Standard deviation calculator
Paste your measurements; get the mean, the spread and the standard deviation, sample and population, in plain English. Free, no signup, nothing leaves your browser.
What standard deviation actually tells you
The standard deviation is the typical distance between a reading and the mean, expressed in the same units as the data. For the twenty shaft diameters in the sample dataset above, the mean is 25.0000 mm and the sample standard deviation is 0.0205 mm, so a normal part sits about two hundredths of a millimetre either side of nominal.
Because it carries the units of the measurement, it is directly comparable to a tolerance, a drawing note or last month’s figure for the same characteristic. A larger standard deviation means a wider process; a smaller one means the readings cluster tightly around the mean, whether or not that mean is where you want it.
Sample vs population standard deviation
Both figures measure the same thing; they differ only in the divisor. The population formula divides by n, the sample formula divides by n minus 1. That correction exists because a sample’s own mean sits closer to the sample than the true process mean would, which makes the raw spread slightly too small. Dividing by n minus 1 compensates.
In manufacturing, inspection data is almost always a sample: you measured twenty parts out of a run that is still going. The parts you did not measure exist, so quote the sample figure, 0.0205 mm in the sample dataset. The population figure, 0.0200 mm, is right only in the rare case where the pasted values are literally every item in a closed batch and nothing remains to measure.
Standard deviation, variance and coefficient of variation
Variance is the standard deviation squared, so it holds exactly the same information in squared units: 0.0205 mm gives a sample variance of 0.000421 mm². Variance is convenient in the arithmetic behind capability studies and Gage R&R, because variances from independent sources can be added; standard deviation is what you quote to people, because it stays in millimetres.
CV = s ÷ mean × 100, so 0.0205 ÷ 25.0000 × 100 = 0.08%
The coefficient of variation expresses the spread as a percentage of the mean, which strips out the units. That makes it the one figure you can use to compare characteristics on different scales (a 25 mm diameter against a 3 kg weight against a 180 °C temperature) and to answer questions like which feature on the part is relatively the most variable.
Mean, median and the shape of your data
The median is the middle reading; the mean is the balance point. When they sit close together, as they do in the sample dataset at 25.0000 mm, the distribution is roughly symmetric and a single standard deviation describes the spread honestly. When the mean sits well above the median, a few large readings are pulling it up (a right skew), often a sign of a tool digging in, a fixture slipping, or two streams of parts mixed into one column.
Skew matters before you go further. Capability indices and the familiar percentage-within-limits rules assume a roughly normal distribution, so a strongly skewed dataset should be investigated rather than summarised. Read the histogram alongside the numbers: one lonely bar at the far end tells you more than any single statistic.
From spread to capability
Standard deviation on its own says how much a process varies, not whether that is acceptable. Add the specification limits and the same spread becomes capability: Cp compares the tolerance width to the spread, and Cpk also accounts for how far the mean has drifted off centre. If you have your upper and lower limits to hand, take these same readings to the Cp/Cpk calculator and you will have the capability picture in a few seconds.