The square root of the variance; spread expressed in the same units as the data.
Example
Common mistakes
Learn it in the course
Measures of spread: range, SD and quartiles · Two couriers with the same average delivery time, and why only one keeps the promise: measures of spread from range and variance to standard deviation (calculated step by step), quartiles, IQR and the coefficient of variation.
Sample vs population and the normal distribution · Sample vs population: why we study samples to learn about all users, why the bell-shaped normal distribution shows up everywhere, and how the central limit theorem ties them together.
95% confidence intervals · Building a 95% confidence interval from its formula and margin of error, so you report a range of plausible values instead of one number.
Paired t-test: before and after · Comparing the same users before and after a change with a paired t-test or the Wilcoxon signed-rank test.
Simple linear regression · Simple linear regression: predicting one metric from another with a line of best fit, and checking the fit with R squared.
The standard deviation (SD) measures how far values typically sit from the mean. A small SD means the values are packed tightly around the mean; a large one means they are scattered. Unlike the variance, it is in the same units as the data (hours, dollars, orders), which is why it is the spread measure people actually report: "delivery takes 24 ± 8 hours".
Step by step, for a sample of n values:
Compute the mean xˉ.
Subtract it from every value to get the deviationsxi−xˉ. They always sum to zero, so you can't simply average them.
Square each deviation and add the squares up.
Divide by n−1. That is the sample variance s2.
Take the square root.
s=n−1∑i=1n(xi−xˉ)2
If your data is the whole population rather than a sample, divide by n instead. In product analytics you almost always have a sample, so n−1 is the default: STDEV.S in Sheets or Excel, as opposed to STDEV.P.
For roughly bell-shaped data (see normal distribution) about 68% of values fall within one SD of the mean and about 95% within two. On skewed data that rule breaks down, and because deviations are squared, a few outliers can inflate the SD a lot. There the interquartile range is the safer partner of the median. To compare spread across metrics on different scales, divide the SD by the mean to get the coefficient of variation. And don't confuse the SD, the spread of individual values, with the standard error, the uncertainty of a mean.
Daily orders from CatChow's paid social ads over five days: 32, 24, 34, 28, 32.
Mean: xˉ=(32+24+34+28+32)/5=150/5=30 orders.
Deviations: +2,−6,+4,−2,+2 (sum: 0).
Squares: 4,36,16,4,4; sum of squares =64.
Sample variance: s2=64/(5−1)=16 orders².
Standard deviation: s=16=4 orders.
Report it as 30 ± 4 orders a day: a typical day lands about 4 orders away from 30. If these five days were the entire population you cared about, you would divide by 5 instead: 64/5=12.8≈3.58. With only five values the population version comes out about 11% smaller, which is why it matters which function your dashboard uses.
Confusing SD with standard error. SD describes how individual days or orders vary; the standard error describes how precisely you know the mean, and it shrinks as n grows.
Applying the 68/95 rule to skewed data. CatChow's 90-day spend per new customer has a mean of $50 and an SD of about $47.7, so "mean ± 2 SD" goes down to about −$45, a spend nobody can have.
Mixing STDEV.S and STDEV.P. On small samples the population version is noticeably smaller. Pick the sample version unless you really have every observation.
Reporting the mean without the SD. Two couriers can both average 24 hours while one ranges from 14 to 38. Show "24 ± 8 hours", not just "24".
Comparing raw SDs of metrics on different scales. An SD of $150 in ad spend and 9 in orders can't be compared directly; use the coefficient of variation.
