Measures of Dispersion: Mean Deviation, Variance, Standard Deviation, Coefficient of Variation

TAKEAWAYS:

• Measures of dispersion tell us how scattered or clustered data points are around a central value (like the mean).
• Mean deviation is the average of absolute differences from the mean, but variance and standard deviation use squared differences to avoid cancellation.
• Variance is the average of squared deviations, and standard deviation is its square root (in the same units as the data).
• The coefficient of variation compares dispersion relative to the mean, useful for comparing datasets with different units.
• Shortcut formulas for variance and standard deviation simplify calculations when the mean is known.
• Always check units and whether the data is grouped or ungrouped before choosing a formula.

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### What is Dispersion?
Dispersion means how spread out the values in a dataset are. For example, consider two groups of students’ marks:

```figure
{"type":"bar","labels":["Group A (80, 82, 78, 81, 83)","Group B (50, 60, 70, 90, 100)"],"values":[5,50],"ylabel":"Spread of Marks (Variance)","caption":"Group B shows higher dispersion despite identical means (80).","color":["#4CAF50","#F44336"]}

• Group A: 80, 82, 78, 81, 83
• Group B: 50, 60, 70, 90, 100

Both groups have the same mean (80), but Group B’s marks are more spread out. Dispersion helps us understand this spread.

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Types of Measures of Dispersion

There are four main measures:

1. Mean Deviation (MD)
2. Variance (σ²)
3. Standard Deviation (σ)
4. Coefficient of Variation (CV)

We will study each one in detail.

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1. Mean Deviation (MD)

Mean deviation is the average of the absolute differences between each data point and the mean.

Formula:

For ungrouped data: [ MD = \frac{\sum |x_i - \bar{x}|}{n} ] where:

• (x_i) = each data point
• (\bar{x}) = mean of the data
• (n) = number of data points

Example:

Find the mean deviation of the following data: Data: 10, 12, 14, 16, 18

Step 1: Calculate the mean ((\bar{x})). [ \bar{x} = \frac{10 + 12 + 14 + 16 + 18}{5} = \frac{70}{5} = 14 ]

Step 2: Find the absolute deviations from the mean. [ |10 - 14| = 4, \quad |12 - 14| = 2, \quad |14 - 14| = 0, \quad |16 - 14| = 2, \quad |18 - 14| = 4 ]

Step 3: Sum the absolute deviations. [ 4 + 2 + 0 + 2 + 4 = 12 ]

Step 4: Divide by the number of data points. [ MD = \frac{12}{5} = 2.4 ]

For Grouped Data:

If data is in frequency distribution, use: [ MD = \frac{\sum f_i |x_i - \bar{x}|}{N} ] where:

• (f_i) = frequency of each class
• (N) = total frequency

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2. Variance (σ²)

Variance is the average of the squared deviations from the mean. Squaring removes negative signs and emphasizes larger deviations.

Formula:

For ungrouped data: [ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} ]

Example:

Using the same data: 10, 12, 14, 16, 18

Step 1: Mean ((\bar{x})) = 14 (from above).

Step 2: Calculate squared deviations. [ (10 - 14)^2 = 16, \quad (12 - 14)^2 = 4, \quad (14 - 14)^2 = 0, \quad (16 - 14)^2 = 4, \quad (18 - 14)^2 = 16 ]

Step 3: Sum the squared deviations. [ 16 + 4 + 0 + 4 + 16 = 40 ]

Step 4: Divide by (n). [ \sigma^2 = \frac{40}{5} = 8 ]

Shortcut Formula:

[ \sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 ]

Verification: [ \frac{10^2 + 12^2 + 14^2 + 16^2 + 18^2}{5} - 14^2 = \frac{100 + 144 + 196 + 256 + 324}{5} - 196 = \frac{1020}{5} - 196 = 204 - 196 = 8 ]

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3. Standard Deviation (σ)

Standard deviation is the square root of variance. It is in the same units as the original data.

Formula:

[ \sigma = \sqrt{\sigma^2} ]

Example:

From the previous example, (\sigma^2 = 8), so: [ \sigma = \sqrt{8} \approx 2.83 ]

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4. Coefficient of Variation (CV)

CV compares the dispersion of two datasets with different units or means. It is expressed as a percentage.

Formula:

[ CV = \left( \frac{\sigma}{\bar{x}} \right) \times 100% ]

Example:

Suppose we have two datasets:

• Dataset A: Mean = 50, Standard Deviation = 10
• Dataset B: Mean = 100, Standard Deviation = 20

Calculate CV for both: [ CV_A = \left( \frac{10}{50} \right) \times 100% = 20% ] [ CV_B = \left( \frac{20}{100} \right) \times 100% = 20% ]

Even though Dataset B has a higher standard deviation, its CV is the same as Dataset A because its mean is also higher.

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Comparison Table

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Advantages and Disadvantages

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Solved Example (Grouped Data)

Find the mean deviation and standard deviation for the following frequency distribution:

Step 1: Calculate the mean ((\bar{x})). [ \bar{x} = \frac{\sum f_i x_i}{N} = \frac{(10 \times 2) + (20 \times 3) + (30 \times 5) + (40 \times 4)}{2 + 3 + 5 + 4} = \frac{20 + 60 + 150 + 160}{14} = \frac{390}{14} \approx 27.86 ]

Step 2: Calculate mean deviation. [ MD = \frac{\sum f_i |x_i - \bar{x}|}{N} ] [ = \frac{2|10 - 27.86| + 3|20 - 27.86| + 5|30 - 27.86| + 4|40 - 27.86|}{14} ] [ = \frac{2(17.86) + 3(7.86) + 5(2.14) + 4(12.14)}{14} ] [ = \frac{35.72 + 23.58 + 10.7 + 48.56}{14} = \frac{118.56}{14} \approx 8.47 ]

Step 3: Calculate variance. [ \sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N} ] [ = \frac{2(10 - 27.86)^2 + 3(20 - 27.86)^2 + 5(30 - 27.86)^2 + 4(40 - 27.86)^2}{14} ] [ = \frac{2(319) + 3(61.8) + 5(4.6) + 4(147.4)}{14} ] [ = \frac{638 + 185.4 + 23 + 589.6}{14} = \frac{1436}{14} \approx 102.57 ]

Step 4: Calculate standard deviation. [ \sigma = \sqrt{102.57} \approx 10.13 ]

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NEB Board-Style Questions

1. Short Answer:

   • What is the difference between variance and standard deviation?
   • Why is the coefficient of variation useful?

2. Calculate:

   • Find the mean deviation and standard deviation for the data: 5, 7, 9, 11, 13.

   • Calculate the variance for the following frequency distribution:

     x	f
     10	3
     20	5
     30	2

3. Compare:

   • Two datasets have means 50 and 100, with standard deviations 10 and 20, respectively. Which has higher dispersion? Use CV to justify.

4. Theoretical:

   • Explain why squaring deviations is necessary in variance calculations.

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Exam Tip

1. Always check whether data is grouped or ungrouped before applying formulas.
2. Use shortcut formulas for variance and standard deviation to save time.
3. Label your answers clearly (e.g., "Mean Deviation = ...").
4. Show all steps in calculations, especially for grouped data.
5. Practice with both ungrouped and grouped data to be prepared for any question type.
6. Remember units: Standard deviation has the same units as the original data, but variance does not.

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