Standard Deviation Calculator

How to Use This Standard Deviation Calculator

This calculator makes finding standard deviation quick and accurate. Follow these simple steps:

• Enter your data: Type or paste your numbers into the text area. Separate values with commas, spaces, or line breaks. The calculator accepts any combination of these separators, so you can paste data directly from spreadsheets or other sources.
• Click Calculate: Press the Calculate button or use Ctrl+Enter to process your data. You need at least two numbers to calculate standard deviation.
• Choose your result: The calculator displays both population and sample standard deviation. Use population when you have data for an entire group. Use sample when your data represents a portion of a larger group.
• Copy results: Click any result box to copy the value directly to your clipboard.

The calculator also shows variance (the square of standard deviation), count, mean, sum, and sum of squares to help you verify calculations or use in further analysis.

What is Standard Deviation?

Standard deviation is one of the most important concepts in statistics. It measures how spread out numbers are from their average value (the mean). Think of it as quantifying how much variation or dispersion exists in a data set.

A low standard deviation indicates that data points tend to be close to the mean. For example, if test scores have a mean of 75 and a standard deviation of 5, most scores fall between 70 and 80. A high standard deviation means values are spread over a wider range. If the same test had a standard deviation of 15, scores would be much more scattered, ranging perhaps from 60 to 90 or beyond.

Standard deviation is expressed in the same units as your original data, making it intuitive to interpret. If you are measuring heights in inches, the standard deviation is also in inches. This is a key advantage over variance, which is expressed in squared units.

This measure is used everywhere that data analysis matters. In finance, standard deviation measures investment risk and volatility. In manufacturing, it monitors quality control and process consistency. In science, it quantifies experimental uncertainty. In education, it helps interpret test score distributions. In weather forecasting, it measures temperature variability. Understanding standard deviation gives you a powerful tool for making sense of data in any field.

Population vs Sample Standard Deviation

This calculator provides both types because choosing correctly matters for accurate results:

Population Standard Deviation (represented by the Greek letter sigma): Use this when your data set includes every member of the group you are analyzing. Examples include all employees in a small company, all students in a specific class, or all products manufactured in a particular batch. The formula divides by N, the total number of values.

Sample Standard Deviation (represented by the letter s): Use this when your data is a subset randomly selected from a larger population. This is the more common scenario in research and real-world analysis. Examples include surveying 500 customers to understand thousands, measuring 50 parts to assess a production run of millions, or studying 100 patients to draw conclusions about all patients with a condition. The formula divides by (n-1) instead of n.

The (n-1) denominator in sample standard deviation is called Bessel's correction. It compensates for the fact that a sample tends to underestimate the true population variance. By dividing by a slightly smaller number, we get an unbiased estimate of the population standard deviation.

Standard Deviation Formulas

Population Standard Deviation:

sigma = sqrt( sum((xi - mean)^2) / N )

Sample Standard Deviation:

s = sqrt( sum((xi - mean)^2) / (n - 1) )

Step-by-step calculation:

• Calculate the mean (average) of all values
• Subtract the mean from each value to find the deviation
• Square each deviation (making all values positive)
• Sum all the squared deviations
• Divide by N (population) or n-1 (sample) to get variance
• Take the square root of variance to get standard deviation

Variance is simply the standard deviation squared. While variance is mathematically useful, standard deviation is more interpretable because it uses the same units as the original data.