Descriptive Statistics Calculator (summary)

Understanding the descriptive statistics calculator — concept and principle

A descriptive statistics calculator summarises a data set in one view: centre (mean, median), spread (standard deviation, minimum, maximum, quartiles) and shape (skewness, excess kurtosis), plus the count n. It is the usual first step: spot outliers, check rough normality and report a summary table.

How it is calculated

• Mean x̄ = Σx ÷ n; median = the middle value of the sorted data (the average of the two middle values when n is even).
• SD (sample) s = √(Σ(x − x̄)² ÷ (n − 1)).
• Q1 / Q3 use linear interpolation at position 1 + p(n − 1) in the sorted list, with p = 0.25 and 0.75 (the method of Excel's QUARTILE.INC).
• With central moments m_r = Σ(x − x̄)^r ÷ n: skewness G₁ = √(n(n − 1)) ÷ (n − 2) × m₃ ÷ m₂^1.5.
• Excess kurtosis G₂ = (n − 1) ÷ ((n − 2)(n − 3)) × [(n + 1)(m₄ ÷ m₂² − 3) + 6].

At least 4 values are needed for the kurtosis formula.

The principle

The mean uses every value and balances the data like a see-saw; the median only depends on order, so it resists outliers. Quartiles split the sorted data into four equal parts. Skewness uses cubed deviations, which keep their sign, so a long right tail gives a positive value. Kurtosis uses fourth powers, which weight the tails heavily; subtracting 3 makes a normal distribution score 0. The factors with n are small-sample corrections (adjusted Fisher–Pearson formulas) that reduce bias.

Worked example

1. Sorted data: 11, 12, 14, 15, 15, 16, 18, 19, 22, 25 (n = 10). Mean = 167 ÷ 10 = 16.7.
2. Median = (15 + 16) ÷ 2 = 15.5; SD = √(172.1 ÷ 9) ≈ 4.3729; min / max = 11 / 25.
3. Q1 at position 3.25: 14 + 0.25 × 1 = 14.25. Q3 at 7.75: 18 + 0.75 × 1 = 18.75, giving 14.25 / 18.75.
4. m₂ = 17.21, m₃ ≈ 41.616, m₄ ≈ 717.94. G₁ = √90 ÷ 8 × 41.616 ÷ 17.21^1.5 ≈ 0.6912.
5. m₄ ÷ m₂² ≈ 2.4240, so G₂ = 9 ÷ 56 × (11 × (2.4240 − 3) + 6) ≈ −0.0541.

How to read the result

The mean is above the median and skewness is positive: a mild right tail pulled by 22 and 25. A common rule of thumb treats |skewness| below 0.5 as fairly symmetric, 0.5 to 1 as moderately skewed and above 1 as highly skewed. Excess kurtosis near 0 means tails similar to a normal distribution. The IQR here is 4.5, and the 1.5 × IQR fences (7.5 and 25.5) flag no outliers.

Common mistakes

• Comparing quartiles from different software, which may use other interpolation methods.
• Over-interpreting skewness and kurtosis from small samples; their sampling error is large.

Frequently asked questions

Why do my quartiles differ from another calculator?

There are several standard quartile definitions; this one matches Excel's QUARTILE.INC and R's default type 7.

Is excess kurtosis the same as kurtosis?

No. Excess kurtosis is kurtosis minus 3, so a normal distribution scores 0 instead of 3.

Sources: Joanes, D. N. and Gill, C. A., "Comparing measures of sample skewness and kurtosis", The Statistician 47(1), 1998 · Hyndman, R. J. and Fan, Y., "Sample quantiles in statistical packages", The American Statistician 50(4), 1996