How to Calculate Confidence Interval with Confidence Level and Mean

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. This guide explains how to calculate a confidence interval using the confidence level and mean, including the formula, assumptions, and interpretation of results.

What is a Confidence Interval?

A confidence interval is a statistical range that provides an estimate of the true value of a population parameter. It is calculated from sample data and provides a range of values within which the true population parameter is likely to fall, with a specified level of confidence.

For example, if you calculate a 95% confidence interval for the mean height of a population, you can be 95% confident that the true mean height falls within that range. The confidence level (often 90%, 95%, or 99%) represents the probability that the interval contains the true parameter.

How to Calculate Confidence Interval

To calculate a confidence interval, you need three key pieces of information:

The sample mean (x̄)
The standard error of the mean (SE)
The confidence level (z-value or t-value)

The standard error of the mean is calculated by dividing the standard deviation (s) by the square root of the sample size (n). The confidence level determines the critical value (z or t) from the standard normal or t-distribution tables.

The margin of error is calculated by multiplying the critical value by the standard error. The confidence interval is then calculated by adding and subtracting the margin of error from the sample mean.

Confidence Interval Formula

Confidence Interval = Sample Mean ± (Critical Value × Standard Error)

Where:

Sample Mean (x̄) = Sum of all sample values / Number of samples
Standard Error (SE) = Standard Deviation (s) / √n
Critical Value = z-score for normal distribution or t-value for small samples

The critical value depends on the confidence level and whether you know the population standard deviation. For large samples (n > 30), use the z-score from the standard normal distribution. For small samples, use the t-value from the t-distribution table with n-1 degrees of freedom.

Worked Example

Let's calculate a 95% confidence interval for the mean height of a sample of 25 people, with a sample mean of 170 cm and a standard deviation of 10 cm.

Calculate the standard error: SE = 10 / √25 = 2 cm
Find the critical value for 95% confidence: z = 1.96
Calculate the margin of error: 1.96 × 2 = 3.92 cm
Calculate the confidence interval: 170 ± 3.92 = (166.08 cm, 173.92 cm)

We can be 95% confident that the true mean height of the population falls between 166.08 cm and 173.92 cm.

Interpreting Results

The confidence interval provides a range of plausible values for the population parameter. A 95% confidence interval means that if you took 100 different samples and calculated a 95% confidence interval for each, you would expect about 95 of those intervals to contain the true population mean.

When interpreting results, consider the following:

Narrower intervals indicate more precise estimates
Wider intervals indicate more uncertainty
Always report the confidence level with your interval
Don't interpret as the probability that the true value is within the interval

Note: The confidence interval is based on assumptions about the data distribution and sample size. For small samples, the t-distribution should be used instead of the normal distribution.

FAQ

What is the difference between confidence level and confidence interval?

The confidence level is the percentage that represents the probability that the interval contains the true population parameter. The confidence interval is the actual range of values calculated from the sample data.

When should I use a z-score instead of a t-value?

Use a z-score when you know the population standard deviation and have a large sample size (n > 30). Use a t-value when you don't know the population standard deviation or have a small sample size.

How does sample size affect the confidence interval?

A larger sample size generally results in a narrower confidence interval, indicating a more precise estimate. A smaller sample size leads to a wider interval, reflecting greater uncertainty.