AP Stats Confidence Interval Calculator

Build a confidence interval for a population mean. Enter your sample mean, standard deviation, and sample size, choose 90%, 95%, or 99% confidence, and this calculator gives you the interval and the margin of error.

What Is a Confidence Interval?

A confidence interval is a range of plausible values for a population parameter, such as a mean or a proportion, calculated from sample data. A 95% confidence interval means that if you repeated the sampling many times, about 95% of the intervals you built would contain the true value. The interval is centered on the sample statistic, and its width is the margin of error on each side.

How to Use This Calculator

1) Type the sample mean. 2) Type the standard deviation, using sigma if it is known, or s if it is estimated from the sample. 3) Type the sample size, n. 4) Choose your confidence level. 5) Press Calculate Interval. Read the lower and upper bounds, and use the interpretation sentence in your answer: you are that confident the interval captures the true population mean.

How to Calculate a Confidence Interval

The formula is: Interval = sample mean ± critical value × (standard deviation / square root of n). The part after ± is the margin of error. Critical values used here are 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%.

Worked example: Sample mean = 50, standard deviation = 10, n = 36, 95% confidence. Standard error = 10 / 6 = 1.67. Margin of error = 1.96 × 1.67 = 3.27. Interval = 50 ± 3.27 = (46.73, 53.27). You are 95% confident the interval from 46.73 to 53.27 captures the true population mean.

Tips and Common Mistakes

• Do not say there is a 95% probability the true mean is in this one interval. The 95% refers to the method over many samples, not to one fixed interval.
• A higher confidence level gives a wider interval, because the critical value is larger.
• A larger sample size gives a narrower interval, because the standard error gets smaller.
• Check conditions first: random sample, independence, and a normal population or large enough sample.
• For a small sample with s instead of sigma, a z interval is too narrow. Use a t interval for that case.

Frequently Asked Questions

What does a 95% confidence interval mean?

If you repeated the sampling method many times, about 95% of the intervals built that way would capture the true population mean.

Why is a 99% interval wider than a 95% interval?

Because you need a larger critical value to be more confident, and a larger critical value makes a larger margin of error.

Should I use z or t for a confidence interval for a mean?

Use z if the population standard deviation is known or the sample is large. Use t with df = n - 1 if the population standard deviation is unknown and the sample is small.

What happens to the interval if n doubles?

The standard error shrinks by a factor of the square root of 2, so the interval gets narrower, but not by half.

Can a confidence interval include a negative number?

Yes, if the data allow it. The interval simply shows the plausible range for the population mean; it is not limited to positive values.

What is the margin of error?

The margin of error is critical value × standard error. It is half the width of the confidence interval.

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