Mean Deviation (M.D.) – Computation & Properties 📊

Mean Deviation is one of the most important absolute measures of dispersion. It tells us how much, on average, the values deviate from a central value (Mean, Median, or Mode).

It uses absolute deviations (ignoring +/– signs), making it simpler and more intuitive than variance or standard deviation.

Definition

Mean Deviation is the average of the absolute deviations of all observations from a central value (Mean/Median/Mode).

It is usually calculated from:

• Median (preferred – gives minimum M.D.)
• Mean or Mode (if required)

Formulas

For Ungrouped Data

MD = (Σ |X – A|) / N

Where:

• X = individual values
• A = central value (Mean/Median/Mode)
• N = number of observations

For Grouped Data

MD = (Σ f |X – A|) / Σf

Where:

• f = frequency
• X = class midpoint (for continuous data)

Solved Examples

Example 1 — Mean Deviation from Mean (Ungrouped)

Marks: 10, 12, 15, 17, 21

Step 1: Mean

Mean = (10 + 12 + 15 + 17 + 21) / 5 = 75 / 5 = 15

Step 2: Deviations |X – Mean|

| X | |X – 15| | |----|---------| | 10 | 5 | | 12 | 3 | | 15 | 0 | | 17 | 2 | | 21 | 6 |

Sum of deviations = 5 + 3 + 0 + 2 + 6 = 16

Step 3: Mean Deviation

MD = 16 / 5 = 3.2

Example 2 — Mean Deviation from Median (Ungrouped)

Data: 8, 10, 12, 15, 20, 25

Step 1: Median

N = 6 → Median = average of 3rd and 4th values = (12 + 15) / 2 = 13.5

Step 2: Deviations from Median

| X | |X – 13.5| | |----|-----------| | 8 | 5.5 | | 10 | 3.5 | | 12 | 1.5 | | 15 | 1.5 | | 20 | 6.5 | | 25 | 11.5 |

Sum = 30

Step 3: MD

MD = 30 / 6 = 5

Example 3 — Mean Deviation for Grouped Data

Class intervals and frequency:

Step 1: Compute the Mean

ΣfX = (4×5) + (6×15) + (10×25)
     = 20 + 90 + 250 = 360
Σf = 4 + 6 + 10 = 20
Mean = 360 / 20 = 18

Step 2: Compute |X – Mean|

| X | |X – 18| | f | f|X – Mean| | |----|---------|---|-------------| | 5 | 13 | 4 | 52 | | 15 | 3 | 6 | 18 | | 25 | 7 |10 | 70 |

Σf|X – Mean| = 52 + 18 + 70 = 140

Step 3: Mean Deviation

MD = 140 / 20 = 7

Properties of Mean Deviation

• Uses absolute deviations, not squared ones
• M.D. from Median is minimum
• Easy to calculate and interpret
• Less affected by extreme values compared to Standard Deviation
• More representative than Range but less than Standard Deviation

Merits ✔️

• Simple to understand
• Uses all observations
• Better than Range for measuring consistency
• Useful for preliminary dispersion analysis

Limitations ❌

• Ignores algebraic signs
• Not suitable for advanced statistical techniques
• Cannot be used in correlation, regression, or hypothesis testing
• Not as accurate as Standard Deviation

Uses of Mean Deviation in Business & Economics

• Measuring consistency in worker productivity
• Analysing variation in sales, profits, or expenses
• Studying stability of economic indicators
• Comparing dispersion between two datasets

Summary ✨

• Mean Deviation = Average of absolute deviations
• MD = Σ|X – A| / N (Ungrouped)
• MD = Σf|X – A| / Σf (Grouped)
• M.D. from Median is minimum
• Simple but not mathematically powerful

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