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Confidence Interval Calculator
with Steps, Margin of Error & Graph

Calculate confidence intervals for means and proportions instantly. See step-by-step work, margin of error, standard error, and an interactive distribution visual. All calculations run in your browser — no data sent to any server.

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Enter your values and click Calculate

The confidence interval, margin of error, distribution visual, and step-by-step work will appear here.

How to Calculate a Confidence Interval

The formulas and a worked example for each type of confidence interval.

Mean Confidence Interval

x̄ ± t* × (s / √n)

Use when population σ is unknown

x̄ ± z* × (σ / √n)

Use when population σ is known

  • x̄ = sample mean
  • s / σ = sample / population standard deviation
  • n = sample size
  • t* / z* = critical value from t or normal distribution

Worked Example (95% CI for mean, unknown σ)

x̄ = 72, s = 12, n = 36

1. SE = 12 / √36 = 2.00

2. df = 36 − 1 = 35

3. t* ≈ 2.030 (for 95%, df=35)

4. ME = 2.030 × 2.00 = 4.06

5. CI = 72 ± 4.06 = (67.94, 76.06)

Proportion Confidence Interval

p̂ ± z* × √(p̂(1−p̂) / n)

Normal approximation method

  • p̂ = sample proportion (successes / trials)
  • n = sample size (number of trials)
  • z* = critical value from normal distribution

Worked Example (95% CI for proportion)

84 successes out of 200 trials

1. p̂ = 84 / 200 = 0.420

2. SE = √(0.42 × 0.58 / 200) = 0.0349

3. z* = 1.960 (for 95%)

4. ME = 1.960 × 0.0349 = 0.0684

5. CI = 0.42 ± 0.068 = (0.352, 0.488)

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How it works

1

Choose your parameter

Select whether you want to estimate a population mean or a population proportion. Toggle between known and unknown population standard deviation for means.

2

Enter your data

Input your sample statistics — mean, standard deviation, and sample size for means; or successes and trials for proportions. Select your confidence level.

3

Get instant results

See the confidence interval, margin of error, standard error, critical value, and a distribution visual. Export to PDF with one click.

Frequently asked questions

What is a confidence interval?

A confidence interval is a range of values that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval for a mean means that if many samples were taken and an interval computed from each, about 95% of those intervals would contain the true population mean.

How does the Best Answer Hub Confidence Interval Calculator work?

Select whether you want to calculate a confidence interval for a mean or a proportion. Enter your sample statistics — sample mean, standard deviation, and sample size for means; or sample proportion and sample size for proportions. Choose a confidence level (90%, 95%, 99%, or custom). The calculator instantly returns the lower and upper bounds, margin of error, standard error, and critical value, along with step-by-step work and a plain-English interpretation.

When should I use a z-interval versus a t-interval?

Use a z-interval when the population standard deviation (σ) is known and the sample size is large (typically n ≥ 30). Use a t-interval when the population standard deviation is unknown and must be estimated from the sample standard deviation (s). The t-distribution has heavier tails than the normal distribution, which accounts for the extra uncertainty from estimating σ. The Best Answer Hub Confidence Interval Calculator automatically selects the appropriate method based on your input.

What is the margin of error?

The margin of error (ME or E) is the maximum likely distance between the sample statistic and the true population parameter. It equals the critical value multiplied by the standard error. A smaller margin of error means a more precise estimate. You can reduce the margin of error by increasing the sample size or accepting a lower confidence level.

How do I interpret a 95% confidence interval?

A 95% confidence interval does not mean there is a 95% probability that the true value lies inside this specific interval. It means that the method used to construct the interval would capture the true parameter in approximately 95% of repeated samples. The Best Answer Hub Confidence Interval Calculator includes a plain-English interpretation with every result to help avoid this common misconception.

Why is my confidence interval wider than I expected?

A wider interval usually means one of three things: (1) higher variability in the data (larger standard deviation), (2) a smaller sample size, or (3) a higher confidence level. The Best Answer Hub Confidence Interval Calculator shows the margin of error and standard error breakdown so you can see exactly what is driving the width.

Can I calculate a confidence interval for a proportion?

Yes. Switch to "Proportion" mode and enter either the number of successes and total trials, or the sample proportion directly. The Best Answer Hub Confidence Interval Calculator uses the normal approximation method for proportions: p̂ ± z* × √(p̂(1−p̂)/n). This works well when both np̂ and n(1−p̂) are at least 10.

What confidence level should I use?

95% is the most common choice in scientific research and is a good default. 90% is sometimes used in exploratory work or when a narrower interval is preferred. 99% is used when you need very high certainty, though it produces wider intervals. The Best Answer Hub Confidence Interval Calculator supports all three as quick presets, plus any custom level you need.

What is the standard error?

The standard error (SE) measures the variability of a sample statistic. For a mean, SE = σ/√n (or s/√n). For a proportion, SE = √(p̂(1−p̂)/n). It represents how far the sample statistic is expected to deviate from the true parameter due to random sampling alone. The margin of error is the critical value multiplied by the standard error.

Does the Best Answer Hub Confidence Interval Calculator show step-by-step work?

Yes. The calculator displays every step of the computation — identifying the formula, substituting values, calculating the standard error, finding the critical value, computing the margin of error, and building the final interval. This makes it ideal for checking homework, studying for exams, or documenting statistical analysis.

Can I export my confidence interval calculation to PDF?

Yes. The Best Answer Hub Confidence Interval Calculator includes a one-click PDF export that captures all inputs, formulas, step-by-step work, the distribution visual, and the interpretation. No account or signup is required.

What sample size do I need for a valid confidence interval?

For means, there is no strict minimum, but the Central Limit Theorem ensures the sampling distribution is approximately normal for n ≥ 30. For smaller samples with unknown σ, the t-interval is used. For proportions, the normal approximation works well when both np̂ ≥ 10 and n(1−p̂) ≥ 10. The Best Answer Hub Confidence Interval Calculator includes a validation warning when these conditions may not be met.

Is my data private when using this calculator?

Yes. All calculations run entirely in your browser using client-side JavaScript. No data is uploaded, stored, or logged on any server. Sensitive research data can be calculated with complete privacy.

Can I compare two confidence intervals to see if groups differ?

Yes. If two confidence intervals do not overlap, the groups are likely statistically different at that confidence level. However, overlapping intervals do not always mean the groups are the same — the correct approach is to calculate a confidence interval for the difference between the two means or proportions. The Best Answer Hub Confidence Interval Calculator focuses on single-parameter intervals; for two-group comparisons, a dedicated T-Test Calculator (coming soon) will provide the proper statistical test.

Is the Best Answer Hub Confidence Interval Calculator free?

Yes. The entire Statistics Calculator Suite is free with no hidden fees, premium tiers, subscription plans, or advertisements.

Explore the rest of the Statistics Calculator Suite

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Built & maintained by Shahbaz Ali Malik Last updated: