Paste your sample (separated by spaces, new lines or semicolons, or a column from Excel) or a frequency table. The calculator builds the ordered series, the frequency table and a grouped table using Sturges' rule, finds the sample statistics and draws the histogram, frequency polygon and ogive. The solution is shown step by step, like in a textbook.
What is descriptive statistics of a sample
A sample is the result of n observations of a random variable. Processing starts with the ordered series: the values are listed in ascending order. Equal values are merged and counted, which gives the frequency table (nᵢ is the frequency, wᵢ = nᵢ/N the relative frequency, plus cumulative frequencies). If there are many continuous values, the series is grouped into classes.
How many classes to use
The number of classes is usually chosen by Sturges' rule k = 1 + 3.322·lg N, where N is the sample size. For 50 observations this gives about 7 classes, for 100 about 8. The class width is h = (xmax − xmin)/k. The rule is a guideline: use the slider to take more or fewer classes and see how the histogram changes.
Main characteristics
- Mean x̄ = Σxᵢnᵢ/N is the centre of the sample.
- Variance D = Σ(xᵢ − x̄)²nᵢ/N is the mean squared deviation. The corrected variance s² = N/(N − 1)·D is an unbiased estimate of the population variance and is used in estimation and hypothesis testing.
- Coefficient of variation V = s/x̄·100% is the spread as a percentage of the mean.
- The mode is the most frequent value, the median is the middle of the ordered series.
- Skewness A = μ₃/σ³ and kurtosis E = μ₄/σ⁴ − 3 show how the distribution differs from the normal one.
Charts
The histogram has bars over the classes with height nᵢ/(N·h), the polygon is a broken line through the class midpoints, and the ogive (cumulative curve) plots the cumulative relative frequency, an estimate of the distribution function.
FAQ
What is the difference between D and s²?
D is divided by N and describes this particular sample. s² is divided by N − 1 and is the corrected (unbiased) estimate of the population variance. For large N the difference is small. The calculator shows both.
How do I choose the number of classes?
By default Sturges' rule k = 1 + 3.322·lg N is used and rounded. It is a guideline, not a law: if the histogram looks too ragged or too coarse, change k with the slider. Make sure most classes contain several observations.
How do I paste data from Excel?
Select a column of cells in Excel, copy it and paste it into the input field: line breaks and decimal commas are parsed automatically. For a frequency table choose the Frequency table mode and paste two columns: values and frequencies.
How are quartiles found, and why do they differ from Excel?
Here the quartiles are the medians of the lower and upper halves of the ordered series. Excel and other programs use interpolation (several different rules), so results for small samples can differ slightly. For large samples the difference is negligible.
What do skewness and kurtosis mean?
Skewness A = μ₃/σ³ is zero for a symmetric distribution, positive for a long right tail and negative for a long left tail. Excess kurtosis E = μ₄/σ⁴ − 3 is zero for the normal law: E > 0 means a sharper peak, E < 0 a flatter one. For small samples these indicators are unreliable.