Calculate population or sample standard deviation, variance, mean, sum, min, max, median, and confidence intervals. Enter any set of numbers to get a full statistical breakdown with step-by-step workings and a distribution histogram.
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Standard deviation (σ or s) measures how spread out the values in a data set are relative to the mean. A low standard deviation means data points cluster tightly around the mean; a high value means they are widely dispersed.
Population standard deviation (σ) is used when you have data for every member of the group. It divides the sum of squared deviations by N.
Sample standard deviation (s) is used when your data is a random sample drawn from a larger population. It divides by N − 1 (Bessel's correction) to produce an unbiased estimate of the population standard deviation.
Variance is the square of the standard deviation. Margin of error (confidence interval) uses the standard error of the mean (SEM = σ / √N) to express the uncertainty around the sample mean at various confidence levels.
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation means that the values tend to be close to the mean (average) of the set, while a high standard deviation means that the values are spread out over a wider range. It is one of the most widely used statistics in science, finance, quality control, and social research.
There are two versions of standard deviation, and choosing the wrong one is a common mistake that can significantly affect your results.
Use population standard deviation when your data set contains every member of the group you are studying. For example, if you have the test scores of every student in a specific class, you would use population standard deviation. The formula divides the sum of squared deviations by N (the total number of values).
Use sample standard deviation when your data is a subset (sample) drawn from a larger population. This is the more common scenario in research. For example, if you surveyed 200 people to estimate the opinions of a city of 1 million, you would use sample standard deviation. The formula divides by N-1 (Bessel's correction) to account for the fact that a sample tends to underestimate the true population variance.
To calculate standard deviation manually: 1) Find the mean of your data set. 2) Subtract the mean from each data point and square the result. 3) Find the average of those squared differences (divide by N for population, N-1 for sample). 4) Take the square root of that average. The result is your standard deviation.
In finance, standard deviation is used to measure the volatility of an investment. A stock with a high standard deviation of returns is considered riskier than one with a low standard deviation. In manufacturing, standard deviation is central to quality control — a process is considered in control when measurements fall within three standard deviations of the mean (the 99.7% rule, or Six Sigma principle).
In education, standardized test scores are often reported as z-scores, which express how many standard deviations a student's score is above or below the mean. A z-score of +1.5 means the student scored 1.5 standard deviations above average.
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