Formula for Mean Deviation For Ungrouped Data

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.

Meaning of Mean Deviation

In Statistics, the Deviation is defined as the difference between the observed and predicted value of a Data point. As a result, Mean Deviation, also known as Mean Absolute Deviation, is the average Deviation of a Data point from the Data set's Mean, median, or Mode. The term "Mean Deviation" is abbreviated as MAD.

Benefits of using Mean Deviation

The benefits of making use of Mean Deviation are: 

• As it is based on all of the Data values provided, it will provide a more accurate assessment of dispersion.

• It is simple to comprehend and compute.

Drawbacks of using Mean Deviation

As Mean Deviation cannot be further Algebraically treated, it has lower usefulness. The following are some of the other drawbacks of Mean Deviation:

• It can be determined with respect to Mean, median, and Mode, therefore it isn't strictly defined.

• This metric is rarely used to assess Data in sociological studies.

• As we use the absolute value, we ignore both negative and positive indicators. This may result in inaccuracies in the final product.

Example of Mean Deviation

Let's say we have a series of observations with the values 2, 7, 5, 10 and wish to determine the Mean Deviation from the Mean. The Mean of the Data given by 6 is found. After that, we subtract the Mean from each number, sum the absolute values of each result, and obtain 10. Finally, we divide this number by the total number of observations (4) to arrive at 2.5 as the Mean Deviation.

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: 

MD=\[\frac{\sum \mid X-\bar{X}\mid}{N}\]

∑  = Summation

X = Observation / Values 

X¯ = Mean

N = Number of observations

2) Discrete Series: The formula of Mean Deviation from Mean for a discrete series is:  

MD=\[\frac{\sum f\mid X-\bar{X}\mid}{\sum f}\]

∑ = Summation

X = Observation / Values 

X¯ = Mean

f = frequency of observations

3) Continuous Series: The formula to find the Mean Deviation for a discrete series is:  

MD=\[\frac{\sum f\mid X-\bar{X}\mid}{\sum f}\]

∑ = Summation

X = Mid-Value of the class 

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: 

MD=\[\frac{\sum \mid X-{M}\mid}{N}\]

∑ = 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: 

MD=\[\frac{\sum f\mid X-{M}\mid}{\sum f}\]

∑ = 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: 

MD=\[\frac{\sum f\mid X-{M}\mid}{\sum f}\]

∑ = 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:  

MD=\[\frac{\sum \mid X-{Mode}\mid}{N}\]

∑ = 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: 

MD=\[\frac{\sum f\mid X-{Mode}\mid}{\sum f}\]

∑ = 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:  

MD=\[\frac{\sum f\mid X-{Mode}\mid}{\sum f}\]

∑ = 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 

MD=MD=\[\frac{\sum \mid X-{M}\mid}{N}\]

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

=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=\[\frac{Sum of all the terms}{total number of terms}\]

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

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

=9

Next, we have to find the Mean Deviation

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

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

=2.75

3. Find the mean data deviation values for 5, 3,7, 8, 4, 9.

Answer: The Data values are 5, 3, 7, 8, 4, 9, and so on.

The process for calculating the Mean Deviation is well known.

To begin, calculate the Mean of the Data:

5+3+7+8+4+9/6 is the average.

\[\frac{36}{6}\] = \[\frac{36}{6}\]

(= 6)

As a result, the average value is 6.

Subtract each Mean from the Data value, ignoring any minus symbols that may appear.

(Ignore”-”)

6 + 5 = 1

3 – 6 equals 3

1 = 7 – 6

2 = 8 – 6

2 = 4 – 6

3 = 9 – 6

The resulting data set is now 1, 3, 1, 2, 2, 3.

Finally, calculate the Mean value for the Data set you've gathered.

As a result, the standard deviation is

/6 (1+3 + 1+ 2+ 2+ 2+3)

12/6 =

+ 2

As a result, the Mean Deviation for the numbers 5, 3,7, 8, 4, 9 is 2.