Confidence Interval Calculator Given N S and C

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain an unknown population parameter with a certain degree of confidence. 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.

Key Components

• Sample size (n): The number of observations in your sample
• Standard deviation (s): A measure of how spread out the numbers in your sample are
• Confidence level (c): The percentage that represents how confident you want to be that the interval contains the true parameter (typically 90%, 95%, or 99%)

Types of Confidence Intervals

The most common confidence intervals are for means, but they can also be calculated for proportions, differences between means, and other parameters. This calculator focuses on confidence intervals for means.

How to Calculate a Confidence Interval

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

The critical t-value depends on your confidence level and degrees of freedom (n-1). For common confidence levels:

For sample sizes less than 30, you should use the t-distribution rather than the normal distribution. This calculator automatically adjusts for sample size when calculating the interval.

Using the Calculator

To use the calculator on the right side of this page:

1. Enter your sample size (n) - the number of observations in your sample
2. Enter your sample standard deviation (s) - a measure of how spread out your numbers are
3. Select your desired confidence level (c) from the dropdown
4. Click "Calculate" to see your confidence interval

Interpreting Results

When you get a confidence interval, you can interpret it as follows: "We are X% confident that the true population mean falls within this range."

Example Interpretation

Suppose you calculate a 95% confidence interval for the mean height of adults in a city and get the result [66.2 inches, 68.8 inches]. This means you can be 95% confident that the true average height of all adults in the city is between 66.2 inches and 68.8 inches.

Common Misinterpretations

• Don't say "There is a 95% probability that the true mean is within this interval." The confidence level refers to the method, not the probability of the true mean being in the interval.
• Don't say "All future samples will have means within this interval." The interval is about the population parameter, not future samples.

Common Mistakes

When working with confidence intervals, it's easy to make several common mistakes:

1. Using the wrong distribution

For small sample sizes (n < 30), you must use the t-distribution rather than the normal distribution. This calculator automatically handles this adjustment.

2. Misinterpreting the confidence level

Remember that the confidence level is about the method, not the probability that the true mean is in the interval. Multiple confidence intervals calculated from the same data may or may not contain the true mean.

3. Ignoring sample size

Smaller sample sizes result in wider confidence intervals because there's more uncertainty with fewer data points. Always consider sample size when interpreting confidence intervals.