Confidence interval calculator

This confidence interval calculator helps you determine the confidence interval based on four inputs: sample size, sample mean, standard deviation, and confidence level. The calculations assume that the sample mean follows a normal distribution.

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What is a confidence interval?

A confidence interval (CI) is a range of values used to estimate an unknown population parameter, such as the population mean (μ). It gives a range of plausible values where we expect the true value to fall with a certain level of confidence.

Formula for confidence interval

The confidence interval for a population mean when the standard deviation is known is:

CI = \bar{x} \pm Z \times \left(\frac{\sigma}{\sqrt{n}}\right)

where:

• \bar{x} = Sample mean
• Z = Critical value (Z-score) based on the confidence level
• \sigma = Standard deviation
• n = Sample size

Example calculation

Let’s say we have:

• Sample size (n ) = 40
• Sample mean (\bar{x}) = 15
• Standard deviation (\sigma ) = 9
• Confidence level = 99.9% (which gives Z = 3.291)

Compute standard error (SE)

SE = \frac{\sigma}{\sqrt{n}} = \frac{9}{\sqrt{40}} = 1.423

Calculate margin of error (MOE)

MOE = Z \times SE = 3.291 \times 1.423 = 4.683

Compute confidence interval

CI = 15 \pm 4.683 \\
CI = [10.317, 19.683]

Interpreting the confidence interval

A 99.9% confidence interval of [10.317, 19.683] means:

• We are 99.9% confident that the true population mean falls within this range.
• It does not mean that there’s a 99.9% probability the mean is inside this range. Instead, it means that if we repeatedly take samples and compute intervals, 99.9% of them would contain the true mean.

Key takeaways

✅ Higher confidence levels (e.g., 99%) result in wider intervals, increasing certainty but reducing precision.
✅ Larger sample sizes reduce the interval width, improving accuracy.
✅ Confidence intervals do not predict individual values, only population parameters.

Z-values for several commonly used confidence levels

Here are the Z-values (critical values) for several commonly used confidence levels:

How these Z-values are found

The Z-value represents the number of standard deviations away from the mean that captures the central (Confidence level)% of a standard normal distribution.

To get the Z-score:

• Find \alpha:

\alpha = 1 - \frac{\text{Confidence level}}{100}

• Look up Z_{\alpha/2}​ in a standard normal table (or use the inverse normal function).

Example

For 95% confidence:

• \alpha = 1 - 0.95 = 0.05
• Since the confidence interval is two-tailed, we take \alpha/2 = 0.025 in each tail.
• From the Z-table, the Z-score for 0.975 cumulative probability is 1.960.