Introduction to Measures of Dispersion
While measures of central tendency (mean, median, mode) describe the centre of data, measures of dispersion describe how spread out the data is.
Common Measures of Dispersion
- Range
- Mean Deviation
- Variance
- Standard Deviation
- Coefficient of Variation
Range
Range = Maximum value - Minimum value
Limitation: Range depends only on two extreme values and ignores the rest of the data. It is not a reliable measure for dispersed data.
Example: For data 2, 5, 8, 12, 20: Range = 20 - 2 = 18.
Mean Deviation
Mean deviation about a measure of central tendency A (mean, median, or mode):
For Ungrouped Data
MD = sum_(i=1)^n |x_i - A|/n
For Grouped Data (Discrete Frequency)
MD = sum_(i=1)^n f_i |x_i - A|/(sum f_i)
For Grouped Data (Continuous)
MD = (sum f_i |x_i - A|)/(sum f_i) where x_i are class midpoints.
Mean Deviation about Mean
MD(bar(x)) = (sum |x_i - bar(x)|)/n
Mean Deviation about Median
MD(M) = (sum |x_i - M|)/n
Mean deviation is minimum when taken about the median.
Variance and Standard Deviation
Variance (Ungrouped Data)
sigma^2 = sum_(i=1)^n (x_i - bar(x))^2 / n
Standard Deviation (Ungrouped Data)
sigma = sqrt(sigma^2) = sqrt(sum_(i=1)^n (x_i - bar(x))^2 / n)
Variance (Grouped Data - Discrete)
sigma^2 = (sum f_i (x_i - bar(x))^2) / (sum f_i)
Variance (Grouped Data - Continuous)
Same formula, using class midpoints as x_i.
Shortcut Formula
sigma^2 = (sum f_i x_i^2)/(sum f_i) - bar(x)^2
This is computationally more efficient and avoids calculating deviations individually.
Coefficient of Variation (CV)
CV = (sigma/|bar(x)|) * 100%
The coefficient of variation is a relative measure of dispersion, useful for comparing two or more datasets with different units or means.
Interpretation: Lower CV implies more consistency (less variability) in data.
Comparison of Two Series
- Series with lower CV is more consistent and stable.
- Series with higher CV is more variable.
Standard Deviation of Combined Series
If two series have n_1, n_2 observations with means bar(x)_1, bar(x)_2 and standard deviations sigma_1, sigma_2:
Combined mean: bar(x) = (n_1 bar(x)_1 + n_2 bar(x)_2)/(n_1 + n_2)
Combined variance: sigma^2 = (n_1(sigma_1^2 + d_1^2) + n_2(sigma_2^2 + d_2^2))/(n_1 + n_2) where d_1 = bar(x)_1 - bar(x), d_2 = bar(x)_2 - bar(x).
Worked Examples
Example 1: Find the mean deviation about the mean for: 10, 20, 30, 40, 50.
Solution: bar(x) = (10+20+30+40+50)/5 = 30.
MD = (|10-30| + |20-30| + |30-30| + |40-30| + |50-30|)/5 = (20+10+0+10+20)/5 = 60/5 = 12.
Example 2: Find variance and standard deviation for: 3, 7, 11, 15, 19.
Solution: bar(x) = (3+7+11+15+19)/5 = 55/5 = 11.
sigma^2 = ((3-11)^2 + (7-11)^2 + (11-11)^2 + (15-11)^2 + (19-11)^2)/5
= (64+16+0+16+64)/5 = 160/5 = 32
sigma = sqrt(32) = 4sqrt(2) approx 5.66
Example 3: Two groups have means 50 and 60, and SDs 10 and 12. Find CV of each.
Solution: CV_1 = 10/50 * 100 = 20%. CV_2 = 12/60 * 100 = 20%. Both groups have equal relative variability.
Comparison Table: Measures of Dispersion
| Measure | Unit | Affected by Outliers | Best Used When |
|---|---|---|---|
| Range | Same as data | Highly | Quick estimate |
| Mean Deviation | Same as data | Moderately | Distribution is symmetric |
| Standard Deviation | Same as data | Moderately | Most statistical analysis |
| Coefficient of Variation | Percentage | Moderately | Comparing different datasets |
Common Mistakes
- Absolute values in mean deviation: Do not drop modulus sign — deviations are always positive.
- SD vs variance: Standard deviation has same unit as data; variance has squared units.
- n vs (n-1): For population data, divide by n. For sample data, divide by (n-1).
- CV comparison: CV is meaningful only when comparing; it has no standalone interpretation.
ISC Exam Focus
- Theory (70%): Definitions, formulas for variance and SD, combined variance derivation.
- Application (30%): Numerical problems calculating mean deviation, variance, SD, CV.
- ISC typically asks: "Find the mean deviation about the mean/median for the following data."
- 4-6 mark questions involving grouped frequency data calculations.
Self-Test Questions
Q1: Find the range of data: 15, 22, 8, 31, 42, 10. Answer: Maximum = 42, Minimum = 8. Range = 34.
Q2: Find the mean deviation about the mean: 4, 8, 12, 16, 20.
Answer: bar(x) = 12. MD = (8+4+0+4+8)/5 = 24/5 = 4.8.
Q3: Find the standard deviation: 5, 10, 15, 20, 25.
Answer: bar(x) = 15. sigma^2 = (100+25+0+25+100)/5 = 250/5 = 50. sigma = sqrt(50) = 5sqrt(2).
Q4: For two groups, Group A: mean = 40, SD = 8; Group B: mean = 80, SD = 12. Which is more consistent?
Answer: CV_A = 8/40 * 100 = 20%. CV_B = 12/80 * 100 = 15%. Group B is more consistent (lower CV).
Q5: Calculate variance for the frequency distribution: x: 2, 4, 6, 8; f: 3, 5, 7, 5.
Answer: bar(x) = (6+20+42+40)/20 = 108/20 = 5.4. sigma^2 = (3(2-5.4)^2 + 5(4-5.4)^2 + 7(6-5.4)^2 + 5(8-5.4)^2)/20 = (34.68+9.8+3.36+33.8)/20 = 81.64/20 = 4.082.
Q6: If CV of a dataset is 25% and mean is 40, find SD.
Answer: CV = sigma/|bar(x)| * 100 => 25 = sigma/40 * 100 => sigma = (25*40)/100 = 10.
