Describing a dataset
Collecting numerical observations is merely the first step in quantitative inquiry. Once numbers are gathered, descriptive statistics reduce complex tables into manageable summaries of central location and dispersion. Understanding what each metric communicates—and what it conceals—is vital for responsible interpretation.
Measures of Central Tendency
The three classic measures of center answer the question: where do the observations typically cluster?
- Arithmetic Mean: The sum of observations divided by the sample size n. While mathematically convenient and sensitive to every point, the mean is vulnerable to distortion by extreme outliers.
- Median: The physical middle observation when values are sorted in ascending order. If n is even, it is the average of the two middle observations. The median is robust against severe skew.
- Mode: The value occurring with the greatest frequency. A dataset may be unimodal, bimodal, multimodal, or possess no repeated mode.
Measures of Dispersion and Spread
Center alone can be deeply misleading. Two datasets can share an identical mean of 50 while representing radically different realities: one might span tightly between 49 and 51, while the other oscillates between 0 and 100.
| Metric | Formula | Primary Utility | Outlier Sensitivity |
|---|---|---|---|
| Range | Max - Min | Quick rough boundary check | Extremely sensitive |
| Sample Variance (s²) | ∑(xi - x̄)² / (n - 1) | Foundation for inferential tests | High (squared errors) |
| Standard Deviation (s) | √s² | Spread expressed in original units | High |
| Interquartile Range (IQR) | Q3 - Q1 | Middle 50% spread; box plot bounds | Very robust |
Bessel's Correction: Why We Divide by (n - 1)
When computing variance for an entire known population, the divisor is N. However, when working with a sample to estimate true population dispersion, using n systematically underestimates the variance because the sample points lie closer to their own sample mean than to the true population mean.
Dividing by n - 1 (Bessel's correction) restores un-biasedness to the estimator. As sample sizes grow into the thousands, the practical difference between n and n - 1 approaches zero, but in small bench testing (e.g., n = 10), omitting this correction introduces an error of roughly 10%.
Interpreting Skewness and Shape
In a perfectly symmetrical distribution, mean, median, and mode coincide at the center. When a distribution is positively skewed (skewed right), a tail of unusually high numbers pulls the mean above the median. In income distributions, website response times, or natural disaster losses, the median consistently offers the more honest depiction of everyday experience.
Continue to Probability in practice to see how random events converge over repeated trials.