Descriptive Statistics

Descriptive statistics summarise a data set with a few numbers: where its centre is, and how spread out it is.

Measures of central tendency

For the data 4, 8, 6, 5, 3, 8, 9, 5, 8, 4:

The mean is pulled by extreme values (outliers); the median is not. For a moderately skewed distribution, the empirical relation mode ≈ 3 median − 2 mean holds.

Measures of spread

• Range = maximum − minimum = 9 − 3 = 6.
• Population variance σ² = Σ(xᵢ − x̄)² / n = 40 / 10 = 4, standard deviation σ = 2.
• Sample variance s² = Σ(xᵢ − x̄)² / (n − 1) = 40 / 9 ≈ 4.44. Dividing by n − 1 corrects the bias that comes from measuring deviations from the sample's own mean. Read questions carefully to see which is meant.
• A convenient formula: σ² = (Σxᵢ²)/n − x̄² (the mean of the squares minus the square of the mean).

Shifting and scaling

If every value is transformed to yᵢ = a·xᵢ + b:

Adding a constant shifts the centre but not the spread; multiplying scales both.

Correlation and regression

For paired data (xᵢ, yᵢ), the Pearson correlation coefficient

r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √(Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)²)

measures linear association, from −1 (perfect decreasing line) through 0 (no linear relation) to +1 (perfect increasing line). It is unchanged by shifting either variable and by scaling with positive factors. Correlation does not imply causation.

The least-squares regression line of y on x is ŷ = a + bx with slope b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² and intercept a = ȳ − b x̄; it always passes through (x̄, ȳ).

Python's statistics module computes all of these:

import statistics as st

data = [4, 8, 6, 5, 3, 8, 9, 5, 8, 4]
print("mean", st.mean(data), "| median", st.median(data), "| mode", st.mode(data))
print("population variance", st.pvariance(data), "| sample variance", round(st.variance(data), 4))
print("population std dev", st.pstdev(data))
scaled = [3 * v + 2 for v in data]
print("after y = 3x + 2: mean", st.mean(scaled), "| population variance", st.pvariance(scaled))

x, y = [1, 2, 3, 4, 5], [2, 4, 5, 4, 5]
fit = st.linear_regression(x, y)
print(f"correlation r = {st.correlation(x, y):.4f}, regression line y = {fit.intercept:.1f} + {fit.slope:.1f}x")

mean 6 | median 5.5 | mode 8
population variance 4 | sample variance 4.4444
population std dev 2.0
after y = 3x + 2: mean 20 | population variance 36
correlation r = 0.7746, regression line y = 2.2 + 0.6x

Practice questions