Why dispersion matters

Two sets can have the same mean but completely different spread. Scores 49, 50, 51 and scores 10, 50, 90 both have mean 50, yet the second set is much more variable. Dispersion quantifies that variability and helps interpret wages, labour-market distributions, financial risk and examination scores.

Range and coefficient of range

Range = maximum − minimum. It is fast but ignores interior observations and reacts strongly to outliers. The coefficient of range = (maximum − minimum)/(maximum + minimum) for positive values where the denominator is meaningful. Example: 12, 15, 18, 25 → range 13; coefficient = 13/37 ≈ 0.351.

Quartiles and quartile deviation

Quartiles divide ordered observations into quarters under a specified convention. Interquartile range (IQR) = Q₃ − Q₁. Quartile deviation (semi-IQR) = (Q₃ − Q₁)/2. A common coefficient of quartile deviation is (Q₃ − Q₁)/(Q₃ + Q₁). Different textbooks use different positional conventions in small ungrouped datasets; check the method assumed by the question.

Mean deviation

Mean absolute deviation about a chosen centre a is Σ|x−a|/n for ungrouped observations and Σf|x−a|/N for grouped data. The median minimises total absolute deviations in one dimension. Unlike standard deviation, mean absolute deviation does not square deviations.

Population variance and standard deviation

For the entire population: σ² = Σ(x−μ)² / N and σ = √σ². For a sample estimate using the conventional Bessel correction, s² = Σ(x−x̄)² /(n−1). A common exam trap is mixing population and sample denominators. Variance has squared units; standard deviation has the original measurement unit.

Worked example: variance and SD

For population values 2, 4, 6, mean μ=4. Squared deviations = 4, 0, 4. Population variance = 8/3 ≈ 2.667; SD = √(8/3) ≈ 1.633. If the same three observations were being used to estimate a population variance from a random sample, sample variance would be 8/(3−1)=4, with sample SD=2.

Coefficient of variation (CV)

CV = (standard deviation / mean) × 100%, conventionally meaningful when mean is nonzero and on a suitable positive ratio scale. CV compares relative dispersion when scales or means differ. Example: Factory A mean wage ₹400, SD ₹40 → CV 10%; Factory B mean wage ₹500, SD ₹75 → CV 15%. A has the lower relative variation, despite a lower mean. CV is not appropriate to every variable or near-zero mean.

Shortcut formula

Population variance can also be written σ² = Σx²/N − (Σx/N)². For frequency tables, use Σfx²/N − (Σfx/N)². Beware severe rounding errors if calculations use many decimal places.

Interpreting distributions

Standard deviation is affected by extreme values because it squares deviations. IQR is more robust for heavily skewed salary data. A box plot marks median, quartiles and potential outliers; it does not prove those observations are mistakes. A normal distribution has mean and median equal by symmetry, but observing mean≈median does not prove normality.

Practice MCQs

Q1. Variance of a constant series 7,7,7 is (A) 7 (B) 0 (C) 1 (D) 49. Answer: B. Every deviation from the mean is zero.

Q2. A series has mean 80 and SD 8. CV equals (A) 8% (B) 10% (C) 80% (D) 100%. Answer: B. 8/80×100=10%.

Q3. Which measure keeps the same unit as original observations? (A) Variance (B) Standard deviation (C) Squared deviation (D) Covariance. Answer: B. SD is the square root of variance.

FAQs

Is the standard deviation always negative?

No. It is nonnegative; zero indicates no variation.

Can a dataset have zero range but nonzero SD?

No. Zero range implies all observations are identical.

Should sample variance always divide by n?

The familiar unbiased sample variance estimator divides by n−1; always read the question’s definition.