How to use this calculator
- Enter or choose numbers.
- The mean updates as you type. Use Copy result or Share to keep it.
Method
Mean x̄ = Σx ÷ n; s = √(Σ(x − x̄)² ÷ (n − 1)); σ = √(Σ(x − x̄)² ÷ n); SE = s ÷ √n; Quartile at position 1 + (n − 1)p in sorted data, interpolated; G₁ = n ÷ ((n − 1)(n − 2)) × Σ((x − x̄) ÷ s)³
- n
- number of values
- x̄
- mean
- s, σ
- sample and population standard deviation
- p
- 0.25 for Q1, 0.5 for the median, 0.75 for Q3
Worked example
- Mean = 40 ÷ 8 = 5. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32.
- σ = √(32 ÷ 8) = 2; s = √(32 ÷ 7) ≈ 2.1381.
- Q1 sits at position 1 + 7 × 0.25 = 2.75 → 4; Q3 at 6.25 → 5 + 0.25 × (7 − 5) = 5.5; IQR = 1.5.
The data have mean 5, median 4.5, population SD 2 (sample SD 2.14) and an interquartile range of 1.5.
What the result means
Use the sample standard deviation when your numbers are a sample from a bigger group (the usual case) and the population figure only when you have every member. Positive skewness means a long tail of high values.
Assumptions and limits
- Quartiles use linear interpolation between sorted values (Hyndman–Fan type 7), the method of Excel’s QUARTILE.INC and R’s default.
- Other quartile conventions (for example Tukey’s hinges or QUARTILE.EXC) can give slightly different Q1 and Q3 for small data sets.
- Skewness is the adjusted Fisher–Pearson coefficient used by Excel’s SKEW, and needs at least 3 values.
Questions
Why do different tools give different quartiles?
There are at least nine published definitions. They agree for large data sets but can differ for small ones; this calculator names the one it uses so you can match it.
What is the standard error for?
It estimates how much the sample mean would vary from sample to sample. A 95% confidence interval for the mean is roughly mean ± 2 × SE for moderate samples.