Statistics: Measures of Dispersion, Skewness and Bivariate Data
Statistics: Measures of Dispersion, Skewness and Bivariate Data tests spread, shape and association. You must compute range, variance, standard deviation and coefficient of variation, interpret skewness, and analyse paired data using covariance, correlation and regression ideas.
Core rules
- Range = maximum - minimum. Variance uses squared deviations from the mean: σ^2=Σ(x_i-mean)^2/n for a population-style data set; if the syllabus specifies sample variance, divide by n-1.
- Standard deviation σ=√variance. Coefficient of variation CV=(σ/mean)×100%, used to compare variability when units or means differ.
- Karl Pearson's coefficient of skewness is (mean-median)/σ or (mean-mode)/σ. Positive means a right tail; negative means a left tail; zero suggests symmetry.
- Covariance measures the direction of linear association: cov(X,Y)=Σ(x_i-x̄)(y_i-ȳ)/n.
- Correlation r=cov(X,Y)/(σ_x σ_y), with -1≤r≤1. Its sign matches the regression slope; it measures linear association only.
- Regression line of y on x is y-ȳ=b_yx(x-x̄), where b_yx=cov(X,Y)/σ_x^2.
Worked example
For 2,4,6,8,10, mean=6. Squared deviations: 16,4,0,4,16; sum=40. Variance=40/5=8, so σ=√8≈2.828. CV=(2.828/6)×100≈47.1%. The data are symmetric, so Pearson skewness is 0.
Mark-losing traps
- Dividing by n-1 when the question expects n, or vice versa.
- Forgetting to square deviations before summing.
- Treating CV as having units; it is a percentage.
- Reading correlation as proof of causation.
- Using r to describe a curved relationship.