 # 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. Some of the important methods that can be used to calculate the mean deviation are Individual Data Series [used when data is provided individually with no range and frequency], Discrete Data Series [used when data is provided along with their frequencies], and Continuous Data Series [used when data provided along with their frequencies are based on the ranges]. Mean deviation definition is composed of two words, “mean” and “deviation” defines the average of the difference between the expected and the obtained Mathematical values in a statistical dataset. In simple words, it is the difference in the value of the observation from the average obtained from the specific set of data.

### Mean Deviation Formula

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 = $\sum$ $\left |X - \overline{X} \right |$ ÷ 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. The mean absolute deviation formula is given as:

MAD = $\sum$ $\left | x_{i} - \overline{x} \right |$ ÷ n

1. ### 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) =  $\frac{\sum𝑓\left |x - \mu \right |}{N}$

M.D (median) = $\frac{\sum𝑓\left |x - median \right |}{N}$

1. ### 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. = $\frac{\sum𝑓\left |x - Me \right |}{N}$

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 found later on.

Thus, 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.

### The 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.

### Solved Example

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

A) 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.

1. What is Mean Deviation’s definition?

Answer. There are different ways in which the mean deviation can be defined. Their meaning however remains the same. One of the definitions states that, mean deviations are the varying measure that is equal to the mean of the absolute values of the deflection from a specified value within a dataset.

One of the other definitions states that, in distribution of statistical data, the mean deviation is the average of the values of the intermediate data or numbers and the mean of that dataset.

Mean Deviation is extensively used in Statistics, where Data Scientists analyse the data and enhance the performance of a company.

2. What is Mean Absolute Deviation?

Answer. Also known as the Average Absolute Deviation, it is the average value that is calculated for the absolute deviations. The absolute deviations of all the endpoints from their central point are calculated and then their average is taken. Absolute deviation means that the negative numbers of deviations are converted into positive numbers. For example, the absolute value of 5 is 5 and that of -4 is 4. The absolute mean is carried out by converting the calculated mean to its absolute value once all the calculations have been made. It is used to determine the mean distance between the calculated mean and each of the data points. The mean absolute deviation formula is given as:

MAD = ∑ |xi -  x̅| ÷ n