A poll says average sleep is 7.2 hours. How sure are we about that number? A confidence interval gives a range that shows how much the average could shift from sample to sample.
The Confidence Interval Calculator builds that range around a sample mean. You enter the mean, the standard deviation, the sample size, and a confidence level. It suits students, researchers, and analysts.
Table of Contents
- What the Calculator Does
- How to Use It, Step by Step
- The Confidence Interval Formula
- Worked Example
- How to Read the Results
- How T Differs From Z at Small Samples
- Limitations and Assumptions
- FAQs About the Confidence Interval Calculator
What the Calculator Does
The tool finds a confidence interval for a mean using the Student t distribution. Results update as you type, with three decimals.
- Confidence Interval Band: the main result, shown as a lower value to an upper value.
- Margin of Error (MOE): how far the interval reaches on each side of the mean.
- Standard Error (SE): the standard deviation divided by the square root of n.
- Lower Bound and Upper Bound: the two ends of the range.
- Critical t Value and Degrees of Freedom (n - 1): the multiplier and the count that set it.
- Copy Confidence Interval copies a text summary, and Reset restores the sample values, including the 95% level.
How to Use It, Step by Step
- Enter the Sample Mean (x̄). The sample value is 50, and a blank field gives a message.
- Enter the Sample Standard Deviation (s), which must be above 0. The sample value is 8.5.
- Enter the Sample Size (n) as a whole number from 2 to 1,000,000,000. The sample value is 100.
- Choose a Confidence Level: 90%, 95%, or 99%. The sample value is 95%.
- Read the band in the large box, then check the six cards below it. A red message names any field that needs fixing, and the results show N/A until you do.
- Press Copy Confidence Interval to paste the summary into a report.
The Confidence Interval Formula
The tool adds and subtracts a margin of error from your sample mean. The margin comes from the standard error and a Student t critical value. That value depends on n - 1 degrees of freedom.
Mean x̄±t (n - 1 df)×s ÷ √n=Interval
- df = n - 1, and SE = s ÷ √n.
- t cuts off an upper tail of 5% at 90%, 2.5% at 95%, and 0.5% at 99%.
- MOE = t × SE.
- Lower = x̄ - MOE. Upper = x̄ + MOE.
The two tails together hold the missing confidence. The page computes t itself, so no table lookup is needed. Its values agree with standard statistical software.
Worked Example
Tomas measured the commute time of 25 coworkers. The mean was 72.4 minutes, and the standard deviation was 11.3 minutes.
| Result | Calculation | Output |
|---|
| Degrees of Freedom | 25 - 1 | 24 |
| Critical t Value at 95% | t with 24 df | 2.0639 |
| Standard Error | 11.3 ÷ √25 | 2.260 |
| MOE at 95% | 2.0639 × 2.26 | ±4.664 |
| Lower Bound | 72.4 - 4.664 | 67.736 |
| Upper Bound | 72.4 + 4.664 | 77.064 |
| Band | Lower to upper | 67.736 to 77.064 |
At 90% confidence, t is 1.7109 and the same data give 68.533 to 76.267. At 99%, t is 2.7969 and the band is 66.079 to 78.721. Higher confidence always means a wider range.
How to Read the Results
A 95% interval means the method captures the true mean in about 95% of repeated samples. It does not mean a 95% chance for this one range.
A smaller margin of error means a more precise estimate. You can shrink it with a larger sample or lower confidence. The standard error falls as n grows, but only by the square root.
The default inputs give 48.313 to 51.687, with a margin of 1.687 and a standard error of 0.850. Their critical t is 1.9842 with 99 degrees of freedom.
How T Differs From Z at Small Samples
You usually know only the sample spread, not the true spread. The t value pays for that doubt, and it grows as the sample shrinks.
| Sample Size n | df | t at 95% |
|---|
| 2 | 1 | 12.7062 |
| 5 | 4 | 2.7764 |
| 25 | 24 | 2.0639 |
| 100 | 99 | 1.9842 |
| 1,000 | 999 | 1.9623 |
The z value for 95% is 1.96. By n = 100 the two nearly agree, and by n = 1,000 they differ in the third decimal.
Small samples matter most. With mean 10, s = 2, and n = 5, the tool gives 7.517 to 12.483. A z multiplier would have given a falsely narrow 8.247 to 11.753.
To find the inputs, use the average calculator and the standard deviation calculator. To plan how many people to survey, try the sample size calculator.
The z-score calculator shows how far a value sits from its mean. The normal distribution calculator covers the bell curve behind both methods. More tools are in the math and science calculators hub.
Limitations and Assumptions
- It builds an interval for a mean, not for a proportion, median, or difference.
- It uses the t critical value with n - 1 degrees of freedom, so small samples are handled. It has no finite population correction.
- It assumes a random sample and data that are roughly normal, or a big n. Very skewed small samples can still mislead.
- The standard deviation must be above 0, and n must be a whole number from 2 to 1,000,000,000. Mean and standard deviation are limited to 1,000,000,000,000 in size.
- Only 90%, 95%, and 99% confidence levels are offered.
- Results show three decimals and are not rounded to match your data units.
FAQs About the Confidence Interval Calculator
What does a 95% confidence interval mean?
If you repeated the study many times, about 95% of the intervals would contain the true mean. It describes the method, not one range.
Does this calculator use a z or a t distribution?
It uses the t distribution with n - 1 degrees of freedom, shown in the Critical t Value card. That is the usual choice when you know only the sample standard deviation.
What happens when I increase the sample size?
The standard error shrinks, so the interval narrows. Quadrupling n cuts the margin of error in half.
Why does 99% confidence give a wider range?
To be more sure of capturing the true mean, you must accept a wider range. For Tomas's data, t rises from 2.0639 to 2.7969.
Should I enter the sample or the population standard deviation?
Enter the sample standard deviation s, as the field label says. It is the usual input when the population value is unknown.
Is the interval the same as the margin of error?
No. The margin of error is the half width. The interval runs from the mean minus that margin to the mean plus it.
This calculator gives general estimates only and is not professional advice. Reviewed for general accuracy by Prof. Dr. Khalil Mudassar, PhD.