Enter numbers separated by commas or spaces. Get standard deviation, mean, variance, 95% CI, frequencies, and a quick histogram—with clear steps you can expand.
Input
Separate with commas, spaces, or new lines. Non-numeric tokens are ignored. Results update automatically.
Result summary
Paste or type numbers to see mean, variance, and standard deviation.
Step-by-step explanation
Enter data to see steps.
Confidence interval for the mean
A confidence interval is a range of plausible values for the population mean based on your sample. Wider intervals reflect more uncertainty. Below, SE = s / √n (sample standard deviation).
Need at least 2 values for a confidence interval.
95% row: uses Student's t when n < 30 and z ≈ 1.96 when n ≥ 30. Other levels use normal critical values (common quick approximation).
Frequency table
No data yet.
What is standard deviation?
Standard deviation measures how spread out numbers are around the mean. Small standard deviation: values cluster near the average. Large standard deviation: values are more dispersed. It is the square root of the variance, so it is in the same units as your original data.
How to calculate standard deviation?
Find the mean, subtract it from each value, square each difference, average those squares (using n or n−1 depending on population vs sample), then take the square root. This calculator does those steps for you and shows a readable breakdown in the collapsible section.
Sample vs population standard deviation
Use population when your list is the entire group you care about. Use sample when the list is a subset and you want to estimate variability in the wider population—then variance divides by n − 1 instead of n.
Examples
Data 2, 4, 4, 4, 5, 5, 7, 9 — mean 5; population SD ≈ 2.
Test scores that are all close to 75 have a smaller SD than scores spread from 40 to 100.
FAQs
What is the difference between sample and population standard deviation?
Population divides by n (whole group). Sample divides by n−1 (Bessel’s correction) to better estimate spread when data are only a sample from a larger population.
What does a confidence interval tell me?
It gives a range that plausibly contains the true population mean, based on your sample mean and spread. The 95% row is the one most often reported.
Why is my histogram empty?
You need at least one numeric value. If all values are identical, you will see a single bar.
