Enter data points and see standard deviation, mean, and normal distribution curve with interactive visualization.
Paste or type a list of numbers and this tool returns the mean and standard deviation, then draws a normal distribution curve built from those two values so you can see where your data sits against the bell shape. The curve is the part most people misread: it is the smooth normal model fitted from your mean and standard deviation, not a picture of your actual data. Skewed entries, outliers, or tiny samples make the curve spread wider than most of your points. Also watch the sample-versus-population setting - it changes the number.
A concrete input and expected output from the current implementation.
Input
2, 4, 4, 4, 5, 5, 7, 9
Expected output
Mean 5.0. Standard deviation 2.0 with the population formula (divide by n), or 2.14 with the sample formula (divide by n - 1). Bell curve centered at 5.0, and when the standard deviation is 2.0 the one-standard-deviation band spans 3.0 to 7.0.
The eight entries sum to 40, so the mean is 5. The squared deviations from 5 are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32: 32/8 = 4, whose square root is 2, and 32/7 = 4.57, whose square root is 2.14.
The mean is the sum of the entries divided by their count. Each entry's distance from the mean is squared, the squares are averaged, and the standard deviation is the square root of that average. Squaring first stops positive and negative deviations from canceling out, and the square root returns the result to the original units of the data. Dividing by n rather than n - 1 selects the population or sample version of the calculation.
The curve is the normal probability density function with your mean and standard deviation as its two parameters: it is centered on the mean, the standard deviation sets how wide the hump is, and the area under the curve between mean minus one standard deviation and mean plus one standard deviation is about 68 percent of the total. Your data points are drawn beneath the curve so you can judge how well the normal model fits them.
That depends on the setting the tool offers. The population formula divides by n and the sample formula divides by n - 1, and the sample version is the common default because most entered data is a sample of something larger. If your entries cover the entire group you care about, use the population version; the two converge as the number of entries grows.
Probably not. For roughly normal data about two-thirds of points fall within one standard deviation, but small samples routinely show all points inside it, and the curve describes the fitted model rather than a requirement on your data. The more telling signs are strong skew or points lying far beyond three standard deviations.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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