Descriptive Statistics

Measures of central tendency answer: what's typical?

• Mean is the arithmetic average. Sensitive to outliers — one billionaire moves the average household income of a small town enormously.
• Median is the middle value when data is sorted. Robust to outliers — the billionaire doesn't change the median household income.
• Mode is the most frequently occurring value. The only measure of central tendency that makes sense for purely categorical data.

When you have a skewed distribution — income, web session length, time-to-failure — the gap between mean and median tells you something. Mean substantially greater than median? Long right tail. Mean substantially less? Long left tail.

Measures of dispersion answer: how spread out is the data?

• Standard deviation is the average distance of data points from the mean.
• Variance is the square of the standard deviation. Useful in statistical derivations.
• Range is maximum minus minimum. Dominated by extremes.
• Interquartile range (IQR) is Q3 − Q1. Captures the spread of the middle half of your data, robust to outliers — which is why box plots use it.

A note on the formulas: the population standard deviation divides by N. The sample standard deviation divides by n − 1. That n − 1 is Bessel's correction — it compensates for the systematic underestimation that happens when you use the sample mean instead of the true population mean.

Measures of distribution shape describe asymmetry and tailedness.

• Skewness measures asymmetry. Positive skewness = longer right tail (right-skewed). Negative = longer left tail. Symmetric distributions like the normal have skewness near zero.
• Kurtosis measures tailedness — how prone the distribution is to extreme values. Higher kurtosis = heavier tails = more outliers relative to a normal distribution.