Sample Confidence Interval Normal Distribution Calculator

This calculator helps you determine the confidence interval for a sample mean when the population standard deviation is known. Confidence intervals provide a range of values that are likely to contain the true population mean with a specified level of confidence.

What is a Sample Confidence Interval?

A sample confidence interval is a range of values that is likely to contain the true population mean. It's calculated from a sample of data and provides a measure of the uncertainty associated with the sample mean.

When working with normally distributed data, the confidence interval is typically calculated using the sample mean, sample standard deviation, sample size, and the desired confidence level. The most common confidence levels are 90%, 95%, and 99%.

Note: This calculator assumes you know the population standard deviation. If you only have sample data, you should use a t-distribution instead of a normal distribution.

How to Calculate a Sample Confidence Interval

The formula for calculating a confidence interval for a normally distributed population is:

Confidence Interval = Sample Mean ± (Z × (Population Standard Deviation / √Sample Size))

Where:

Sample Mean - The average of your sample data
Z - The Z-score corresponding to your desired confidence level
Population Standard Deviation - The standard deviation of the entire population
Sample Size - The number of observations in your sample

The Z-score is a value from the standard normal distribution that corresponds to the desired confidence level. For example:

90% confidence level: Z = 1.645
95% confidence level: Z = 1.960
99% confidence level: Z = 2.576

Interpreting the Results

The confidence interval provides a range of values that is likely to contain the true population mean. For example, if you calculate a 95% confidence interval of [4.2, 5.8], you can be 95% confident that the true population mean falls between 4.2 and 5.8.

If the confidence interval is wide, it indicates that there is more uncertainty in your estimate. If the interval is narrow, it suggests that your sample provides a more precise estimate of the population mean.

Worked Example

Suppose you have a sample of 30 measurements with a mean of 5.2 and a population standard deviation of 1.5. You want to calculate a 95% confidence interval.

Using the formula:

Confidence Interval = 5.2 ± (1.960 × (1.5 / √30))

First calculate the standard error:

Standard Error = 1.5 / √30 ≈ 0.2887

Then calculate the margin of error:

Margin of Error = 1.960 × 0.2887 ≈ 0.5656

Finally, calculate the confidence interval:

Lower Bound = 5.2 - 0.5656 ≈ 4.6344

Upper Bound = 5.2 + 0.5656 ≈ 5.7656

So the 95% confidence interval is approximately [4.63, 5.77].

FAQ

What is the difference between a confidence interval and a confidence level?: The confidence level is the percentage that represents how certain you are that the interval contains the true population mean. The confidence interval is the actual range of values calculated from your sample data.
Can I use this calculator for non-normal data?: No, this calculator assumes your data follows a normal distribution. For non-normal data, you should use a t-distribution or other appropriate method.
What if I don't know the population standard deviation?: If you only have sample data, you should use the sample standard deviation and a t-distribution instead of a normal distribution.
How do I choose the right confidence level?: Common choices are 90%, 95%, and 99%. Higher confidence levels provide wider intervals, while lower levels provide narrower intervals. The choice depends on your specific requirements for precision and certainty.
What does it mean if my confidence interval is very wide?: A wide confidence interval indicates that there is more uncertainty in your estimate. This could be due to a small sample size or a large population standard deviation.