Descriptive statistics calculator

Type or paste a data set —an Excel column too, with a decimal point or comma— and get every usual measure of a statistics course, with the formulas and the steps.

What you can do

Central tendency: mean, median, mode, geometric mean and harmonic mean.
Dispersion: range, sample (n − 1) and population (N) variance and standard deviation, coefficient of variation, standard error and mean absolute deviation.
Position: quartiles, interquartile range and percentiles, by linear interpolation as in Excel.
Shape: skewness and kurtosis; outliers by the Tukey fences.
Data with frequencies, absolute, relative and cumulative frequency table, histogram and box plot.

Guide: descriptive statistics step by step

Descriptive statistics summarize a data set with a few numbers: where its center is, how spread out it is and what shape its distribution has. This guide explains each measure, when to use it and works a complete example by hand.

Measures of central tendency

Mean (x̄): the sum of the data divided by n. It is sensitive to extreme values.
Median: the middle value of the sorted data (with even n, the average of the two middle ones). Extreme values do not affect it.
Mode: the most frequent value; there can be several or none.

If the mean and the median are far apart, the distribution is skewed or there are outliers.

Measures of spread

Range: maximum minus minimum.
Variance: the average of the squared deviations from the mean. The sample variance divides by n − 1 and the population variance by n. With a sample, to estimate the variability of the whole population, use n − 1.
Standard deviation: the square root of the variance, in the same units as the data.
Coefficient of variation: CV = 100 · s / x̄. It lets you compare the spread of data on different scales.
Standard error of the mean: SE = s / √n. It measures how much the mean would vary from one sample to another.

Quartiles and outliers

Quartiles split the sorted data into four parts: Q1 has 25 % of the data below it and Q3, 75 %. The interquartile range is IQR = Q3 − Q1. There are several methods for computing quartiles, and with few data they do not all agree. The tool uses Excel's PERCENTILE.INC / QUARTILE.INC: the position of quantile p is 1 + p·(n − 1) in the sorted data, interpolating between neighbors. By Tukey's rule, a value is an outlier if it lies below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Worked example

Data: 4, 8, 6, 5, 3, 7, 9, 5 (n = 8).

Sorted: 3, 4, 5, 5, 6, 7, 8, 9.
Mean: 47 / 8 = 5.875.
Median: (5 + 6)/2 = 5.5. Mode: 5. Range: 9 − 3 = 6.
Squared deviations: their sum is Σ(x − x̄)² = 28.875.
Sample variance: 28.875 / 7 = 4.125; standard deviation s = √4.125 ≈ 2.031. (The population values would be 28.875/8 ≈ 3.609 and σ ≈ 1.900.)
CV = 100 · 2.031 / 5.875 ≈ 34.6 %; SE = 2.031 / √8 ≈ 0.718.
Quartiles: Q1 is at position 1 + 0.25·7 = 2.75, between the 2nd value (4) and the 3rd (5): Q1 = 4 + 0.75·(5 − 4) = 4.75. Q3 is at position 6.25: Q3 = 7 + 0.25·(8 − 7) = 7.25. IQR = 2.5.
Outliers: the fences are 4.75 − 3.75 = 1 and 7.25 + 3.75 = 11; no value falls outside.

The calculator gives the same values, plus the skewness and kurtosis, adjusted as in Excel (here, 0.224 and −0.886).

How to use the calculator

Type or paste the data. If a value repeats many times, you can give its frequencies separately, in the same order. Choose the decimal separator and whether the data are a sample or the whole population.
Read the measures of center, spread, position (quartiles and percentiles) and shape, the outliers, the histogram, the box plot and the frequency table.
Go through the steps to see each calculation: sorting, summing, the deviations table and the quartile interpolation. The detailed PDF solution has the full derivation.

Common mistakes

Dividing by n when the data are a sample. It underestimates the population variance.
Computing the median without sorting the data.
Comparing quartiles computed with different methods: your textbook or calculator may use another rule.
Summarizing very skewed data with the mean (incomes, waiting times); there the median represents the center better.

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