Measures of Dispersion

Measures of Dispersion

Dispersion (variability/spread) measures how spread out data values are from the centre. Two datasets can have the same mean but very different spreads. Dispersion reveals consistency, risk, and reliability — essential for quality control, financial analysis, and decision-making.

Why Measure Dispersion

Consider two employees with average monthly sales of Rs 100,000. Employee A's sales range from Rs 95,000 to Rs 105,000 (consistent). Employee B's range from Rs 50,000 to Rs 150,000 (volatile). The mean alone doesn't distinguish them. Dispersion measures quantify this difference, helping managers assess reliability, control quality, evaluate risk, and compare variability across datasets.

Range

Range = Maximum value − Minimum value. The simplest measure but uses only two extreme values, ignoring the distribution of all other data points. Heavily influenced by outliers. Coefficient of Range = (Max − Min)/(Max + Min) — a relative measure for comparing datasets with different units or scales. Range is useful for quick assessment and quality control (acceptable range of product dimensions).

Quartile Deviation

Quartiles divide sorted data into four equal parts. Q₁ (25th percentile), Q₂ (median, 50th), Q₃ (75th percentile). Interquartile Range (IQR) = Q₃ − Q₁ covers the middle 50% of data. Quartile Deviation (QD) = (Q₃ − Q₁)/2 (semi-interquartile range). Coefficient of QD = (Q₃ − Q₁)/(Q₃ + Q₁). QD is not affected by extreme values, making it better than range for skewed data. Used for identifying outliers: values below Q₁ − 1.5×IQR or above Q₃ + 1.5×IQR are potential outliers.

Mean Deviation

Mean deviation (MD) is the average of absolute deviations from the mean (or median): MD = Σ|x − x̄|/n. For grouped data: MD = Σf|x − x̄|/Σf. MD from median is minimum (property). Coefficient of MD = MD/Mean (or MD/Median). MD considers all values and is easy to understand, but the absolute value function makes it mathematically less tractable than standard deviation.

Standard Deviation

Variance (σ²) is the average of squared deviations from the mean: σ² = Σ(x − x̄)²/n. Standard deviation (σ) is the square root of variance: σ = √[Σ(x − x̄)²/n]. For grouped data: σ = √[Σf(x − x̄)²/Σf]. Shortcut formula: σ = √[Σfx²/Σf − (Σfx/Σf)²]. Standard deviation is the most important and widely used measure of dispersion.

Coefficient of Variation

Coefficient of Variation (CV) = (σ/x̄) × 100%. CV is a relative measure — it expresses standard deviation as a percentage of the mean. This allows comparing variability of datasets with different units. Lower CV indicates more consistency. CV is essential for comparing risk in finance.

Variance Properties

Var(x + c) = Var(x) — adding a constant doesn't change variance. Var(cx) = c²Var(x) — multiplying by a constant squares the effect. Combined variance formula allows computing from subgroup variances. Useful in portfolio analysis and quality control.

Summary

Dispersion measures — range, quartile deviation, mean deviation, standard deviation, and coefficient of variation — quantify data spread and variability. Standard deviation is the most important. CV enables comparison across datasets. Together with central tendency, dispersion provides a complete numerical summary.

Worked Example: Standard Deviation

The daily sales (in Rs thousands) of a shop for 7 days are: 45, 50, 38, 55, 42, 60, 48. Calculate the standard deviation and coefficient of variation.

Solution:

Step 1: Mean x̄ = (45+50+38+55+42+60+48)/7 = 338/7 = 48.29

Step 2: Calculate deviations and squared deviations:

Step 3: Variance σ² = Σ(x−x̄)²/n = 341.40/7 = 48.77

Step 4: Standard Deviation σ = √48.77 = Rs 6.98 thousand

Step 5: Coefficient of Variation CV = (σ/x̄) × 100 = (6.98/48.29) × 100 = 14.45%

Interpretation: The average daily sales is Rs 48,290 with a standard deviation of Rs 6,980. The CV of 14.45% indicates moderate variability — sales are reasonably consistent. If another shop has CV = 30%, it has more volatile sales despite possibly having the same average.

Worked Example: Comparing Two Datasets Using CV

A company wants to compare the consistency of two sales teams:

Although Team B has a higher standard deviation (4 > 3), its CV is lower (16% < 20%), meaning Team B is actually more consistent relative to its average. This is why CV is essential — comparing standard deviations alone would be misleading when means differ.

Business Applications of Dispersion

Quality Control: A manufacturing process producing bolts should have low dispersion — if the target diameter is 10mm with tolerance ±0.1mm, standard deviation must be small enough that virtually all bolts fall within specification. Financial Risk: In finance, standard deviation of stock returns measures volatility (risk). Investors compare CV across stocks to assess risk per unit of expected return. HR Management: Salary dispersion within a company indicates pay equity. High dispersion may suggest discrimination or inconsistent pay policies. Market Research: If customer satisfaction scores have high dispersion, it means experiences are inconsistent — some customers are very happy, others very unhappy. Low dispersion means consistent experience.

Exam Tips for Dispersion

Tip 1: Always state the formula before substituting values — examiners give marks for correct formula even if calculation has errors. Tip 2: For grouped data, use class midpoints in calculations. Tip 3: When comparing two datasets, always use CV (not SD) if means are different. Tip 4: Note that Σ(x−x̄) always equals zero — that is why we square the deviations. Tip 5: The shortcut formula σ = √[Σx²/n − (x̄)²] is faster for computation. Tip 6: Remember the property: if every value is multiplied by a constant k, SD is multiplied by |k| but if a constant is added, SD doesn’t change.