Mathisfun Confidence Interval Calculator

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. This calculator helps you determine the confidence interval for a sample mean.

What is a Confidence Interval?

A confidence interval provides an estimated range of values which is likely to contain the population parameter. For example, if you calculate a 95% confidence interval for the average height of adults in a city, you can be 95% confident that the true average height falls within that range.

The confidence level is typically expressed as a percentage, such as 90%, 95%, or 99%. A higher confidence level means a wider interval, while a lower confidence level means a narrower interval.

How to Calculate a Confidence Interval

The formula for calculating a confidence interval for a sample mean is:

Confidence Interval = Sample Mean ± (Critical Value × Standard Error)

Where:

Sample Mean - The average of your sample data
Critical Value - The z-score or t-score from the appropriate distribution table
Standard Error - Standard Deviation / √Sample Size

The critical value depends on the confidence level and whether you know the population standard deviation:

For large samples (n > 30) or when the population standard deviation is known, use the z-distribution.
For small samples (n ≤ 30) and unknown population standard deviation, use the t-distribution.

This calculator uses the t-distribution for most cases as it's more conservative and accounts for uncertainty in the population standard deviation.

Interpreting Confidence Intervals

When you say you have a 95% confidence interval, it means that if you took 100 different samples and calculated a 95% confidence interval for each, approximately 95 of those intervals would contain the true population parameter.

It's important to note that a 95% confidence interval does not mean there's a 95% probability that the true parameter is in the interval. Instead, it reflects the long-run success rate of the method.

Key Points:

Confidence intervals are not about the data, but about the method used to estimate the parameter.
Narrower intervals indicate more precise estimates.
Wider intervals indicate more uncertainty in the estimate.

Worked Example

Let's say you want to estimate the average height of all students in a school. You take a random sample of 25 students and find:

Sample Mean (x̄) = 165 cm
Sample Standard Deviation (s) = 8 cm

You want to calculate a 95% confidence interval for the population mean height.

Using the calculator:

Enter the sample mean: 165
Enter the sample standard deviation: 8
Enter the sample size: 25
Select confidence level: 95%
Click Calculate

The calculator will show you the confidence interval, which might look something like: 162.1 cm to 167.9 cm.

This means you can be 95% confident that the true average height of all students in the school falls between 162.1 cm and 167.9 cm.

Common Confidence Interval Scenarios

Here's a comparison of confidence intervals for different sample sizes and confidence levels:

Sample Size	90% Confidence Interval	95% Confidence Interval	99% Confidence Interval
10	±1.812 × SE	±2.262 × SE	±3.250 × SE
20	±1.372 × SE	±1.729 × SE	±2.576 × SE
30	±1.190 × SE	±1.440 × SE	±2.042 × SE

Note: SE = Standard Error = Standard Deviation / √Sample Size

FAQ

What does a 95% confidence interval mean?

A 95% confidence interval means that if you took 100 different samples and calculated a 95% confidence interval for each, approximately 95 of those intervals would contain the true population parameter.

How do I choose the right confidence level?

The confidence level depends on how certain you need to be about your estimate. Common choices are 90%, 95%, and 99%. Higher confidence levels result in wider intervals.

Can I use this calculator for proportions?

This calculator is specifically for means. For proportions, you would use a different formula involving the sample proportion and standard error of the proportion.

What if my sample size is very small?

For small sample sizes (typically n ≤ 30), the calculator uses the t-distribution which accounts for more uncertainty in the population standard deviation.