# ✅ Mean Deviation Formula ⭐️⭐️⭐️⭐️⭐️

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## Mean Deviation

In statistics and mathematics, the deviation is a measure that is used to find the difference between the observed value and the expected value of a variable. In simple words, the deviation is the distance from the center point. Similarly, the mean deviation is used to calculate how far the values fall from the middle of the data set. In this article, let us discuss the definition, formula, and examples in detail.

## Mean Deviation Definition

The mean deviation is defined as a statistical measure that is used to calculate the average deviation from the mean value of the given data set. The mean deviation of the data values can be easily calculated using the below procedure.

Step 1: Find the mean value for the given data values

Step 2: Now, subtract the mean value from each of the data values given (Note: Ignore the minus symbol)

Step 3: Now, find the mean of those values obtained in step 2.

## Mean Deviation Formula

The formula to calculate the mean deviation for the given data set is given below.

Mean Deviation = [Σ |X – µ|]/N

Here,

Σ represents the addition of values

X represents each value in the data set

µ represents the mean of the data set

N represents the number of data values

| | represents the absolute value, which ignores the “-” symbol

## Mean Deviation for Frequency Distribution

To present the data in the more compressed form we group it and mention the frequency distribution of each such group. These groups are known as class intervals.

Grouping of data is possible in two ways:

1. Discrete Frequency Distribution
2. Continuous Frequency Distribution

In the upcoming discussion, we will be discussing mean absolute deviation in a discrete frequency distribution.

Let us first know what is actually meant by the discrete distribution of frequency.

### Mean Deviation for Discrete Distribution Frequency

As the name itself suggests, by discrete we mean distinct or non-continuous. In such a distribution the frequency (number of observations) given in the set of data is discrete in nature.

If the data set consists of values x1,x2, x3………xn each occurring with a frequency of f1, f2… fn respectively then such a representation of data is known as the discrete distribution of frequency.

To calculate the mean deviation for grouped data and particularly for discrete distribution data the following steps are followed:

Step I: The measure of central tendency about which mean deviation is to be found out is calculated. Let this measure be a.

If this measure is mean then it is calculated as,

If the measure is median then the given set of data is arranged in ascending order and then the cumulative frequency is calculated then the observations whose cumulative frequency is equal to or just greater than N/2 is taken as the median for the given discrete distribution of frequency and it is seen that this value lies in the middle of the frequency distribution.

Step II: Calculate the absolute deviation of each observation from the measure of central tendency calculated in step (I)

StepIII: The mean absolute deviation around the measure of central tendency  is then calculated by using the formula

### Mean Deviation Examples

Example 1:

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

Solution:

Given data values are 5, 3, 7, 8, 4, 9.

We know that the procedure to calculate the mean deviation.

First, find the mean for the given data:

Mean, µ = ( 5+3+7+8+4+9)/6

µ = 36/6

µ = 6

Therefore, the mean value is 6.

Now, subtract each mean from the data value, and ignore the minus symbol if any

(Ignore”-”)

5 – 6 = 1

3 – 6 = 3

7 – 6 = 1

8 – 6 = 2

4 – 6 = 2

9 – 6 = 3

Now, the obtained data set is 1, 3, 1, 2, 2, 3.

Finally, find the mean value for the obtained data set

Therefore, the mean deviation is

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

= 12/6

= 2

Hence, the mean deviation for 5, 3,7, 8, 4, 9 is 2.

Example 2:

In a foreign language class, there are 4 languages, and the frequencies of students learning the language and the frequency of lectures per week are given as:

Calculate the mean deviation about the mean for the given data.

Solution: The following table gives us a tabular representation of data and the calculations

## Frequently Asked Questions on Mean Deviation

### What does the mean deviation tell us?

The mean deviation gives information about how far the data values are spread out from the mean value.

### Mention the procedure to find the mean deviation.

The procedure to find the mean deviation are:
Step 1: Calculate the mean value for the given data
Step 2: Subtract the mean from each data value (Distance)
Step 3: Finally, find the mean for the distance

### What are the advantages of using the mean deviation?

The advantages of using mean deviation are:
It is based on all the data values provided, and hence it will give a better measure of dispersion.
It is easy to understand and calculate.

