Mean Deviation 

Mean Deviation, also known as Average Deviation, is a statistical measure that represents the average of the absolute differences between each data point and a central value—typically the mean or median. It shows how spread out or dispersed the values in a data set are. Unlike standard deviation, mean deviation uses absolute values, making it easier to interpret. This measure helps in understanding the consistency and variability of data. It is widely used in economics, business, and various scientific fields to assess data dispersion in a more intuitive and less sensitive way to outliers compared to variance or standard deviation.

1.0What is Mean Deviation?

Mean Deviation, also known as Average Deviation, measures the average distance between each data point and a central value (usually the mean or median). Unlike variance and standard deviation, it doesn't square the differences, making it easier to interpret.

In simple terms, Mean Deviation tells you how spread out the values in a dataset are.

2.0Mean Deviation Formula

The general formula for Mean Deviation is:

$\text{Mean Deviation}=\frac{\sum |x_i-A|}{n}$

Where:

• $x_i$ = each data point
• A = mean or median of the data
• n = total number of observations
• $|x_i-A|$ = absolute deviation from the mean or median

3.0Mean Deviation Formula for Ungrouped Data

When dealing with ungrouped data (raw data without frequency), the Mean Deviation Formula is:

$MD=\frac{\sum |x_i-\bar{x}|}{n}$

Where: $\bar{x}$ = arithmetic mean of the data

4.0Mean Deviation Formula for Grouped Data

For grouped data, where data is organized in classes, the Formula for Mean Deviation for Grouped Data is:

$M.D.=\frac{\sum f_i|x_i-\bar{x}|}{\sum f_i}$

Where:

• $f_i$ = frequency of each class
• $x_i$ = mid-point of each class
• $\bar{x}$ = mean of the grouped data

This formula gives us the Mean Deviation for Grouped Data, accounting for how frequently each value (or group) occurs.

5.0Solved Examples on Mean Deviation

Example 1: Find the Mean Deviation about the Mean for the data: 3, 5, 7, 9, 11

Solution:

1. Step 1: Find the Mean

$\text{Mean}=\frac{3+5+7+9+11}{5}$

$\text{Mean}=\frac{35}{5}$

$\text{Mean}=7$

2. Step 2: Find the deviations from the mean

$|3-7|=4,|5-7|=2,|7-7|=0,|9-7|=2,|11-7|=4$

3. Step 3: Mean Deviation about Mean

$\text{M.D.}=\frac{4+2+0+2+4}{5}=\frac{12}{5}=2.4$

Answer: Mean Deviation = 2.4

Example 2: Find the Mean Deviation about the Median for the data: 2, 4, 6, 8, 10, 12

Solution:

1. Step 1: Find the Median

Number of observations = 6 (even)

$\text{Median}=\text{Average of 3rd and 4th terms}=\frac{6+8}{2}=7$

2. Step 2: Deviations from the median

$|2-7|=5,|4-7|=3,|6-7|=1,|8-7|=1,|10-7|=3,|12-7|=5$

3. Step 3: Mean Deviation about Median

$\text{M.D.}=\frac{5+3+1+1+3+5}{6}=\frac{18}{6}=3$

Answer: Mean Deviation = 3

Example 3: Find the Mean Deviation about the Mean for the following data:

Solution:

1. Step 1: Calculate Mean

$\overline{x}=\frac{(2)(1)+(4)(2)+(6)(3)+(8)(4)}{1+2+3+4}$

$\overline{x}=\frac{2+8+18+32}{10}$

$\overline{x}=\frac{60}{10}$

$\overline{x}=6$

2. Step 2: Find |x – Mean| and multiply by frequency

∣2 − 6∣ = 4 ⇒ 1 × 4 = 4

∣4 − 6∣ = 2 ⇒ 2 × 2 = 4

∣6 − 6∣ = 0 ⇒ 3 × 0 = 0

∣8 − 6∣ = 2 ⇒ 4 × 2 = 8

3. Step 3: Mean Deviation

$M.D.=\frac{4+4+0+8}{10}=\frac{16}{10}=1.6$

Answer: Mean Deviation = 1.6

Example 4: Let the data set be {1, 2, 3, ..., 11}. Find the mean deviation about the median.

Solution:

• This is an AP of 11 terms from 1 to 11.
• Median (odd terms) = middle value = 6

Now compute:

$\text{M.D.}=\frac{{\sum }_{i=1}^{11}|x_i-6|}{11}$

Deviations:

|1 - 6| = 5, |2 - 6| = 4, ..., |11 - 6| = 5 

So, deviations:

5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5

Sum = 2(5 + 4 + 3 + 2 + 1) + 0 = 2(15) = 30

$M.D=\frac{30}{11}$

Answer: $\frac{30}{11}$

Example 5: Let $x_1,x_2,...,x_n$ be a set of n real numbers such that the mean deviation about mean is minimum. Which of the following is necessarily true?

A. All $x_i$ are equal

B. Data is symmetric about the mean

C. Mean = Median

D. Mean deviation is zero

Solution:

• Mean Deviation is minimum when all values are equal, because then all deviations are 0.
• So, Mean Deviation = 0 only if all $x_i$'s are equal

Correct Option: A

6.0Why Use Mean Deviation in Statistics?

• Simple to Understand: Easier for beginners to grasp compared to variance.
• Less Affected by Outliers: Especially when calculated from the median.
• Useful in Decision Making: In business, finance, and quality control.

7.0Sample Questions in Mean Deviation

1.  What is the formula for Mean Deviation?

Answer:

• About Mean : $M.D.(\text{about mean})=\frac{1}{n}{\sum }_{i=1}^n\left|x_i-\overline{x}\right|$
• About Median: $M.D.(\text{about median})=\frac{1}{n}{\sum }_{i=1}^n\left|x_i-M\right|$

2. Is Mean Deviation used for grouped data?

Answer: Yes. The formula is modified using frequencies:

$M.D.=\frac{\sum f_i|x_i-A|}{\sum f_i}$