Confidence Interval Calculator (Mean)

Where it is used

• Teachers: A teacher enters every mark from a class test to see the average score and the range the class's true ability likely falls within.
• Small-business owners: A cafe owner types in the satisfaction ratings customers gave this week to find the average and check how dependable that average really is.
• HR staff: An HR officer enters staff survey scores about workplace happiness so they can report the average with a sensible margin around it.

When to use the t distribution

When you estimate a confidence interval for an average from a sample, the t distribution is the correct choice, especially for smaller samples. It is slightly wider than the normal distribution to account for the extra uncertainty of estimating the standard deviation from the data.

When should you use it?

Use it whenever you have a list of numbers that all measure the same thing, such as test marks, ratings out of ten, ages, or waiting times in minutes. It works best when you only asked or measured some of the people, but you want to say something sensible about the whole group. The calculator gives you the average of your numbers plus a trustworthy range around it. If your data is words or yes and no answers rather than numbers, this is not the right tool.

What does the result mean?

The calculator shows two things: the average of the numbers you entered, and a range below and above it. At the usual 95% setting, you can be about 95 percent confident that the true average for the whole group sits somewhere inside that range. A narrow range means your average is precise and dependable. A wide range means there is more uncertainty, usually because you have few responses or your numbers are very spread out. There is no single perfect width, but narrower is generally better.

Mistakes to avoid

Do not type words, categories, or yes and no answers, because this tool only works with number measurements. Do not mix different things in one list, such as ages and prices together, since the average would be meaningless. Avoid drawing strong conclusions from only a handful of numbers, as the range will be very wide and unreliable. Remember to include the decimal point where it matters, and remove any currency symbols or units. Finally, do not confuse the confidence range with the highest and lowest values in your data.

How to use this calculator

1. Enter your sample mean in Sample mean.
2. Enter the sample standard deviation and the number of observations in their respective boxes.
3. Choose the confidence level, such as 95%.
4. Read the confidence interval around the mean. It describes sampling uncertainty for the population mean under the model assumptions, rather than the spread of individual scores.

Worked example

For the ten ratings 8, 7, 9, 6, 8, 7, 9, 8, 7, 9, the mean is 7.8 and the sample standard deviation is about 1.033. Enter those values and a sample size of 10. The 95% t interval is approximately 7.06 to 8.54, under the model assumptions.

Frequently asked questions