### What is the mean deviation about a median?

The mean deviation about the median is similar to the mean deviation about mean. Instead of calculating the mean for the given set of data values, find the median value by arranging the data values in the ascending order and then find the middle value. After finding the median, now subtract the median from each data value, and finally, take the average

### What are the three different ways to find the mean deviation?

The mean deviation can be calculated using
Individual Series
Discrete series
Continuous series

## Mean Deviation

Mean deviation is used to compute how far the values in a data set are from the center point. Mean, median, and mode all form center points of the data set. In other words, the mean deviation is used to calculate the average of the absolute deviations of the data from the central point. Mean deviation can be calculated for both grouped and ungrouped data.

Mean deviation is a simpler measurement of variability as compared to standard deviation. When we want to find the average deviation from the data’s center point, the mean deviation is used. In this article, we will take an in-depth look at mean deviation, its formula, examples as well as the merits and demerits.

## What is Mean Deviation?

Mean Deviation falls under average absolute deviation. The average absolute deviation can be defined as the average of the absolute deviations from the central point of the data. The central point can be computed by using either mean, median, or mode.

### Mean Deviation Definition

The difference between the observed value of a data point and the expected value is known as deviation in statistics. Thus, mean deviation or mean absolute deviation is the average deviation of a data point from the mean, median, or mode of the data set. Mean deviation can be abbreviated as MAD.

### Mean Deviation Example

Suppose we have a set of observations given by {2, 7, 5, 10} and we want to calculate the mean deviation about the mean. We find the mean of the data given by 6. Then we subtract the mean from each value, take the absolute value of each result and add them up to get 10. Finally, we divide this value by the total number of observations (4) to get the mean deviation as 2.5.

### Mean Deviation Formula

Depending upon the type of data available as well as the type of the central point, there can be several different formulas to calculate the mean deviation. Given below are the different mean deviation formulas.

### Mean Deviation Formula for Ungrouped Data

Data that is not sorted or classified into groups and remains in raw form is known as ungrouped data. To calculate the mean deviation for ungrouped data the formula is as follows:

### Mean Deviation Formula for Grouped Data

When data is organized and classified into groups it is known as grouped data. Grouping of data is done by continuous and discrete frequency distributions. The mean deviation formulas for grouped data are given below:

Mean Deviation for Continuous Frequency Distribution

Such a type of grouped data consists of class intervals. The frequency of repetition of an observation within each interval is given by the continuous frequency distribution. The mean deviation formula is as follows:

The mean is also known as the expected value of a data set. The simple definition of mean is given as the sum of all observations divided by the total number of observations. The formulas for mean deviation about the mean are given below:

cf stands for cumulative frequency preceding the median class, l is the lower value of the median class, h is the length of the median class and f is the frequency of the median class.

Mode is defined as that value that occurs most frequently in a given data set. The formulas to calculate mean deviation about mode are as follows:

l is the lower value of the modal class, h is the size of the modal class, f is the frequency of the modal class, f1 is the frequency of the class preceding the modal class, and f2 is the frequency of the class succeeding the modal class.

## How to Calculate Mean Deviation?

Regardless, of whether the mean deviation about the mean, median or mode needs to be determined, the general steps remain the same. The only difference will be in the formulas used to calculate the mean, median or mode depending upon the type of data available to us. Suppose the mean deviation about the mean has to be determined for the data set {10, 15, 17, 15, 18, 21}. Then the below-given steps can be followed.

• Step 1: Calculate the value of the mean, mode, or median of the given data values. Here, we find the mean given by 16.
• Step 2: Subtract the value of the central point (here, mean) from each data point. (10 – 16), (15 – 16), …, (21 – 16) = -6, -1, 1, -1, 2, 5.
• Step 3: Now take the absolute of the values obtained in step 2. The values are 6, 1, 1, 1, 2, 5
• Step 4: Take the sum of all the values obtained in step 3. This gives 6 + 1 + 1 + 1 + 2 + 5 = 16
• Step 5: Divide this value by the total number of observations. This results in the mean deviation. As there are 6 observations hence, 16 / 6 = 2.67 which is the mean deviation about the mean.

