Free Online Calculators
Standard deviation & variance for population & sample
This calculator computes the standard deviation and variance for both a full population (σ) and a sample (s) — just enter your numbers and get results instantly. Standard deviation measures how spread out values are around the mean. For samples, the formula divides by n−1 (Bessel's correction) rather than n to give an unbiased estimate.
Whether you're a student crunching numbers for a statistics class, a researcher analyzing experimental data, or a business analyst measuring performance variability, understanding how spread out your data is can be just as important as knowing its average. Our Standard Deviation Calculator gives you instant, accurate results for both population and sample standard deviation, along with variance and mean — all displayed side by side so you can compare them at a glance. No spreadsheet software required, no manual formula gymnastics — just paste or type your numbers and get the full statistical picture in seconds.
Standard deviation is a measure of how much individual values in a dataset differ from the mean. A low standard deviation means your data points cluster tightly around the average; a high standard deviation means they're spread widely apart. This single number is one of the most powerful tools in statistics because it tells you about consistency, risk, and reliability.
For example, two investment portfolios might both earn an average return of 8% per year, but one might fluctuate between 6% and 10% while another swings from −5% to 21%. The standard deviation immediately reveals which portfolio carries more risk — even though their means are identical.
Our calculator also outputs variance, which is simply the square of the standard deviation. Variance is used heavily in more advanced statistical techniques like ANOVA and regression analysis, so having it readily available is a bonus.
This calculator computes two versions of standard deviation simultaneously, and understanding the difference is essential for using statistics correctly.
Use this when your dataset includes every member of the group you care about — for instance, all test scores from a single classroom, or the heights of every player on one sports team. The formula divides by n (the total number of values):
Use this when your dataset is a sample drawn from a larger population — for example, surveying 500 voters to estimate how an entire country thinks, or measuring 30 manufactured parts to estimate quality across thousands. The only difference is dividing by n − 1 instead of n. This adjustment, known as Bessel's correction, compensates for the fact that a sample tends to underestimate the true variability of the full population:
Behind the scenes, this calculator uses the Welford one-pass algorithm rather than the naive two-pass approach. The naive method — calculating the mean first, then summing squared deviations — can suffer from significant floating-point rounding errors when dealing with very large numbers or datasets where values are clustered close together. Welford's algorithm processes each number incrementally, maintaining running totals that keep precision high regardless of the data. The result: you can trust the output even with hundreds of large or unusual values.
A teacher records the following quiz scores for a class of 8 students: 72, 85, 90, 68, 95, 78, 88, 74. Since these are all the scores in this specific class (not a sample from a larger group), she uses the population standard deviation. The mean works out to approximately 81.25, and σ ≈ 8.96. This tells her the typical student scored within about 9 points of the class average — useful for deciding whether to adjust the difficulty of future quizzes.
An engineer randomly selects 10 bolts from a production run of 50,000 and measures their diameter in millimeters: 10.02, 9.98, 10.01, 10.03, 9.97, 10.00, 10.02, 9.99, 10.01, 10.00. Because this is a sample, he uses the sample standard deviation (s ≈ 0.019 mm). If the manufacturing tolerance allows ±0.05 mm, a standard deviation well under that threshold confirms the process is running within acceptable limits.
An investor tracks the monthly returns (%) of a stock over the past year: 2.1, −1.3, 3.5, 0.8, −2.0, 4.1, 1.2, −0.5, 3.0, 2.8, −1.8, 1.9. These 12 months represent a sample of the stock's overall long-term behavior, so she uses sample standard deviation (s ≈ 1.96%). A higher s compared to a benchmark fund signals greater volatility — meaning potentially higher reward but also higher risk.
Population standard deviation (σ) is used when your data represents every member of the group you're studying. Sample standard deviation (s) is used when your data is a subset of a larger group. The mathematical difference is that σ divides by n, while s divides by n − 1 (Bessel's correction). Using the wrong version can lead to slightly misleading conclusions, especially with small datasets — so choose based on whether your numbers are the whole story or just a part of it.
Variance and standard deviation both measure spread, but they're used in different contexts. Standard deviation is expressed in the same units as your original data (e.g., dollars, seconds, centimeters), making it intuitive for interpretation. Variance, being the squared version, is essential in advanced statistical methods such as ANOVA, linear regression, and statistical hypothesis testing, where working with squared terms simplifies the underlying mathematics. Our calculator provides both so you're ready for either situation.
The calculator is designed to handle datasets of virtually any practical size — from as few as 2 values up to several thousand. For very large datasets, the Welford algorithm ensures results remain numerically stable and accurate. If you're working with an extremely large dataset (tens of thousands of values), copying from a spreadsheet and pasting directly into the input field is the fastest approach. The calculator processes even large inputs in a fraction of a second.