Confidence Interval Calculator Given N S and C

A confidence interval is a range of values that is likely to contain the true population parameter with a specified level of confidence. This calculator helps you determine the confidence interval for a population mean when you know the sample size (n), sample standard deviation (s), and confidence level (c).

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 mean height of adults in a city, you can be 95% confident that the true mean height falls within that range.

The confidence level (c) represents the probability that the interval will contain the true parameter. Common confidence levels are 90%, 95%, and 99%. The width of the confidence interval depends on the sample size, standard deviation, and the desired confidence level.

How to Use This Calculator

Enter the sample size (n) - the number of observations in your sample.
Enter the sample standard deviation (s) - a measure of how spread out the numbers in your sample are.
Select the confidence level (c) - the percentage of confidence you want for your interval.
Click "Calculate" to see the confidence interval.

Note: This calculator assumes a normal distribution of the sample data. For small sample sizes (n < 30), the t-distribution is used instead of the normal distribution.

Formula and Assumptions

The confidence interval for a population mean is calculated using the following formula:

Confidence Interval = x̄ ± t*(s/√n) where: x̄ = sample mean t = critical t-value (for small samples) or z-value (for large samples) s = sample standard deviation n = sample size

Assumptions:

The sample data is randomly selected from the population.
The sample size is large enough (typically n ≥ 30) for the normal distribution to apply.
The population standard deviation is unknown and must be estimated from the sample.

Worked Example

Suppose you have a sample of 25 students with an average height of 170 cm and a standard deviation of 10 cm. You want to calculate a 95% confidence interval for the population mean height.

Given:

Sample size (n) = 25
Sample standard deviation (s) = 10
Confidence level (c) = 95%

Calculation:

Determine the critical t-value for 24 degrees of freedom (n-1) and 95% confidence.
t* = 2.064 (from t-distribution tables)
Margin of error = t* × (s/√n) = 2.064 × (10/5) = 4.128
Confidence interval = 170 ± 4.128 = (165.872, 174.128)

Result: You can be 95% confident that the true population mean height falls between 165.87 cm and 174.13 cm.

Interpreting Results

The confidence interval provides a range of plausible values for the population parameter. For example, if you calculate a 95% confidence interval of (165.87, 174.13) for the mean height:

This means that if you took 100 different samples of 25 students and calculated a 95% confidence interval for each, approximately 95 of those intervals would contain the true population mean height.
The confidence level does not indicate the probability that the true parameter is within the interval. It refers to the long-run success rate of the method.
A narrower confidence interval indicates more precise estimates, which can be achieved by increasing the sample size or reducing the confidence level.

Tip: To increase the precision of your confidence interval, collect a larger sample or reduce the confidence level.

FAQ

What does a 95% confidence interval mean?: It means that if you were to take 100 different samples and compute a 95% confidence interval for each, approximately 95 of those intervals would contain the true population parameter.
How does sample size affect the confidence interval?: A larger sample size results in a narrower confidence interval, providing more precise estimates of the population parameter.
What if my sample size is small (n < 30)?: For small sample sizes, the calculator uses the t-distribution instead of the normal distribution, which accounts for the increased uncertainty in the estimate.
Can I use this calculator for proportions instead of means?: No, this calculator is specifically designed for calculating confidence intervals for population means. For proportions, you would need a different calculator.
What if my data is not normally distributed?: The confidence interval formula assumes a normal distribution. For non-normal data, consider using bootstrapping methods or transformations to achieve normality.