Sample and population spread
Variance averages squared deviations from the mean. Population variance divides their sum by n; sample variance divides by n − 1. Standard deviation is the square root of variance and uses the original measurement units.
Choose population when the data is the whole group you want to describe. Choose sample when it is used to estimate a larger group. The displayed standard error is the selected standard deviation divided by √n; interpreting it as uncertainty in a mean also needs appropriate sampling assumptions.
Example: 2, 4, 4, 4, 5, 5, 7, 9
The mean is 5, median 4.5 and mode 4. Squared deviations sum to 32. Population variance is 32 ÷ 8 = 4 and population standard deviation is 2. Sample variance is 32 ÷ 7 ≈ 4.571429 and sample standard deviation is about 2.13809.
Using type-7 linear interpolation, Q1 = 4 and Q3 = 5.5, so IQR = 1.5. The 1.5 × IQR fences are 1.75 and 7.75. The value 9 is flagged; a flag calls for investigation, not automatic deletion.
Quartiles, shape and limits
Quartiles interpolate at positions (n − 1)p in the sorted data, counting positions from zero. Other quartile conventions can give different answers for small data sets. Skewness and excess kurtosis use population central moments; they are undefined when there is no spread.
The geometric mean is shown only when all observations are positive. A coefficient of variation is meaningful for a ratio scale with a meaningful zero; avoid interpreting it for negative means or arbitrary origins such as Celsius temperature.
Enter 2–10,000 valid observations, each within ±10⁵⁰. Commas separate observations rather than thousands. Invalid tokens are rejected. Histogram bins are a visual summary and do not change the underlying statistics.