## Mean Deviation and Standard Deviation

Both mean deviation and standard deviation help to measure the variability of data. Given below is the table of differences between mean deviation and standard deviation.

## Merits and Demerits of Mean Deviation

Mean deviation is a statistical measure and hence, has its merits and demerits. It is utilized in checking the spread of data with respect to the central value.

### Merits of Mean Deviation

Mean deviation is a useful measure as it can remove the shortcomings of other types of statistical measures. Some of the merits are given below:

• It is easy to calculate and simple to understand.
• It does not get extremely affected by outliers.
• It is widely used in business and commerce.
• It has the least sample fluctuations as compared to other statistical measures.
• It is a good comparison measure as it is based on the deviations from the mid-value.

### Demerits of Mean Deviation

Mean deviation is not capable of further algebraic treatment hence, this can lead to reduced usability. Other demerits of mean deviation are listed below:

• It is not rigidly defined as it can be calculated with respect to mean, median, and mode.
• Sociological studies rarely use this measure to analyze data.
• Negative and positive signs are ignored because we take the absolute value. This can lead to inaccuracies in the result.

Important Notes on Mean Deviation

• Mean deviation is a statistical measure used to give the average value of the absolute deviation with respect to the central point of the data.
• Mean deviation can be calculated about the mean, median, and mode.
• The general formula to calculate the mean deviation for ungrouped

## Examples on Mean Deviation

To find the median class,
N/2 = 33/2 = 16.5
The cf value that is nearest to 16.5 but higher than it is 18.
Thus, median class is 25 – 35.

Example 2: Find the mean deviation about the mean for the given data

 x f 12 7 9 3 6 8 18 1 10 2

Solution:

## FAQs on Mean Deviation

### What is Mean Deviation in Statistics?

In statistics, the mean deviation is used to give the spread of data about the central point (mean, median or mode). It is a type of average absolute deviation.

### What is the Formula of Mean Deviation?

The formulas for mean deviation are listed below:

### Is Mean Deviation a Measure of Central Tendency?

Mean deviation is not a measure of central tendency. However, it gives the spread of data about the different measures of central tendency such as mean, median, or mode.

### Is Mean Deviation a Measure of Dispersion?

Mean deviation is a measure of dispersion. It helps to determine the variability of data with respect to the central value of the given data set.

### Is Mean Deviation and Standard Deviation the Same?

No, mean deviation and standard deviation are not the same. Mean deviation calculates the absolute deviations from the central point of the data. However, standard deviation calculates the square of deviations from the mean of the given data.

Mean deviation is very easy to understand and does not get affected greatly by outliers or extreme points in the data. On the other hand, it is not rigidly defined and ignores the negative sign of the data by taking the absolute value.

### Mean Deviation

How far, on average, all values are from the middle.

Calculating It

Find the mean of all values … use it to work out distances … then find the mean of those distances!

In three steps:

• 1. Find the mean of all values
• 2. Find the distance of each value from that mean (subtract the mean from each value, ignore minus signs)
• 3. Then find the mean of those distances

Like this:

Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the mean:

## Formula

The formula is:

Let’s do our example again, using the proper symbols:

Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the mean:

Note: the mean deviation is sometimes called the Mean Absolute Deviation (MAD) because it is the mean of the absolute deviations.

## What Does It “Mean” ?

Mean Deviation tells us how far, on average, all values are from the middle.

Here is an example (using the same data as on the Standard Deviation page):

Example: You and your friends have just measured the heights of your dogs (in millimeters):

Example: Dogs

Deviations left of mean: 224 + 94 = 318

Deviations right of mean: 206 + 76 + 36 = 318

If they are not equal … you may have made a msitake!

## What is Mean Deviation

Deviation in statistics is a measure that refers to the difference between the observed and expected values of a variable. In layman’s terms, a deviation is a distance from the centre point. Mean, median, and mode are all data set centre points. Similarly, the mean deviation is used to calculate the distance between the values in a data collection and the centre point.

Mean deviation is a statistical measure that computes the average deviation from the average value of a given data collection. The mean deviation can be calculated using various data series, such as – continuous data series, discrete data series and individual data series.

