Confidence Interval for Mean Calculator

Calculator

Sample Mean

Standard Deviation

Sample Size

Deviation Type

Critical Value Method

Confidence Level

Custom Confidence Level

Interval Type

Population Size Optional

Hypothesized Mean Optional

Decimal Places

Raw Data Optional

Example Data Table

Case	Sample Mean	Standard Deviation	Sample Size	Confidence	Method
Exam scores	72.4	8.6	36	95%	T interval
Delivery time	4.8	1.2	64	99%	T interval
Machine output	101.3	3.5	100	90%	Z interval

Formula Used

For a two sided interval, the calculator uses this structure:

Confidence interval = sample mean ± critical value × standard error

Standard error = standard deviation ÷ square root of sample size

If finite population correction is used, standard error is multiplied by:

FPC = square root of ((population size - sample size) ÷ (population size - 1))

The t method uses degrees of freedom equal to sample size minus one. The z method uses the normal critical value.

How to Use This Calculator

Enter the sample mean, standard deviation, and sample size.
Or paste raw observations into the raw data box.
Select whether the standard deviation is sample based or known.
Choose a confidence level or enter a custom percentage.
Select two sided, lower bound, or upper bound output.
Add population size only when sampling without replacement.
Press Calculate to view the interval above the form.
Use CSV or PDF buttons to export the same report.

Confidence Interval for Mean Guide

Why This Interval Matters

A confidence interval for a mean gives a practical range for an unknown population average. It is more useful than a single estimate because sampling always carries random error. This calculator supports summary data and raw values. You can enter the sample mean, standard deviation, and sample size. You can also paste observations separated by commas, spaces, or lines. The tool then computes the sample mean and sample standard deviation automatically.

Choosing the Right Method

The calculator chooses the common method for your data. Use the t method when the population standard deviation is unknown. That is the normal case for small samples and many surveys. Use the z method when the population standard deviation is known, or when your course asks for a normal critical value. The auto option selects z for known population deviation and t for sample deviation.

What Changes the Width

The margin of error depends on three items. It uses the critical value, the standard error, and any finite population correction. A larger confidence level gives a wider interval. A larger sample size usually makes the interval narrower. A larger standard deviation makes it wider. If you sample without replacement from a small finite population, the correction can reduce the standard error.

Bounds and Reports

This calculator also supports one sided bounds. A two sided interval gives a lower and upper limit. A lower confidence bound says the mean is likely above a value. An upper confidence bound says the mean is likely below a value. These options help with quality checks, research reports, and planning targets.

Good Interpretation

Always check assumptions before using the result. The sample should be collected fairly. Observations should be independent, unless your study design says otherwise. For small samples, the data should be roughly normal without strong outliers. For large samples, the mean is often stable by the central limit theorem. Still, extreme skew can affect interpretation. Treat the interval as an estimate, not as a guaranteed promise. Export the CSV or PDF report when you need a clean record for assignments, audits, or team decisions.

Reporting Tips

When reporting the answer, state the confidence level, method, sample size, and interval limits. Include units whenever the mean has units. This makes the result easier to review, compare, and reproduce later during future reviews too.

FAQs

What is a confidence interval for a mean?

It is a range built from sample data. It estimates where the population mean may fall. The range includes sampling uncertainty, so it is more useful than one sample average alone.

Should I use the t method or z method?

Use the t method when the population standard deviation is unknown. Use the z method when the population standard deviation is known or required by your assignment.

What does a 95% confidence level mean?

It means the method would capture the true mean in about 95% of repeated samples. It does not guarantee that one single interval contains the mean.

Why does a larger sample make a narrower interval?

A larger sample lowers the standard error. A lower standard error reduces the margin of error. That usually creates a narrower interval around the sample mean.

Can I paste raw data values?

Yes. Paste values separated by commas, spaces, semicolons, or line breaks. The calculator will compute the sample mean, standard deviation, and sample size automatically.

What is finite population correction?

It adjusts the standard error when sampling without replacement from a limited population. It is useful when the sample is a meaningful share of the population.

What is a one sided confidence bound?

A one sided bound gives only a lower or upper limit. It is useful when you only need to show that a mean is above or below a target.

Why is my interval very wide?

A wide interval can come from a small sample, high standard deviation, or high confidence level. More data or less variability usually narrows the result.