# Statistics Calculator

> Paste a list of numbers to get a full summary: mean, median, mode, quartiles and IQR, sample and population standard deviation, standard error and skewness.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/statistics-calculator

One paste gives the descriptive statistics you need to report a data set: centre, spread, quartiles and shape, with the method for each named.

Also called: Descriptive statistics calculator, Summary statistics calculator.

## Inputs
- Numbers (default 2, 4, 4, 4, 5, 5, 7, 9)

## Results
- Mean
- Median
- Mode
- Sample standard deviation (s)
- Population standard deviation (σ)
- Sample variance (s²)
- Population variance (σ²)
- Standard error of the mean
- First quartile (Q1)
- Third quartile (Q3)
- Interquartile range
- Minimum
- Maximum
- Range
- Count
- Sum
- Coefficient of variation
- Skewness (adjusted, G₁)

## Formula

```
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
```

Where:
- 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
1. Mean = 40 ÷ 8 = 5. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32.
2. σ = √(32 ÷ 8) = 2; s = √(32 ÷ 7) ≈ 2.1381.
3. 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.

**Result:** 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.

## Related calculators
- [Standard Deviation Calculator](https://calculators.geekmidst.com/calculators/standard-deviation-calculator)
- [Average Calculator](https://calculators.geekmidst.com/calculators/average-calculator)
- [Sample Size Calculator](https://calculators.geekmidst.com/calculators/sample-size-calculator)
- [Percent Error Calculator](https://calculators.geekmidst.com/calculators/percent-error-calculator)
- [Interquartile Range Calculator](https://calculators.geekmidst.com/calculators/interquartile-range-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/statistics-calculator