Descriptive Statistics: Univariate and Bivariate Data, Correlation and Regression
This domain tests summary measures for one variable (mean, median, mode, variance, standard deviation, coefficient of variation, quartiles), and for two variables the scatterplot, covariance, Pearson correlation coefficient and the least-squares regression line, together with their interpretation.
Univariate measures
- Mean: x̄ = Σxᵢ/N, or Σxᵢfᵢ/N with frequencies. For grouped data use the class mark.
- Median: the central value of the ordered data (mean of the two central values when N is even); it is resistant to outliers, the mean is not.
- Variance: σ² = Σxᵢ²/N − x̄², and σ = √σ². The PAU normally uses the population (N) form.
- Coefficient of variation CV = σ/x̄, expressed as a percentage; it compares dispersion between variables with different units or means.
- Quartiles Q₁, Q₂, Q₃ support the boxplot and detection of outliers.
Bivariate measures
- Covariance σ_xy = Σxᵢyᵢ/N − x̄·ȳ; its sign indicates the direction of the relation, but its size depends on the units.
- Pearson r = σ_xy/(σ_x·σ_y), with −1 ≤ r ≤ 1; |r| near 1 means a strong linear fit, r near 0 a weak one.
- Regression line of y on x: ŷ = a + bx, with b = σ_xy/σ_x² and a = ȳ − b·x̄. The line always passes through (x̄, ȳ).
- r² is the proportion of the variance of y explained by the linear model.
Worked example
Data: x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5. Then x̄ = 3, ȳ = 4, Σxᵢyᵢ/N = 13.2, so σ_xy = 13.2 − 12 = 1.2; Σxᵢ²/N = 11, so σ_x² = 11 − 9 = 2; Σyᵢ²/N = 17.2, so σ_y² = 1.2. Hence b = 1.2/2 = 0.6 and a = 4 − 0.6·3 = 2.2, giving ŷ = 2.2 + 0.6x. Correlation r = 1.2/√(2·1.2) ≈ 0.77, so the fit is good but not perfect (r² = 0.6).
Traps that cost marks
- Mixing the regression line of y on x with the line of x on y; they are different lines unless r = ±1.
- Predicting far outside the observed range of x, where the linear model need not hold.
- Reading correlation as causation, or describing r without its sign and magnitude.
- Forgetting that multiplying a variable by a negative constant changes the sign of r but not its absolute value.