Summarizing a Pile of Numbers
Descriptive statistics summarize and organize a data set without drawing broader inferential conclusions beyond it. Two families of measures do most of the work. Measures of central tendency describe a data set's typical or center value: the mean (the arithmetic average, sum divided by count) is the most commonly used, but is sensitive to extreme outliers; the median (the middle value when data is ordered) is more robust to outliers and skewed data; the mode (the most frequent value) is the only measure of central tendency that works for purely categorical (non-numeric) data.
| Measure | Definition | Best used when... |
|---|---|---|
| Mean | Sum divided by count | Data is roughly symmetric, no extreme outliers |
| Median | Middle value when ordered | Data is skewed or has outliers |
| Mode | Most frequent value | Data is categorical, or multiple peaks matter |
Measures of variability describe how spread out a data set is around its center. The range (highest value minus lowest) is simple but sensitive to just two extreme points. Variance measures the average squared distance of each point from the mean, and the standard deviation (SD) — the square root of variance — expresses that spread back in the data's original units, making it far more interpretable than variance alone:
A small standard deviation means data points cluster tightly around the mean; a large standard deviation means they are spread widely.
| Measure | What it captures |
|---|---|
| Range | Total spread, highest minus lowest |
| Variance | Average squared distance from the mean |
| Standard deviation | Variance's square root, in original units |
Common pitfall: always defaulting to the mean without checking for outliers or skew first. A single extreme value (like one billionaire's income in a small sample) can drag the mean far from what most people would consider "typical" — the median is far more robust in exactly this situation.