Standard Deviation Calculator

Calculate standard deviation, variance, mean, median, mode, and range from your data set with steps

Data Input

Separated by comma, space, or newline

Quick Examples

Enter Your Data

Enter a set of numbers separated by commas to calculate standard deviation, variance, mean, median, mode, and range.

About Standard Deviation

What is Standard Deviation?

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

Population vs. Sample

Population standard deviation (σ) is used when your data represents an entire population. Sample standard deviation (s) is used when your data is a sample of a larger population, dividing by (n-1) instead of n to provide an unbiased estimate. Use sample standard deviation in most real-world scenarios.

Real-World Applications

Standard deviation is widely used in finance to measure investment risk, in manufacturing for quality control, in research for data analysis, and in education for grading on a curve. It helps determine how reliable the mean is as a representation of the data.

Interpreting Results

In a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This is known as the empirical rule or 68-95-99.7 rule.

Variance Formula & Standard Deviation Formula

Mean (Average): x̄ = Σx / n
Population Variance: σ² = Σ(x - x̄)² / n
Sample Variance: s² = Σ(x - x̄)² / (n - 1)
Population Standard Deviation: σ = √[Σ(x - x̄)² / n]
Sample Standard Deviation: s = √[Σ(x - x̄)² / (n - 1)]

How to Use the Standard Deviation Calculator

Calculate standard deviation in four simple steps

1

Enter Your Data

Type or paste your data set into the text area. Numbers can be separated by commas, spaces, or new lines.

2

Choose Type

Select "Population" if your data represents the entire group, or "Sample" if it represents a subset of a larger population.

3

Calculate

Click the "Calculate Standard Deviation" button to compute all statistical measures.

4

Review Results

View the results including mean, variance, standard deviation, median, mode, and range with detailed steps.

Standard Deviation Calculator FAQ

When should I use population vs. sample standard deviation?
Use population standard deviation (σ) when your data set includes every member of the population you are studying. Use sample standard deviation (s) when your data is a subset (sample) of a larger population. In practice, sample standard deviation is used more often because we rarely have data for an entire population. The key difference is that sample standard deviation divides by (n-1) instead of n.
What does a high standard deviation mean?
A high standard deviation means that the data points are spread out over a wide range of values, indicating high variability. For example, if test scores have a high standard deviation, it means students' scores varied widely from the average. Conversely, a low standard deviation means the data points are clustered closely around the mean.
Can standard deviation be negative?
No, standard deviation cannot be negative. Since it is calculated as the square root of variance (which is the average of squared deviations), it is always a non-negative number. A standard deviation of zero means all data points are identical.
What is the difference between variance and standard deviation?
Variance is the average of the squared differences from the mean, measured in squared units. Standard deviation is the square root of the variance, expressed in the same units as the original data. Standard deviation is generally more useful because it is in the same units as the data, making it easier to interpret.
How many data points do I need for a reliable standard deviation?
While you can calculate standard deviation with as few as 2 data points, a minimum of 30 data points is generally recommended for statistical reliability. For small samples (less than 30), the sample standard deviation may not accurately estimate the population standard deviation. Larger samples provide more reliable and stable estimates.

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