Statistics

Standard Deviation Calculator

A statistical calculator that computes standard deviation, variance, and mean from a dataset. Perfect for analyzing data spread and variability in statistical analysis.

Quick Tips

Click to show tips

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Exam Scores

Analyse variability in a set of student exam scores

Key values: 12 data points · population · moderate spread

Stock Prices

Sample analysis of daily closing prices with outlier detection

Key values: 5 data points · sample · outlier detection

Dataset with Outliers

Identify unusual values in a data set using the IQR rule

Key values: 10 data points · population · 1 outlier

Documentation

Documentation Contents

Standard Deviation Calculator

Measure data spread, compare population vs. sample statistics, and visualize distributions.

This calculator computes mean, median, range, variance, and standard deviation for a dataset. It supports population and sample calculations, optional outlier detection, and multiple visualizations to help you understand how your data is distributed.

Inputs Glossary

A quick reference for the form fields and data entry options.

Input	What to Enter	Notes
Values	Numbers separated by commas, spaces, or new lines	Invalid entries are ignored; at least one valid number is required.
Calculation Type	Population or Sample	Controls whether the denominator is N or N-1.
Detect Outliers	On or Off	Uses the 1.5 x IQR rule to flag outliers.
Text Entry	Paste or type values	Supports commas, spaces, and line breaks.
Tabular Entry	Enter values row by row	Use Apply Data to load the table into the calculator.
Upload CSV	CSV or TXT file	File contents are parsed like text input.
Paste from Spreadsheet	Copy from Excel or Google Sheets	Tabs and new lines are converted into comma-separated values.

How to Use the Calculator

Enter data, choose a calculation type, and review the results.

Select Population or Sample depending on your dataset.
Enter numbers in Text Entry, or switch to Tabular Entry to add values row by row.
Optional: load sample data, paste from a spreadsheet, or upload a CSV/TXT file.
Toggle Detect Outliers to flag unusual values using the IQR rule.
Click Calculate to generate statistics and visualizations.
Use Export to download a CSV or generate a report.

Population vs. Sample

Choose the calculation that matches your dataset.

Population: Use when your data represents the full group. The denominator is N.
Sample: Use when your data is a subset of a larger population. The denominator is N-1 (Bessel's correction).

Sample variance and standard deviation only appear when there are at least 2 values.

Formulas

Core calculations used by the calculator.

Mean

\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

Population Standard Deviation

\sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}}

Sample Standard Deviation

s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}}

Variance is the square of the standard deviation ( $\sigma^2$ or $s^2$ ).

Interpreting Results

What the results cards and summary mean.

Count: Number of valid values used in the calculation.
Mean, Median, Range: Basic measures of central tendency and spread.
Population Variance and Standard Deviation: Always shown.
Sample Variance and Standard Deviation: Shown only when there are at least 2 values and Sample is selected.

Results in the cards are rounded to 4 decimal places.

Visualizations

Explore the distribution using multiple chart types.

Histogram: Frequency distribution with mean and median reference lines. You can choose the bin method (Sturges, Scott, or Freedman-Diaconis).
Dot Plot: Individual values with mean and standard deviation reference lines. Outliers can be highlighted.
Normal Distribution: Normal curve centered at the mean with standard deviation markers.
Box Plot: Quartiles, median, whiskers, and outliers based on IQR.

Visualizations require at least 2 values; otherwise an empty state is shown.

Exports

Download results for reporting or sharing.

Export CSV: Includes full dataset, quartiles, IQR, percentages within 1-3 SD, and detected outliers (if enabled).
Generate Report: Creates an HTML report with summary tables and interpretation notes.

Limitations

Important context for interpreting standard deviation.

Standard deviation is sensitive to outliers, which can inflate the spread.
The 68-95-99.7 interpretation assumes data is roughly normal; skewed datasets may not follow it.
Very small samples can produce unstable variance and standard deviation estimates.
Outlier detection flags values but does not remove them from the calculation.

Frequently Asked Questions

What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides by N, the total number of data points, when the dataset includes every member of the group. Sample standard deviation (s) divides by N-1 (Bessel’s correction) when the dataset is a subset of a larger population. Dividing by N-1 corrects for the bias that arises because a sample tends to underestimate the true variability.

When should I use standard deviation versus variance?

Standard deviation is expressed in the same units as the original data, making it easier to interpret. Variance (σ² or s²) is the square of the standard deviation and is useful in mathematical derivations and ANOVA. Use standard deviation for reporting and communication; use variance when performing further statistical calculations.

What does the 68-95-99.7 rule mean?

For data that follows a normal (bell-curve) distribution, approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This rule is an approximation and does not hold for skewed or multimodal distributions.

How do outliers affect standard deviation?

Outliers increase standard deviation significantly because each value’s contribution is squared in the variance calculation. A single extreme value can inflate the result. Consider using the interquartile range (IQR) as a more robust measure of spread, or investigate whether outliers are data errors before deciding to include or exclude them.

Can standard deviation be zero?

Yes, but only when every value in the dataset is identical. In that case, all deviations from the mean are zero, so the variance and standard deviation are both zero. A standard deviation of zero indicates no variability at all.

How many data points do I need for a reliable standard deviation?

There is no strict minimum, but sample standard deviation requires at least 2 values (since dividing by N-1 is undefined for N=1). In practice, small samples (under 30) produce unstable estimates. For research, power analysis can help determine the sample size needed for your desired precision.

What is the coefficient of variation and when should I use it?

The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage: CV = (s / x̄) × 100%. It is useful for comparing variability between datasets with different units or vastly different means, such as comparing the variability of heights (in cm) with weights (in kg).