Measures of Central Tendency
| Item | Description |
|---|
Mean (xbar = (1/n)*Sum(x_i)) | Arithmetic average — sensitive to outliers |
Median | Middle value (sorted) — robust to outliers, use for skewed data |
Mode | Most frequent value — useful for categorical data, can be multimodal |
When Mean vs Median | Mean for symmetric data; median for skewed data (income, house prices) |
Trimmed Mean | Mean after removing top/bottom k% — compromise between mean and median |
Weighted Mean | Sum(w_i * x_i) / Sum(w_i) — each value has a weight |
Measures of Dispersion
| Item | Description |
|---|
Variance: sigma^2 = (1/n)*Sum(x_i - mu)^2 | Average squared deviation from mean — hard to interpret directly |
Sample Variance: s^2 = (1/(n-1))*Sum(x_i - xbar)^2 | Bessel's correction (n-1) for unbiased estimate from sample |
Standard Deviation: sigma = sqrt(Variance) | Original units — interpretable spread (avg distance from mean) |
Range = Max - Min | Simplest measure — very sensitive to outliers |
IQR = Q3 - Q1 | Interquartile Range — middle 50%, robust to outliers |
MAD = Median(|x_i - median|) | Median Absolute Deviation — robust alternative to std dev |
Measures of Shape
| Item | Description |
|---|
Skewness = E[((X-mu)/sigma)^3] | Asymmetry: >0 right-skewed (long right tail), <0 left-skewed |
Positive Skew Examples | Income, house prices, response times, file sizes |
Negative Skew Examples | Age at death in developed countries, easy test scores |
Kurtosis = E[((X-mu)/sigma)^4] - 3 | Tailedness: >0 heavier tails than normal (leptokurtic) |
Excess Kurtosis | Normal distribution has kurtosis=3, so subtract 3 for excess |
High Kurtosis | Financial returns — more extreme events than normal predicts |
Five-Number Summary
| Item | Description |
|---|
Min, Q1, Median, Q3, Max | Summarizes distribution — the basis of box plots |
Box Plot Outlier Rule | Q1 - 1.5*IQR and Q3 + 1.5*IQR — points beyond are flagged |
Percentiles | Value below which p% of observations fall — Q1=P25, median=P50, Q3=P75 |
Pro Tip: Always report both a measure of center (mean or median) AND a measure of spread (std dev or IQR). One without the other is nearly meaningless.