Everyday central limit theorem problems come down to one relationship between 3 figures. This calculator takes population standard deviation, sample size (n), population mean and turns them into sample standard deviation (s) and sample mean (–x–), with the working shown so you can check the result rather than simply trust it.
What Is the Central Limit Theorem Calculator?
It is built around sample standard deviation (s), sample mean (--x--), and it works the same way whether you are checking a homework answer, confirming a colleague's figure, or working something out for yourself. Everything runs in your browser, so nothing you enter is uploaded anywhere.
How the Central Limit Theorem Calculation Works
Nothing is hidden: each output is worked out from the values above it, which makes it easy to see where a surprising answer came from.
The relationship between population standard deviation, sample size (n), population mean and sample standard deviation (s), sample mean (--x--) is fixed, so the same figures always give the same answer. That makes it a good cross-check: run the numbers a second way and see whether the result agrees.
Getting a Result You Can Trust
For anything that matters, write the inputs down next to the result. A calculation you can reproduce is worth far more than a number you cannot explain, and it takes a moment to record population standard deviation and sample size (n) and population mean alongside sample standard deviation (s), sample mean (--x--).
Frequently Asked Questions
What do I need to work out sample standard deviation (s)?
Enter population standard deviation, sample size (n), population mean. The calculator then works through the relationship and reports sample standard deviation (s), sample mean (--x--).
What does the central limit theorem actually calculate?
It takes population standard deviation and sample size (n) and population mean and converts them into sample standard deviation (s), sample mean (--x--) using the standard relationship for this problem. Each step is applied in turn, so every output is traceable back to the numbers you typed in.
How should I enter population standard deviation?
Use the value you actually have, in the unit shown next to the field. Enter every value in the same unit system. The calculator assumes the figures you supply are consistent, so it cannot correct a mistake that happens before the calculation starts.
Why does a small change in one value move the answer so much?
That happens when an output depends on a ratio, because dividing by a small number magnifies any change in it. Check the units first, then check whether the value you entered is in the range the relationship was designed for.
Can I use this for a quick estimate as well as a precise answer?
Yes. The calculator is built for quick checks, so round your inputs to a sensible number of significant figures and the result will be just as useful. If you need more precision, keep the same method and add digits to the inputs rather than adjusting the method itself.