In This Lesson Measures of Center Measures of Spread Data Visualization Distribution Shape Measures of Center
Mean: x̄ = (Σxᵢ)/n
Median: middle value when sorted
Mode: most frequent value
The mean uses algebraic operations ; it's sensitive to outliers. The median is more robust. For symmetric distributions, mean ≈ median ≈ mode.
Measures of Spread
Variance: σ² = Σ(xᵢ − x̄)²/n
Standard deviation: σ = √(σ²)
Range: max − min
IQR: Q3 − Q1
Variance measures average squared deviation from the mean. Standard deviation has the same units as the data — it's the most widely used spread measure. These connect to the normal distribution via the 68-95-99.7 rule.
Example: Data: 4, 7, 8, 10, 11 Mean = 40/5 = 8
Deviations: −4, −1, 0, 2, 3. Squared: 16, 1, 0, 4, 9
Variance = 30/5 = 6. SD = √6 ≈ 2.45
Data Visualization 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 Histograms: Show distribution shape and frequency (area under the curve connects to integration ) Box plots: Display median, quartiles, and outliers Scatter plots: Show relationships between two variables → regression 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 Bar/pie charts: Compare categorical 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 Distribution Shape 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 Skewness describes asymmetry: right-skewed (mean > median, long right tail), left-skewed (mean < median). Kurtosis describes tail heaviness. The normal distribution has skewness 0 and kurtosis 3 (by convention, "excess kurtosis" = 0).
Descriptive statistics is the first step of
any data analysis. Before fitting
regression models or running hypothesis tests, always visualize and summarize your data. As the saying goes: "Plot your data."