Confidence Interval Calculator for N X and S

This calculator helps you determine the confidence interval for a sample mean when you know the sample size (n), sample mean (x̄), and sample standard deviation (s). Confidence intervals provide a range of values that likely contains the true population mean with a specified level of confidence.

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain the true population parameter (like the mean) with a certain level of confidence. For example, a 95% confidence interval means that if you took 100 samples and calculated 95% confidence intervals for each, about 95 of those intervals would contain the true population mean.

Confidence intervals are essential in statistics because they provide more information than a single point estimate. They help researchers and analysts understand the precision of their estimates and make more informed decisions.

How to Use This Calculator

Enter the sample size (n) - the number of observations in your sample.
Enter the sample mean (x̄) - the average of your sample data.
Enter the sample standard deviation (s) - a measure of how spread out your sample data is.
Select your desired confidence level (typically 90%, 95%, or 99%).
Click "Calculate" to see your confidence interval.

The calculator will display the lower and upper bounds of your confidence interval, along with a visual representation of the distribution.

Formula and Assumptions

The formula for the confidence interval when the population standard deviation is unknown is:

x̄ ± t*(s/√n)

Where:

x̄ = sample mean
t = critical t-value from the t-distribution
s = sample standard deviation
n = sample size

Assumptions:

The sample is randomly selected from the population.
The sample size is large enough (typically n > 30) or the population is normally distributed.
The sample standard deviation is a good estimate of the population standard deviation.

Worked Example

Suppose you have a sample of 25 students with an average height of 170 cm and a standard deviation of 8 cm. You want to calculate a 95% confidence interval for the population mean height.

Sample size (n) = 25
Sample mean (x̄) = 170 cm
Sample standard deviation (s) = 8 cm
Confidence level = 95%

Using the calculator:

Critical t-value ≈ 2.064
Margin of error = 2.064 × (8/√25) ≈ 3.286 cm
Confidence interval = 170 ± 3.286 → (166.714, 173.286) cm

This means we are 95% confident that the true population mean height falls between 166.714 cm and 173.286 cm.

Interpreting Results

When you get a confidence interval, you can interpret it as follows:

If you took many samples and calculated 95% confidence intervals for each, about 95% of those intervals would contain the true population mean.
The width of the interval tells you about the precision of your estimate. Wider intervals indicate less precision.
If the interval is too wide, you might need to collect more data to get a more precise estimate.

Common confidence levels are 90%, 95%, and 99%. Higher confidence levels result in wider intervals, while lower confidence levels result in narrower intervals.

FAQ

What does a 95% confidence interval mean?

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

Why do I need a confidence interval instead of just a point estimate?

A confidence interval provides a range of plausible values for the population parameter, giving you a sense of the precision of your estimate. It helps you understand the uncertainty associated with your sample.

What happens if my sample size is small?

For small sample sizes (typically n < 30), you should use the t-distribution instead of the normal distribution, as shown in this calculator. The t-distribution accounts for the extra uncertainty when the population standard deviation is unknown.