\]

\[\sum\] = Summation

X = Observation / Values 

M = Mode

N = Number of observations


2) Discrete Series: The formula to find the mean deviation from mode for a discrete series is: 

M.D = \[\frac{\sum f|X-Mode|}{\sum f}\]

\[\sum\] = Summation

X = Observation / Values 

M = Mode

f = frequency of observations


3) Continuous Series: The formula to find the mean deviation from mode for a continuous series is:  

M.D = \[\frac{\sum f|X-Mode|}{\sum f}\]

\[\sum\] = Summation

X = Mid-Value of the class 

M = Mode

f = frequency of observations


Mean Deviation Examples

Example 1) Calculate the mean deviation and the coefficient of mean deviation using the data given below:

Test Marks of 9 students are as follows: 86, 25, 87, 65, 58, 45, 12, 71, 35 respectively. 

Solution 1) First we have to arrange them into ascending order, i.e., 12, 25, 35, 45, 58, 65, 71, 86, 87.

Then we have to find out the median so, 

Median = Value of the \[\frac{(N+1)^{th}}{2}\] term

Value of the \[\frac{(9+1)^{th}}{2}\] term = 58

Now we have to calculate the mean deviation 


X

|X - M|     

12

46

25

33

35

23

45

13

58

0

65

7

71

13

86

28

87

29

N = 9

\[\sum\]|X - M| = 460 


M.D = M.D = \[\frac{\sum|X-M|}{N}\]

= \[\frac{460}{9}\]

= 51.11

Lastly, we have to find the coefficient of mean deviation from median so, 

Coefficient of the mean deviation from median = \[\frac{M.D}{M}\]

= \[\frac{51.11}{58}\]

= 0.881


Example 2) Calculate the mean deviation about the mean using the following data

6, 7, 10, 12, 13, 4, 8, 12.

Solution 2) First we have to find the mean of the data that we are provided with

Mean of the given data =  \[\frac{\text{Sum of all the terms}}{\text{total number of terms}}\]

\[\bar{X}\] = \[\frac{6+7+10+12+13+4+8+12}{8}\]

= \[\frac{72}{8}\]

= 9

Next, we have to find the mean deviation


X\[_{i}\]

X\[_{i}\] - \[\bar{x}\]

|X\[_{i}\] - \[\bar{x}\]|

6

6 - 9 = -3

|-3| = 3

7

7 - 9 = -2

|-2| = 2

10

10 - 9 = 1

|1|= 1

12

12 - 9 = 3

|3|= 3

13

13 - 9 = 4

|4|= 4

4

4 - 9 = -5

|-5|= 5

8

8 - 9 = -1

|-1|= 1

12

12 - 9 = 3

|3|= 3



\[\sum_{1}^{8}\] |X\[_{i}\] - \[\bar{x}\]| = 22


Mean deviation about mean = \[\frac{\sum|X_{i} - \bar{X}|}{8}\] 

= \[\frac{22}{8}\]

2.75

FAQ (Frequently Asked Questions)

Question 1: How to Calculate Mean Deviation

Answer: There are a few steps that we can follow in order to calculate the mean deviation.  

Step 1: Firstly we have to calculate the mean, mode, and median of the series.

Step 2: Ignoring all the negative signs, we have to calculate the deviations from the mean, median, and mode like how it is solved in mean deviation examples. 

Step 3: If the series is a discrete one or continuous then we also have to multiply the deviation with the frequency. 

Step 4: Our step 4 will be to sum up all the deviation we calculated

Step 5: This will be the final step and we have to apply the formula to calculate the mean deviation. 

Question 2: What is the Coefficient of Mean Deviation?

Answer: The comparison between the data of two series is done using a coefficient of mean deviation. We can calculate the coefficient of mean deviation by dividing it with the average. If the deviation is from the mean, we will simply divide it by mean. If the deviation is from the median, we will divide it by median and if the deviation is from mode, we will divide it by mode. Let us see the formulas to calculate the mean deviation from the mean, median, and mode.    


The Formulas for the Coefficient of the Mean Deviation

  1. Coefficient of mean deviation from the mean: M.D/X̅

  2. Coefficient of mean deviation from the median: M.D/M 

  3. Coefficient of mean deviation from the mode: M.D/Mode