Confidence interval calculator
Type the sample mean, standard deviation and size (or paste the data) and choose the confidence level. You get the interval, the margin of error and the critical value used.
- Lower limit
- 59.7
- Upper limit
- 65.1
- Margin of error
- 2.71
- t
- 2.09
Show the working
σ is unknown, so estimate it with sn−1 and use the t-distribution with 19 degrees of freedom: t = 2.09.
Standard error = s ÷ √n = 5.80 ÷ √20 = 1.30
x̄ ± t × standard error = 62.4 ± 2.09 × 1.30 = 62.4 ± 2.71
59.7 < μ < 65.1
Answers are rounded to 3 significant figures, which is what IB papers ask for unless a question says otherwise.
How to do it by hand
- Decide which distribution: if the population standard deviation σ is known, use z; if you estimate it from the sample with s (divide by n − 1), use t with n − 1 degrees of freedom.
- Find the critical value that leaves (1 − level) ÷ 2 in each tail, for example z = 1.960 for 95%.
- The interval is x̄ ± critical value × (standard deviation ÷ √n).
Worked example
A random sample of 16 phone batteries lasts a mean of 12.4 hours, with unbiased standard deviation 1.8 hours. Find a 95% confidence interval for the mean lifetime of all the batteries.
Solution
σ is unknown, so estimate it with sn−1 and use the t-distribution with 15 degrees of freedom: t = 2.13.
Standard error = s ÷ √n = 1.80 ÷ √16 = 0.450
x̄ ± t × standard error = 12.4 ± 2.13 × 0.450 = 12.4 ± 0.959
11.4 < μ < 13.4
Answer: 95% confidence interval: 11.4 < μ < 13.4
Every number in this example is worked out by the same code as the tool above, and the code is tested against independent results from Python's SciPy library.
Where you need it
| AA SL | Not in the syllabus |
|---|---|
| AA HL | Not in the syllabus |
| AI SL | Not in the syllabus |
| AI HL | Confidence intervals for a population mean (t, or z when σ is known). |
On your calculator
In the exam you use your own calculator, so practise it too. Our guides give the exact keystrokes for the Casio fx-CG50, TI-Nspire CX II and TI-84 Plus CE, with a worked example to check against:
Common mistakes
- Using z when σ is unknown. With s from the sample, use t with n − 1 degrees of freedom.
- Dividing by n instead of √n in the standard error.
- Saying “there is a 95% probability that μ is in this interval”. The method captures μ in 95% of samples; this one interval either contains it or does not.
Notes and practice
Questions students ask
Why does a 99% interval come out wider than a 95% one?
To be more confident of catching the true mean you need a wider net: the critical value grows from about 1.96 to about 2.58 (for z).
How can I make a confidence interval narrower?
Take a bigger sample. The width shrinks with √n, so four times the sample size halves the width.
What does the confidence level mean?
If you took many samples and made an interval from each, that percentage of the intervals would contain the true mean.