Statistics in simple words
Statistics are short summaries of a list of numbers. Instead of reading every number, you can look at a few that tell the story: the middle, the spread and anything unusual.
The mean (average) and the median (middle number) show where the list is centred. The standard deviation shows how spread out it is. Quartiles cut the sorted list into four equal parts, and the gap between the first and third quartile (the IQR) is the spread of the middle half.
Example: for 2, 4, 4, 4, 5, 5, 7, 9, the total is 40 and there are 8 numbers, so the mean is 40 ÷ 8 = 5. The median is halfway between the 4th and 5th numbers, 4 and 5, which is 4.5.
An outlier is a number much bigger or smaller than the rest. This calculator flags any number more than 1.5 IQRs outside the middle half.
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.
Five-number summary, quartile methods and the box plot
The five-number summary is the minimum, Q1, median, Q3 and maximum. The box plot draws the box from Q1 to Q3 with a line at the median; whiskers run to the most extreme values inside the 1.5 × IQR fences, and anything beyond them is drawn as a separate outlier point.
Choose how quartiles are located. Inclusive (type 7, the default) uses position h = (n − 1)p + 1. Exclusive (type 6, the NIST and Minitab rule) uses h = (n + 1)p. Tukey hinges take the median of the lower and upper halves, including the median in both halves when n is odd. A fractional position is interpolated between its neighbours: h = 2.75 means x₂ + 0.75 × (x₃ − x₂).
For 7, 15, 36, 39, 40, 41 (n = 6): type 7 gives Q1 = 15 + 0.25 × 21 = 20.25 and Q3 = 39 + 0.75 × 1 = 39.75; type 6 gives Q1 = 7 + 0.75 × 8 = 13 and Q3 = 40 + 0.25 × 1 = 40.25; Tukey hinges give Q1 = 15 and Q3 = 40. The median is 37.5 under all three.