Single-Sample Confidence Interval Calculator Using The Z Statistic

This calculator helps you determine a single-sample confidence interval using the z-statistic. Confidence intervals provide a range of values that likely contain the true population mean, based on your sample data. The z-statistic is used when the population standard deviation is known or when the sample size is large enough (n ≥ 30).

What is a Single-Sample Confidence Interval?

A single-sample confidence interval estimates the range within which the true population mean likely falls. It's calculated from a single sample of data and provides a measure of the uncertainty associated with the sample mean.

Key components of a confidence interval:

Sample mean (x̄)
Standard error of the mean (SE)
Z-critical value (from z-table)
Margin of error (ME)

Confidence Interval Formula

Lower Bound = x̄ - (z × SE)

Upper Bound = x̄ + (z × SE)

Where SE = σ/√n

When to Use the Z Statistic

The z-statistic is appropriate when:

The population standard deviation (σ) is known
The sample size is large (n ≥ 30)
The population is normally distributed

Important Note

If the population standard deviation is unknown and the sample size is small (n < 30), use the t-distribution instead of the z-statistic.

How to Calculate a Single-Sample Confidence Interval

Collect your sample data and calculate the sample mean (x̄)
Determine the population standard deviation (σ) or use the sample standard deviation if n ≥ 30
Calculate the standard error of the mean: SE = σ/√n
Find the z-critical value corresponding to your desired confidence level (e.g., 1.96 for 95% confidence)
Calculate the margin of error: ME = z × SE
Determine the confidence interval: x̄ ± ME

Use our calculator to perform these calculations quickly and accurately.

Worked Example

Suppose you want to estimate the average height of adult males in a city. You collect a random sample of 50 men with an average height of 175 cm and a known population standard deviation of 10 cm. Calculate a 95% confidence interval for the true average height.

Sample mean (x̄) = 175 cm
Population standard deviation (σ) = 10 cm
Sample size (n) = 50
Standard error (SE) = 10/√50 ≈ 1.414 cm
Z-critical value (95% confidence) = 1.96
Margin of error (ME) = 1.96 × 1.414 ≈ 2.77 cm
Confidence interval = 175 ± 2.77 → (172.23 cm, 177.77 cm)

This means we're 95% confident the true average height of adult males in the city falls between 172.23 cm and 177.77 cm.

Interpreting Results

When interpreting a confidence interval:

If the interval includes the hypothesized population mean, you can't reject the null hypothesis
If the interval doesn't include zero, the result is statistically significant
Wider intervals indicate more uncertainty in your estimate

Confidence Level	Z-Critical Value	Interpretation
90%	1.645	We're 90% confident the true value falls within this range
95%	1.96	We're 95% confident the true value falls within this range
99%	2.576	We're 99% confident the true value falls within this range

FAQ

What does a 95% confidence interval mean?

A 95% confidence interval means that if you took 100 different samples and calculated a confidence interval for each, approximately 95 of those intervals would contain the true population mean.

Can I use this calculator for small samples?

No, this calculator uses the z-statistic which is appropriate for large samples (n ≥ 30). For small samples, use a t-distribution calculator instead.

What if my population standard deviation is unknown?

If you don't know the population standard deviation, you should use the sample standard deviation and the t-distribution, especially for small samples.