Mean Deviation

Understanding the mean deviation definition and application is essential for students as it is an integral part of their Mathematics syllabus. So, let us dive right into it and analyze the functions of Mean Deviation.

Finding the mean is essentially finding the average of a given set of numbers. Mean Absolute deviation is the average distance between the mean of a set of data.

Methods for calculating Mean Deviation

Some of the important methods that can be used to calculate the mean deviation are stated below.

1. Individual Data Series

Used when data is provided individually with no range and frequency. It is an arrangement of the raw data individually. It gives numeric values for any specific situation.

2. Discrete Data Series

Used when data is provided along with their frequencies. The data in discrete series is presented in a manner such that the exact measurements of observations are clearly shown.

3. Continuous Data Series

Used when data provided along with their frequencies are based on the ranges. However, all the ranges or class intervals may not necessarily have the same frequencies.

Mean deviation definition is composed of two words, “mean” and “deviation”. It defines the average of the difference between the expected and the obtained Mathematical values in a statistical dataset. In other words, it is the difference in the value of the observation from the average obtained from the specific set of data.

Mean Deviation Formula

Mean deviation is a simpler measurement of variability as compared to standard deviation. As formulas are used to carry out calculations, the mean deviation formula helps to find out the value for the deviation within the statistical data. The mean deviation formula is as given below:

MDx =∑|X−X̅|÷ N

Where X-X̅ means Deviation from the arithmetic average

And, N is the total no. of items 

There are several variations that can occur in finding out the average deviation of a set of statistical data. Some of these variations are as follows;

1. Mean Absolute Deviation:

It is used to determine the mean distance between the calculated mean and each of the data points. It provides us with an overview of the variability in the dataset. The mean absolute deviation formula is given as: 

MAD =∑∑|xi−x¯¯¯||xi−x¯|÷ n

2. Mean Deviation for Grouped Data: 

The mean deviation for grouped data is used to find out the value of the mean of data that is divided into groups. It can be found out from the mean as well as from the median of the dataset. The mean deviation formula for grouped data is as given below:

For Grouped Data,

M.D (mean) =∑f|x−μ|/N

M.D (median) =∑f|x−median|/N

3. Mean Deviation for Discrete Distribution Frequency: 

The mean deviation in a set of data that is discrete or not continuous is carried out using a different formula than the mean deviation for continuous data. The formula for it is given as:

M.D. =∑f|x−Me|/N ∑𝑓|x−Me|N

Where “f” stands for different values of frequency

“x” stands for different values of the observations 

“Me” is the Median

And, “N” is the total no. of observations

What is Step Deviation? Understanding the Step Deviation Method Formula

In order to find out a short-cut method to calculate the mean deviation, the step method was discovered. As compared to the traditional method, the step deviation method is easier to calculate by utilizing the step deviation method formula. To find out the actual mean, this method makes use of the assumed mean. 

Any point or preferably the midpoint of the dataset is considered to be the assumed mean. The calculations for the deviations are then carried out about this assumed midpoint. Using these calculations, the actual mean is obtained later on.

Thus, the Step deviation method formula is used D’= (Xi-m)/c and the actual mean is calculated from the formula Actual Mean= m + (∑fD’/N)×c

Where m is the assumed mean

D’ is for the step deviations

And, c is the common factor for reducing the step deviations.

Difference between Mean Deviation for Continuous Distribution Frequency and Discrete Distribution Frequency

Both of these methods are used to find out the mean deviation from a given set of data using the formula. The main difference that separates them is the type of data that is being used as the statistical dataset to find out the mean deviation. Continuous distribution frequency has data that is continuous. For example, the range for the data in the dataset would be as follows; 10-20, 20-30, 30-40, 40-50, and so on.  

Discrete distribution frequency has data that is not continuous and has no relation to the data range before them. For instance, the range for the data in the dataset would be as follows; 11-20, 21-30, 31-40, 41-50, and so on.

As the criteria for data ranges are different for the mentioned distribution frequencies, their formulas are different from each other as well.

Mean Absolute Deviation vs Standard Deviation

These two statistical methods are very similar and both are measures of variability that use the original data units and then compare the data units with the mean.

However, there are some minute differences. Mean Absolute deviation uses average distances of the data points from the mean.

But in the case of standard deviation, the difference of squares between each data point and the mean is taken and then added. This value is then divided by degrees of freedom and then the square root of that value is calculated.

Since standard deviation squares the differences, outliers have a larger impact on standard deviation than Mean absolute deviation.

Uses of Mean Deviation in Real Life.

Mean, variance and mean absolute deviation are all measures to check how spread out the data is. Data that are similar to each other will have a small spread, whereas data that are more different from each other will have a large spread from the individual values.

A lower value of the mean absolute deviation reflects that the data points are close to the central value and the values in the given data set are more consistent. A larger value reflects that the data points are far away from the central point and the values are not much consistent.

Mean deviation has practical application in the field of business and commerce. Mean deviation is easy to calculate and simple to understand. Hence, many working professionals from different industries use mean deviation in their daily lives. Teachers, when giving tests to students, calculate the mean of the results to determine if the average score of the class students is high or low.

Biologists use Mean deviation to compare the weights of different animals to decide on what a healthy weight for that animal should be.

Also, let us understand the use of mean deviation with a real-life scenario. A school has to be built in a town consisting of N individuals. The location should be such that the average distance covered by any resident of the town is minimized. Let X denote the distance of each resident of the town from any reference point and let A be the distance of the proposed site of the hospital from the same reference point. 

Then the distance covered by each resident to the proposed site would be given as

I X-A I and the average distance is the mean deviation.

Now the trick here is to choose A such that the sum of | X-A |is minimized. This ensures that an individual has to walk or cover a minimum average distance to reach the hospital in less time.

This is how the concept of mean absolute deviation can be used in real-life scenarios. Given below are solved examples for a better understanding of the concepts stated above.

Some Solved Examples

1. Using the mean absolute deviation formula, find out the mean of the given dataset having the values 10, 12, 15, 18, 19, 22

Ans: The sum of the given data is 10+12+15+18+19+22=96. The number of values in the dataset is 6. The mean absolute deviation is carried out using the formula:

Mean = Sum of observations/ Total no. of observations

= 96/6

= 16

The mean for the given dataset is 16.

2. Megha enjoys posting pictures of her cat online. Given below is the data of the “likes” the past 7 pictures received.
10, 15, 15, 17, 18, 21, 9.
Find the mean absolute deviation.

Solution.

First, let’s calculate the mean.

The sum of the given data is

10+15+15+17+18+21+9= 105

Here, the total number of observations = 7

Therefore, mean is calculated as,

Mean = Sum of observations/ Total no. of observations

= 105/7

= 15

Therefore the mean is 15. 

Now, let us calculate the distance between each data point and the mean.

Adding the distances obtained in the table above, we get

5+0+0+2+3+6+6= 22

Absolute mean deviation is given as,

Absolute mean deviation= sum of the distances/ no of observations.

= 22/7

= 3.14

Therefore, on average, each picture was about 3 likes away from the mean.