### Mean Deviation Formula

The mean deviation is the mean of the absolute deviations of the observations or values from a suitable average. This suitable average may be the mean, median or mode. We also know it as the mean absolute deviation.

The basic formula to calculate mean deviation for a given data set is as follows:

Step 1 – Calculate the mean, median or mode value of the given data set.

Step 2 – Then we must find the absolute difference between each value in the data set with the mean, ignoring the signs.

Step 3 – We then sum up all the deviations.

Step 4 – Finally, we find the mean or average of those values found in Step 3. The result obtained is the mean deviation.

Let us look at a simple example to understand the working of the above steps. Suppose we have a dataset {2, 4, 8, 10} and we want to calculate the mean deviation about the mean.

Step 1 – We find the mean of the dataset i.e. (2+4+8+10)/4 = 6.

Step 2 – We then subtract each value in the dataset with the mean, get their absolute values i.e. |2-6| = 4, |4-6| = 2, |8-6| = 2, |10-6| = 4

Step 3 – And add them i.e. 4+2+2+4 = 12.

Step 4 – Finally, we divide this sum by the total number of values in the dataset (4) that will give us the mean deviation. The answer is 12/4 = 3.

### Mean Deviation Types

• Individual Data Series – When the given data set is on an individual basis.
• Discrete Data Series – When the data set is given along with their frequencies.
• Continuous Data Series – When the data set is based on ranges along with their frequencies.

We shall now learn more about some important formulas, for example, the mean deviation formula for ungrouped data, ungrouped data as well as for an individual series or a continuous series, etc.

### Mean Deviation Formula for Ungrouped Data

Ungrouped data is data that has not been sorted or categorized into groups and is still in its raw form. Generally, ungrouped data includes individual data series. The formula to calculate mean deviation for ungrouped data is as follows:

### Mean Deviation Formula for Grouped Data

Grouped data is data that has been sorted and classified into groups. Continuous and discrete frequency distributions are used to group data.

• For Discrete Frequency Distribution – The formula to calculate mean deviation is as follows:

h = size of the modal class

f = frequency of the modal class

f1 = frequency of the class preceding the modal class

f2 = frequency of the class succeeding the modal class

### Merits of Mean Deviation

1. Mean deviation is easy to understand and calculate.
2. It gets least affected by extreme values (outliers).
3. Mean Deviation is calculated by considering all the items in the data set.
4. When compared to other statistical measures, it exhibits the fewest sample volatility.
5. It is a useful comparison metric because it is based on deviations from the mean.

### Demerits of Mean Deviation

1. It is not strictly defined because it can be calculated in relation to the mean, median, and mode.
2. Because we take the absolute value, we ignore both negative and positive indications. This can result in inaccuracies in the outcome.
3. It is not a well-defined statistic because the mean deviation from different averages (mean, median, and mode) will differ.
4. It cannot be further algebraically treated.
5. The fluctuations in sampling have a significant impact on it.

### FAQs on Mean Deviation

Question 1. Calculate the mean deviation from the median and the co-efficient of mean deviation from the following data:

Marks of the students: 86, 25, 87, 65, 58, 45, 12, 71, 35.

Solution: Arrange the data in ascending order: 12, 25, 35, 45, 58, 65, 71, 86, 87.

Hence, the mean deviation about the mean is found to be 2.684

Question 3. Calculate the mean deviation for the following data.

Examples

Example 1: What are the advantages of using the mean deviation?

Solution:

The advantages of using mean deviation are:

• It is based on all the data values given, and hence it provides a better measure of dispersion.
• It is easy to understand and calculate.

Example 2: Mention the procedure to find the mean deviation.

Solution:

The procedure to find the mean deviation are:

Step 1: Calculate the mean value for the data given.

Step 2: Then, the mean is subtracted from each data value (distance).

Step 3: Finally, the mean is found for the distance.

Example 3: Find the mean deviation of the following data?

