Standard Deviation Calculator

Run quick descriptive statistics on a list of numbers.

Mean

18

Median

15.5

Mode

—

Std deviation (sample)

13.4907

Variance

182

Count

6

Range

38

Min

4

Max

42

Deviation from Mean

Values Comparison

Values Comparison

#ValueDeviation from Mean% of Total Deviation
#14-1424.14%
#28-1017.24%
#315-35.17%
#416-23.45%
#52358.62%
#6422441.38%

Understanding Standard Deviation

The standard deviation calculator computes key descriptive statistics for any data set including mean, median, mode, variance, standard deviation, range, and quartiles. Standard deviation is one of the most important measures in statistics because it quantifies how spread out your data points are from the average. A low standard deviation means the data points cluster closely around the mean, while a high standard deviation indicates they are spread over a wider range. This calculator computes both population and sample standard deviation, which use slightly different formulas depending on whether your data represents an entire population or a sample drawn from a larger population. The distinction matters because sample standard deviation uses a correction factor to provide an unbiased estimate. Enter your data points separated by commas and receive a comprehensive statistical summary instantly. The calculator also identifies outliers, shows the five-number summary used in box plots, and calculates the coefficient of variation for comparing the spread of different data sets. Use this free tool for quality control analysis, scientific research, financial risk assessment, academic assignments, or any situation where understanding the distribution and variability of your data is important. The clear presentation of results makes statistical analysis accessible to everyone.

Practical Example

Scenario: Let's walk through a practical example of standard deviation and variance to see how this works in practice.

Step 1 — Gather your data: Identify the key values you need for the calculation. Make sure all measurements use consistent units.

Step 2 — Enter your values: Input the numbers into the calculator fields above. Double-check each entry for accuracy.

Step 3 — Review the result: The calculator displays your result instantly. Compare it with your expectations — if the number seems off, verify your inputs.

Pro tip: Run the calculation with slightly different inputs to see how sensitive the result is to each variable. This sensitivity analysis helps you understand which factors matter most for your specific situation.

Frequently Asked Questions

What is standard deviation?

Standard deviation measures how spread out values are around the mean — a low value means values cluster near the mean, high means they're spread.

How is standard deviation calculated?

Take the square root of the variance, where variance is the average of squared differences from the mean: σ = √(Σ(x − μ)² / N).

What's the difference between population and sample SD?

Population SD divides by N; sample SD divides by N − 1 (Bessel's correction) to give an unbiased estimate from a sample.

What if I get a different answer when calculating manually?

First check your order of operations (PEMDAS/BODMAS), then verify your units are consistent. Common errors include rounding too early, sign mistakes, and incorrect formula application. Use this calculator to verify each step of your work.

Are there shortcuts or mental math tricks?

Yes! Many mathematical operations have estimation shortcuts. For example, squaring numbers ending in 5, using the distributive property, or applying benchmark fractions. While shortcuts help with estimates, always use exact calculations for important work.

Disclaimer: This calculator provides estimates for informational purposes only. Actual results may vary. Consult a qualified professional for personalized advice.

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