Statistics & Data Science · pattern F, Dataset
Enter a value, mean and standard deviation for the z score and the probability below, above and between — with the empirical rule figures alongside.
z = (x − μ) ÷ σ, then read the standard normal table
Standardising converts any normal distribution to the standard normal, which is why one table serves every problem. The cumulative probability here uses a numerical approximation accurate to about seven decimal places.
0.84134
The central limit theorem: sums and averages of many independent contributions tend toward a normal distribution regardless of the shape of the originals. This is why measurement errors and sample means are approximately normal, and also why individual quantities often are not — incomes, waiting times and city sizes are all strongly skewed, and treating them as normal produces confident nonsense.
If the number is not the part you are stuck on, that is what the service is for — a specialist who explains the working, not just the answer.
Statistical output here is for coursework and learning. Reported p-values and intervals assume the conditions stated on the page; nothing here checks whether those conditions hold for your data.