Statistics 95 Confidence Interval Calculator

A 95% confidence interval is a range of values that is likely to contain the true population mean with 95% probability. This calculator helps you compute confidence intervals for sample means based on your sample data.

What is a 95% Confidence Interval?

In statistics, a confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. A 95% confidence interval means that if we were to take many samples and compute a 95% confidence interval for each, approximately 95% of these intervals would contain the true population mean.

Confidence intervals are not about the population parameter being in the interval 95% of the time. Instead, they indicate the reliability of our estimate based on the sample data.

The width of the confidence interval depends on several factors:

The sample size (larger samples produce narrower intervals)
The sample standard deviation (higher variability increases interval width)
The confidence level (higher confidence levels produce wider intervals)

How to Calculate a 95% Confidence Interval

The formula for a 95% confidence interval for a population mean is:

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

Where:

Sample Mean (x̄) is the average of your sample data
Critical Value is the z-score from the standard normal distribution for a 95% confidence level (approximately 1.96)
Standard Error (SE) is calculated as Sample Standard Deviation divided by the square root of the sample size

The standard error formula is:

Standard Error = Sample Standard Deviation / √(Sample Size)

For a 95% confidence interval, we use the z-score of 1.96, which corresponds to the point where 95% of the area under the normal curve is within ±1.96 standard deviations from the mean.

Interpreting Confidence Intervals

When you calculate a 95% confidence interval, you can interpret it as follows:

We are 95% confident that the true population mean falls within the calculated interval, based on our sample data.

Important points to remember:

The confidence level (95%) refers to the method's reliability, not the probability that the interval contains the true mean
A 95% confidence interval does not mean there is a 95% probability that the true mean is within the interval
If you took many samples and computed 95% confidence intervals for each, about 95% of those intervals would contain the true population mean

Confidence intervals can be used to:

Compare different groups or treatments
Determine if a population mean is significantly different from a hypothesized value
Assess the precision of your estimate

Worked Example

Let's calculate a 95% confidence interval for a sample of test scores.

Sample Size (n)	Sample Mean (x̄)	Sample Standard Deviation (s)
30	75	10

Step 1: Calculate the standard error

Standard Error = 10 / √30 ≈ 1.83

Step 2: Calculate the margin of error

Margin of Error = 1.96 × 1.83 ≈ 3.59

Step 3: Calculate the confidence interval

Lower Bound = 75 - 3.59 ≈ 71.41

Upper Bound = 75 + 3.59 ≈ 78.59

The 95% confidence interval for this sample is approximately 71.41 to 78.59. This means we are 95% confident that the true population mean test score falls within this range.

FAQ

What does a 95% confidence interval mean?

A 95% confidence interval means that if we were to take many samples and compute a 95% confidence interval for each, approximately 95% of these intervals would contain the true population mean.

How does sample size affect the confidence interval?

Larger sample sizes produce narrower confidence intervals because they provide more information about the population. The standard error decreases as the sample size increases, resulting in a smaller margin of error.

Can I use this calculator for other confidence levels?

This calculator specifically calculates 95% confidence intervals. For other confidence levels, you would need to adjust the critical value (z-score) accordingly.

What if my sample size is small?

For small sample sizes (typically n < 30), it's more appropriate to use the t-distribution rather than the normal distribution. This calculator assumes a large enough sample size for the normal distribution approximation to be valid.

How do I know if my confidence interval is narrow enough?

A narrow confidence interval indicates a more precise estimate. You can make the interval narrower by increasing your sample size or reducing the variability in your sample data.