Confidence Interval Calculator N X S

This confidence interval calculator helps you determine the range within which a population mean likely falls based on a sample mean (x̄), sample size (n), and sample standard deviation (s). It uses the t-distribution for small samples and the normal distribution for large samples (n ≥ 30).

What is a 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. Common confidence levels are 90%, 95%, and 99%.

For example, 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 the population mean (μ) when the population standard deviation (σ) is unknown is:

Confidence Interval = x̄ ± t*(s/√n)

Where:

x̄ = sample mean
t* = critical t-value from t-distribution table
s = sample standard deviation
n = sample size

The critical t-value depends on your confidence level and degrees of freedom (n-1). For large samples (n ≥ 30), you can use the standard normal distribution (z* instead of t*).

Note: This calculator assumes your sample is a simple random sample and that the population is normally distributed or the sample size is large enough (n ≥ 30) to apply the Central Limit Theorem.

Interpreting Confidence Intervals

When you calculate a confidence interval, you're making a statement about the range of values that likely contains the true population parameter. For example:

A 95% confidence interval means you're 95% confident that the true population mean falls within the calculated range.
This doesn't mean there's a 95% probability that any particular value is the true mean.
If you took many samples and calculated 95% confidence intervals for each, about 95% of those intervals would contain the true mean.

Confidence intervals become narrower as your sample size increases, giving you more precise estimates of the population parameter.

Worked Example

Let's calculate a 95% confidence interval for the mean height of students in a school using the following data:

Sample mean (x̄) = 165 cm
Sample size (n) = 50
Sample standard deviation (s) = 8 cm

Degrees of freedom = n - 1 = 49

For a 95% confidence level, the critical t-value is approximately 2.0096

Margin of error = t* × (s/√n) = 2.0096 × (8/√50) ≈ 2.55 cm

Confidence Interval = 165 ± 2.55 = (162.45, 167.55) cm

We can be 95% confident that the true mean height of all students in the school falls between 162.45 cm and 167.55 cm.

Parameter	Value
Sample mean (x̄)	165 cm
Sample size (n)	50
Sample standard deviation (s)	8 cm
Confidence level	95%
Critical t-value	2.0096
Margin of error	2.55 cm
Confidence interval	(162.45, 167.55) cm

FAQ

What does a 95% confidence interval mean?

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. It doesn't mean there's a 95% probability that any particular interval contains the true mean.

When should I use a confidence interval calculator?

Use this calculator when you want to estimate the range within which a population mean likely falls based on sample data. It's particularly useful in research, quality control, and decision-making where you need to understand the uncertainty around your sample estimates.

What assumptions are made in this calculation?

This calculator assumes your sample is a simple random sample, the population is normally distributed, or the sample size is large enough (n ≥ 30) to apply the Central Limit Theorem. It also assumes the sample standard deviation is a good estimate of the population standard deviation.

How does sample size affect the confidence interval?

Larger sample sizes result in narrower confidence intervals because you have more information about the population. The margin of error decreases as the square root of the sample size increases, making your estimates more precise.