Confidence Interval Calculator

A confidence interval gives a range of plausible values for a population mean, based on a sample. A 95% interval is built so that, over many samples, about 95% of such intervals would contain the true mean. This calculator finds the standard error, critical value and margin of error, and shades the interval on a bell curve.

Try:

How to use the confidence interval calculator

  1. Enter the sample mean, standard deviation and sample size.
  2. Choose the confidence level.
  3. Press Calculate to see the interval and margin of error.

Formula

xˉ±z∗σn\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}

Worked example: Mean 72.5, σ 8, n 50, 95%

  • Sample mean x̄: 72.5
  • Standard deviation σ: 8
  • Sample size n: 50
  • Confidence level: 95%

✓ Answer checked

95% CI=(70.2826,  74.7174)95\%\text{ CI} = (70.2826,\; 74.7174)
Margin of error
±2.21745\pm 2.21745
Critical value z*
1.959961.95996
Standard error
1.131371.13137
686970717273747576770.20.40.60.81
The middle 95% of the sampling distribution of the mean, from 70.283 to 74.717.
Step-by-step working (4 steps)

  1. Standard error

    How much the sample mean typically varies.

    SE=σn=850≈1.1313708SE = \frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{50}} \approx 1.1313708

  2. Critical value

    For 95% confidence, 95% of the normal curve lies between −z* and z*.

    z∗≈1.95996z^* \approx 1.95996

  3. Margin of error

    z* × SE.

    1.95996×1.13137≈2.217451.95996 \times 1.13137 \approx 2.21745

  4. Interval

    Mean ± margin of error.

    72.5±2.2174572.5 \pm 2.21745
Formulas used
xˉ±z∗σn\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}
How this was checked
  • ✓ The area between −z* and z* under the normal curve is 95%.

Uses the normal (z) distribution with a known or large-sample standard deviation. For small samples with an estimated σ, a t-interval is slightly wider.

Frequently asked questions

What is the z value for 95% confidence?

About 1.96.

Why is a 99% interval wider than a 95% one?

To be more confident of capturing the true mean, the interval must cover more values.

How do I make the interval narrower?

Use a larger sample. The margin of error shrinks in proportion to 1/√n.

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