Sample Confidence Interval of A Normal Distribution Calculator

A sample confidence interval for a normal distribution provides a range of values that is likely to contain the true population mean with a specified level of confidence. This calculator helps you determine this interval based on your sample data.

What is a Sample 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 a normal distribution, we use the sample mean and standard deviation to calculate this interval.

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 the Confidence Interval

The formula for the confidence interval of a normal distribution is:

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

Where:

Sample Mean - The average of your sample data
Critical Value - The z-score that corresponds to your desired confidence level
Sample Standard Deviation - A measure of how spread out your sample data is
Sample Size - The number of observations in your sample

The critical value is determined by your confidence level. For example:

90% confidence: ±1.645
95% confidence: ±1.960
99% confidence: ±2.576

Interpreting the Results

When you calculate a confidence interval, you're making a statement about the range within which you believe the true population parameter lies. For example, if you calculate a 95% confidence interval of [45, 55] for the mean height of a population, you can be 95% confident that the true population mean height falls between 45 and 55.

Keep in mind that this doesn't mean there's a 95% probability that the true mean is in this interval. Instead, it means that if you took many samples and calculated 95% confidence intervals for each, 95% of those intervals would contain the true mean.

Worked Example

Let's say you have a sample of 30 people with an average height of 50 inches and a standard deviation of 3 inches. You want to calculate a 95% confidence interval for the true population mean height.

Using the formula:

Confidence Interval = 50 ± (1.960 × (3 / √30))

Confidence Interval = 50 ± (1.960 × 0.577)

Confidence Interval = 50 ± 1.13

Final Interval: [48.87, 51.13]

This means you can be 95% confident that the true population mean height falls between 48.87 and 51.13 inches.

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 specified level of confidence. It gives you a measure of the uncertainty in your estimate.
How do I choose the right confidence level?: Common confidence levels are 90%, 95%, and 99%. Higher confidence levels result in wider intervals, while lower confidence levels result in narrower intervals. The choice depends on how precise you need your estimate to be and how much risk you're willing to take.
What if my sample isn't normally distributed?: For small sample sizes (n < 30), the sample should be approximately normally distributed. For larger samples, the Central Limit Theorem often ensures that the sampling distribution of the mean is approximately normal, even if the population isn't.
Can I use this calculator for any type of data?: This calculator is designed for normally distributed data. If your data is not normally distributed, you may need to use alternative methods or transformations.
How does sample size affect the confidence interval?: Larger sample sizes result in narrower confidence intervals because you have more information about the population. Smaller sample sizes result in wider intervals because there's more uncertainty about the population.