Descriptive Statistics Calculator
Paste numbers (comma, space or line-separated) and get every common descriptive statistic.
Descriptive vs Inferential Statistics
This calculator does descriptive statistics — summarizing the numbers you actually have. Inferential statistics (hypothesis tests, confidence intervals, regression) lets you make claims about a larger population from a sample, and requires more careful setup. The Wikipedia article on descriptive statistics covers the theoretical foundations.
The Key Measures and What They Tell You
| Measure | Use Case | Robust to Outliers? |
|---|---|---|
| Mean (average) | Symmetric data, "typical" value | No |
| Median | Skewed data, "middle" value | Yes |
| Mode | Most-common value (categorical or peak) | Yes |
| Range | Spread from min to max | Very sensitive to outliers |
| Standard deviation | Average spread around the mean | No |
| IQR (Q3 − Q1) | Spread of middle 50% | Yes |
U.S. household income
According to the U.S. Census Bureau's 2023 data:
- Mean household income: ~$114,500
- Median household income: ~$80,610
The $34,000 gap is because a small number of very high earners pull the mean upward dramatically. A household making $80k feels nothing like the "average" $114k household — because there are no typical households at $114k. For skewed data like income, wealth, or response times, the median tells you about a real person; the mean is a statistical fiction.
Same logic for tech salaries, marathon times, real estate prices, social media followers — anything with a long right tail.
Sample vs Population Standard Deviation
This calculator returns the sample standard deviation, dividing by n−1 (called Bessel's correction). Use this when your data is a sample from a larger population — which is virtually always the case in real-world use. The "population SD" formula (divide by n instead of n−1) is only correct when you literally have every member of the population. The Wikipedia article on Bessel's correction explains why the −1 is needed.
The 68-95-99.7 Rule (Normal Distributions)
For data that's approximately normally distributed (bell-curve shaped):
- ~68% of values fall within 1 SD of the mean
- ~95% of values fall within 2 SDs of the mean
- ~99.7% of values fall within 3 SDs of the mean
This is also called the empirical rule. It's the basis for "Six Sigma" quality control (defects beyond 6 SD = 3.4 per million opportunities).
Quartiles and IQR
Quartiles divide ordered data into four equal parts. Q1 (25th percentile), Q2 (median = 50th), Q3 (75th percentile). The Interquartile Range (IQR) = Q3 − Q1 measures the spread of the middle 50% of values. Boxplots are built on quartile structure. Outliers are commonly flagged as values beyond Q3 + 1.5×IQR or Q1 − 1.5×IQR.
When to Use Median Instead of Mean
- Skewed distributions: income, wealth, real estate prices, response times
- Data with outliers: a few extreme values would distort the mean
- Ordinal data: rankings where averaging doesn't make sense
For symmetric distributions (height, IQ, test scores, manufacturing tolerances), mean and median agree closely and the mean is preferred because it uses all the data information.