Normally Distributed Confidence Interval Calculator

A normally distributed confidence interval is a range of values that is likely to contain the true population mean with a certain level of confidence. This calculator helps you compute confidence intervals for normally distributed data.

What is a Normally Distributed 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 normally distributed data, we can calculate confidence intervals using the sample mean and standard deviation.

The most common confidence levels are 90%, 95%, and 99%. A 95% confidence interval means that if we took many samples and calculated a 95% confidence interval for each, about 95% of those intervals would contain the true population mean.

How to Calculate a Confidence Interval

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

Confidence Interval = Sample Mean ± (Critical Value × (Sample Standard Deviation / √Sample Size))

Where:

Sample Mean (x̄) is the average of your sample data
Critical Value is the z-score from the standard normal distribution table for your chosen confidence level
Sample Standard Deviation (s) measures the dispersion of your sample data
Sample Size (n) is the number of observations in your sample

The critical value depends on your desired confidence level. For example:

90% confidence: z = 1.645
95% confidence: z = 1.960
99% confidence: z = 2.576

Interpreting Confidence Intervals

When you calculate a confidence interval, you're making a statement about the 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 say:

We are 95% confident that the true population mean falls between 4.2 and 5.8
If we took many samples and calculated 95% confidence intervals, about 95% of those intervals would contain the true population mean
There is a 5% chance that the true population mean is outside this interval

Remember that a confidence interval doesn't tell you the probability that the true mean is in the interval. Instead, it tells you how confident we are that the interval contains the true mean.

Worked Example

Let's say we have a sample of 30 measurements with a mean of 5.2 and a standard deviation of 1.3. We want to calculate a 95% confidence interval.

Identify the critical value for 95% confidence: z = 1.960
Calculate the standard error: 1.3 / √30 ≈ 0.265
Calculate the margin of error: 1.960 × 0.265 ≈ 0.520
Calculate the confidence interval: 5.2 ± 0.520 → [4.68, 5.72]

We are 95% confident that the true population mean falls between 4.68 and 5.72.

FAQ

What does a confidence interval tell me?: A confidence interval provides a range of values that is likely to contain the true population parameter with a certain level of confidence. It doesn't tell you the probability that the true parameter is in the interval.
How do I choose a confidence level?: Common confidence levels are 90%, 95%, and 99%. Higher confidence levels provide wider intervals, while lower confidence levels provide narrower intervals. The choice depends on how precise you need your estimate to be.
What if my data isn't normally distributed?: If your data isn't normally distributed, you might need to use a different method like bootstrapping or a t-distribution confidence interval, especially for small sample sizes.
Can I use this calculator for any sample size?: Yes, this calculator works for any sample size. However, for very small sample sizes (n < 30), you might want to consider using a t-distribution instead of a z-distribution.
How do I know if my confidence interval is accurate?: Your confidence interval is accurate if your sample is representative of the population and your assumptions about the data distribution are correct. You can check this by examining your sample data and considering the context of your study.