7, 5, 1, 3, 6, 4, 10

Solution:

To find the mean deviation,

First, calculate the mean value of the given data

Mean = 5

Now subtract the mean from each data value {ignore negative (-)}

• 7 – 5 = 2
• 5 – 5 = 0
• 1 – 5 = 4
• 3 – 5 = 2
• 6 – 5 = 1
• 4 – 5 = 1
• 10 – 5 = 5

Further find the mean of these values obtained

2, 0, 4, 2, 1, 1, 5

Mean = 2.14

Therefore,

Mean deviation for 7, 5, 1, 3, 6, 4, 10 is 2.14

Example 4: Find the mean deviation of the following data?

11, 9, 7, 3, 2, 8, 10, 12, 15, 13

Solution:

To find the mean deviation,

First, calculate the mean value of the given data

Mean = 8

Now subtract the mean from each data value {ignore negative (-)}

• 11 – 8 = 3
• 9 – 8 = 1
• 7 – 8 = 1
• 3 – 8 = 5
• 2 – 8 = 6
• 8 – 8 = 0
• 10 – 8 = 2
• 12 – 8 = 4
• 15 – 8 = 7
• 13 – 8 = 5

Further find the mean of these values obtained

3, 1, 1, 5, 6, 0, 2, 4, 7, 5.

### 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:

∑ = 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:

∑ = 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:

∑ = 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:

∑ = 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:

∑ = 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:

∑ = 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:

∑ = 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,

Now we have to calculate the Mean Deviation

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

=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.

36/6 = 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.

1.How to calculate mean deviation?

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.

2.What is the coefficient of mean deviation?

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

• Coefficient of Mean Deviation from the Mean: M.D/X̅
• Coefficient of Mean Deviation from the median: M.D/M
• Coefficient of Mean Deviation from the Mode: M.D/Mode

3.What is the mean deviation for discrete distribution frequency?

By discrete, we imply distinct or non-continuous, as the term implies. The frequency (number of observations) supplied in the set of Data is discrete in nature in such a distribution.

The discrete distribution of frequency is a representation of Data that consists of values x1,x2, x3………xn, each occurring with a frequency of f1, f2,… fn, correspondingly.

The steps for calculating the Mean Deviation for grouped Data, and especially for discrete distribution Data, are as follows:

Step I: Determine the measure of central tendency from which the Mean Deviation will be calculated. Let’s call this one a.

Step II: Using the measure of central tendency computed in step one, calculate the absolute Deviation of each observation (I).

Step III: Using the formula, the Mean absolute Deviation around the measure of central tendency is computed.

4.What is the meaning of Mean Deviation for frequency distribution?

We group the Data and note the frequency distribution of each group to offer it in a more compressed format. Class intervals are the names given to these groups.

There are two methods for grouping Data:

• Discrete Frequency Distribution is a type of frequency distribution that is made up of discrete
• Frequency Distribution in Continuous Time.

Discrete Frequency Distribution: By discrete, we imply distinct or non-continuous, as the term implies. The frequency (number of observations) supplied in the set of Data is discrete in nature in such a distribution.

5.What is the Mean Deviation about a median and the different ways to find it?

The Mean Deviation about the median and the Mean Deviation about the Mean are comparable. Find the median value by organising the Data values in ascending order and then finding the middle value instead of computing the Mean for the given set of Data values. After you’ve found the median, remove it from each Data item before calculating the average.

The following are the three methods for calculating Mean Deviation:

• Individual Series can be used to calculate the Mean Deviation.
• A succession of discrete events
• A series that never ends

Example

Consider this example when calculating the average deviation from the mean.

A basketball player played 5 games so far this season. The scoring numbers from each game are 23, 30, 31, 15 and 46.

The first step is calculating the mean. You do this by adding the points and dividing the result by the five games.

*23+30+31+15+46=145*

*145/5=29*

Now that you determined that the player has scored an average of 29 points per game, you need to calculate the deviation from the mean for each game.

*23-29=6*

*30-29=1*

*31-29=2*

*15-29=14*

46-29=17

Next, you need to calculate the sum of all variations.

*6+1+2+14+17=40*

The average deviation is the sum of all deviations divided by the total number of entries.

*Average deviation=40/5=8*

The average deviation from the mean regarding the points scored in the first five games of the season is 8.

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