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Statistics Calculator

Analyse number lists with mean, median, variance, standard deviation, quartiles, percentiles, outliers, data quality and charts.

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Inputs

Enter values

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Statistics StudioEvaluate up to six data series with key figures, spread, charts and outliers

Data series

Start with one series and only add the series you actually want to compare.

1/6

Data series 1

Name (optional)

Optional name for the first data series.

Values

Separate values with line breaks, spaces, semicolons or CSV commas. German decimals like 12,5 remain decimals.

Load example

For illustration only – your own calculation is what counts.

Explanation

How Statistics Calculator works

The statistics studio does not just summarise a number list into individual key figures, but reveals structure, spread, data quality and outliers.

Up to six named data series can be entered in parallel. Each active series receives the same key figures; removed or empty series are ignored.

The input is first parsed robustly. Valid values enter the analysis, invalid tokens are counted and shown in data quality.

Histogram, frequency chart, cumulative distribution and box plot show whether the values are clustered, skewed or contain notable outliers.

Key figures and spread

MeanSum / Count

The mean divides the sum of all valid values by their count.

Variance population nSum of squared deviations / n

Use this variance when your values represent the complete data set.

Variance sample n-1Sum of squared deviations / (n - 1)

Use this variance when your values are a sample from a larger population.

Standard deviationSquare root of variance

The standard deviation brings the spread back into the unit of the input values.

Quartiles, percentiles and outliers

IQRQ3 - Q1

The interquartile range describes the width of the middle 50% of values.

Tukey outlier fencesQ1 - 1.5 x IQR and Q3 + 1.5 x IQR

Values outside these fences are marked as outliers in the studio.

Practice

Practical examples

For illustration only – your own calculation is what counts.

One data series

A single measurement series is analysed with key figures, charts and interpretation.

Result: One data series is analysed with mean, median, spread, box plot and histogram.

Two similar series

Two closely matched series show how small differences become visible in comparison.

Result: The comparison view shows that both series are close together.

Four different series

Four series with different locations and spreads are compared compactly.

Result: Four series are displayed with box plots, histograms and key-figure-based comparison.

Outliers

A measurement series contains extreme values that are flagged by the IQR/Tukey rule.

Result: The value 80 is detected as an outlier by IQR/Tukey and explained in the interpretation.

Right-skewed distribution

Many smaller values and a few high values pull the mean upwards.

Result: The mean is above the median because high values pull the series to the right.

Left-skewed distribution

A few low values and many higher values pull the mean downwards.

Result: The mean is below the median because low values pull the series to the left.

Notes

Common mistakes

Confusing mean and median
The mean is interpreted as the 'typical value' even though for skewed distributions the median is the more robust centre. Example: income distribution with a few very high salaries.
For symmetric distributions, mean and median are similar. For skewed distributions or outliers, the median is more robust. The calculator shows both values and flags skewness.
Confusing population and sample variance
The variance is calculated with n instead of n-1 even though the data is only a sample. This leads to a systematic underestimation of spread.
Variance n (population): when all values of the population are present. Variance n-1 (sample): when the values are only a subset. When in doubt, use n-1.
Automatically removing outliers
Values flagged as outliers by Tukey are deleted without review. Sometimes outliers are valid extreme values (e.g. particularly high measurements).
First verify outliers: measurement error, data entry error or genuinely extreme values? The calculator flags outliers but does not automatically remove them.
Drawing broad conclusions from few values
With 5-10 data points, standard deviation and quartiles are calculated and interpreted as reliable. Small samples carry large uncertainties.
The calculator computes the key figures correctly, but their meaningfulness depends on sample size. With fewer than 20 values, outlier detection and quartiles should be interpreted with caution.

FAQ

Frequently asked questions

How do I enter multiple numbers?

You can separate values with spaces, semicolons, line breaks, tabs or CSV-style commas. German decimals like 12,5 are treated as decimal numbers.

Can I compare multiple data series?

Yes. You can add up to six data series with their own names. Only created series are visible; empty or removed series are not analysed.

What is the difference between variance n and n-1?

Variance n treats your values as the complete population. Variance n-1 is the sample variance and corrects upwards when your values are only a sample from a larger population.

What is the difference between standard deviation n and n-1?

The standard deviation is the square root of the respective variance. Standard deviation n belongs to the population, standard deviation n-1 to the sample.

What does median mean?

The median is the middle value of the sorted data series. It is often more robust than the mean when individual very high or very low values occur.

What do quartiles mean?

Quartiles divide the sorted data series into four parts. Q1 is at 25%, Q2 is the median, Q3 is at 75%.

What does IQR mean?

The IQR is the interquartile range Q3 minus Q1. It describes the width of the middle 50% of values and is used for the outlier check.

What are outliers?

The calculator uses the Tukey rule: values below Q1 - 1.5 x IQR or above Q3 + 1.5 x IQR are flagged as outliers.

When can the mean be misleading?

When a few extreme values occur or the distribution is strongly skewed, the mean can deviate significantly from the median. In that case, the median is often the more stable centre.

Limits

Limitations

The calculator computes descriptive statistics (location, spread, distribution). Inferential methods such as hypothesis tests, confidence intervals or regressions are not offered.
Time series analysis (trends, seasonality, autocorrelation) is not included. The order of input values is not interpreted.
A maximum of six data series can be compared in parallel. For larger data sets, specialised statistics tools are more appropriate.
No formal normality test (Shapiro-Wilk, Kolmogorov-Smirnov) is performed. The skewness indication is a hint, not a statistical test.

Sources

Sources and references

DIN 55350 – Begriffe der Qualitätssicherung und StatistikDeutsches Institut für Normung (DIN)
German standard for defining statistical terms: mean, standard deviation, variance, median, quartiles and outliers. Foundation for consistent calculations in quality assurance and statistics.
View sourceRetrieved: 06/15/2025 · Verified on: 07/22/2026
Statistische Methoden und QualitätsstandardsStatistisches Bundesamt (Destatis)
Methodological foundations of the Federal Statistical Office. Describes the application of descriptive statistics to official data and the interpretation of location and spread measures.
View sourceRetrieved: 06/15/2025 · Verified on: 07/22/2026 · Primary source
Exploratory Data Analysis (Tukey, 1977)John W. Tukey / Addison-Wesley
Foundational work on exploratory data analysis. Defines the box plot, the IQR method for outlier detection and the principle of robust statistics.
Retrieved: 06/15/2025 · Verified on: 07/22/2026
ISO 3534 – Statistik: Begriffe und FormelzeichenInternationale Organisation für Normung (ISO)
International standard for statistical terms, symbols and definitions. Establishes the distinction between population and sample parameters.
View sourceRetrieved: 06/15/2025 · Verified on: 07/22/2026

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