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Confidence Interval for Mean Calculator

Calculate the confidence interval for a sample mean.

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Confidence Interval
Margin of Error (±)4.16

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Assumptions

  • The sample size must be positive and the sample standard deviation cannot be negative.
  • This calculator uses z-scores for common confidence levels, which is most appropriate for larger samples or known-population-style approximations.
  • The interval describes uncertainty around the mean estimate, not the spread of individual observations.

Sensitivity preview

Sample Mean -10%: Confidence Interval 85.84 to 94.16

Sample Mean Current: Confidence Interval 95.84 to 104.16

Sample Mean +10%: Confidence Interval 105.84 to 114.16

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About this Confidence Interval

In statistics, it's rare to know the exact mean of an entire population. Instead, we use a sample to estimate it. A Confidence Interval (CI) gives us a range of values where the true population mean is likely to exist with a certain degree of confidence.

Who this calculator is for

  • Students and researchers estimating a population mean from sample data.
  • Survey, product, and operations teams reporting a range instead of a single point estimate.
  • Anyone translating sample mean and standard deviation into an interval with margin of error.

When to use it

  • Use it after calculating a sample mean and standard deviation for a dataset.
  • Use it when you need a margin of error for a report, experiment, or quality-control summary.
  • Use it to compare how interval width changes with different sample sizes or confidence levels.

How to use this calculator

  1. Enter the Sample Mean (average of your data).
  2. Enter the Sample Standard Deviation.
  3. Enter the Sample Size (n).
  4. Select the Confidence Level (95% is most common).

Key assumptions

  • The sample size must be positive and the sample standard deviation cannot be negative.
  • This calculator uses z-scores for common confidence levels, which is most appropriate for larger samples or known-population-style approximations.
  • The interval describes uncertainty around the mean estimate, not the spread of individual observations.

Accuracy and testing

Review formula sources, default test cases, assumptions, unit handling, rounding rules, and changelog for this calculator.

View accuracy page

How this calculator works

CI = Mean ± (Z × (s / √n)) * Z: Z-score corresponding to the confidence level (1.96 for 95%). * s: Standard Deviation. * n: Sample Size. * s / √n: Standard Error of the Mean.

Inputs explained

Sample Mean

The arithmetic average of your sample data.

Confidence Level

The probability that the interval contains the true parameter. 95% is the standard for scientific research.

Sample Size

The number of observations in your sample. Larger samples result in narrower (more precise) intervals.

Understanding your results

The result is a range (e.g., 95 to 105). This means we are X% confident that the true population mean falls somewhere between these two numbers. The Margin of Error is the 'plus or minus' amount.

Example scenarios

Test Scores

Mean 80, SD 10, Size 30, Conf 95%. Interval: 76.42 to 83.58. We are 95% confident the class average is in this range.

Height

Mean 170cm, SD 5, Size 100, Conf 99%. Interval: 168.7 to 171.3. A larger sample size creates a tighter range.

Key Features

  • Margin of Error Calculation
  • Upper & Lower Bounds
  • Standard Confidence Levels

Tips & Best Practices

  • To reduce the Margin of Error (get a narrower interval), you must increase the sample size.
  • This calculator assumes a normal distribution and uses Z-scores (best for n > 30). For very small samples, a T-distribution should technically be used.

Glossary of Terms

Standard Error
The standard deviation of the sampling distribution of a statistic (Standard Deviation / √n).
Margin of Error
The radius of the confidence interval.

Frequently Asked Questions

What does 95% confidence really mean?
It means if you repeated this experiment 100 times and calculated the interval each time, 95 of those intervals would contain the true population mean.
Why does the interval get wider with higher confidence?
To be MORE certain (e.g. 99%) that the true mean is inside the range, you must cast a wider net, resulting in a broader interval.
When should I use 90% vs 95%?
Use 95% for most scientific or business analysis. Use 90% if you can accept more risk of being wrong in exchange for a narrower range. Use 99% for high-stakes decisions (like medical safety).