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Confidence Intervals

Confidence Interval Calculator

Use this confidence interval calculator to find confidence intervals for a mean, proportion, difference between two means, paired mean difference, or difference between two proportions.

1. Choose a confidence interval

2. Choose data entry format

3. Enter the data

3. Enter the summary statistics

3. Enter the data

3. Enter the sample proportion

3. Enter the data

3. Enter the summary statistics

3. Enter the paired observations

3. Enter the pairwise differences

3. Enter the paired-difference summary statistics

3. Enter the data

3. Upload the dataset

Upload a CSV, TXT, XLSX, or XLS file with column headings.
Upload a dataset to select the variables for your confidence interval.
Paired Data Format

4. Set the confidence level

Common confidence levels are 90%, 95%, and 99%.

How to Use the Confidence Interval Calculator

This calculator allows you to calculate confidence intervals for a population mean, population proportion, difference between two independent means, paired mean difference, or difference between two independent proportions.

To use the calculator:

  1. Choose the confidence interval you want to calculate.
  2. Select the data entry format available for that interval.
  3. Enter your data, summary statistics, or upload your dataset.
  4. Enter the confidence level, such as 90%, 95%, or 99%.
  5. Select the number of decimal places
  6. Click Calculate.

The calculator returns the lower and upper confidence limits together with other related statistics such as the point estimate, standard error, critical value, margin of error where applicable, and the method used. It also includes a direct interpretation statement of the constructed confidence interval.

Tip. The calculator allows you to enter raw data for the supported intervals or upload CSV, TXT, XLS, and XLSX datasets when working with larger datasets.

What Is a Confidence Interval?

A confidence interval is a range of values calculated from sample data that is used to estimate an unknown population parameter, such as a population mean, proportion, or difference between two population parameters.

For example, suppose a sample gives a 95% confidence interval for the population mean of 68.4 to 75.7. This means that “we are 95% confident that the true population mean lies between 68.4 and 75.7”.

More precisely, the 95% confidence level describes the statistical procedure. If we repeatedly took samples and constructed confidence intervals in the same way, approximately 95% of those intervals would contain the true population mean.

The width of the interval reflects the precision of the estimate. Narrower intervals give a more precise estimate, while wider intervals indicate greater uncertainty.

Confidence Interval Formula

The general confidence interval formula is: CI = Point Estimate ± Critical Value × Standard Error.

Where:

  • CI is the confidence interval
  • Point Estimate is the sample statistic (such as the sample mean or sample proportion) used to estimate an unknown population parameter
  • Critical Value is a multiplier that determines the width of your margin of error. It measures how many standard errors you need to extend away from your sample statistic to capture the true population parameter with your desired level of confidence.
  • Standard Error is a measure of the expected variability or sampling error of the point estimate from sample to sample, which depends on sample size (n) and data variability.

Sometimes the formula can be written as: Confidence Interval = Point Estimate ± Margin of Error because Critical Value × Standard Error is the margin of error.

The exact point estimate, standard error, and critical value depend on the parameter being estimated. However, some methods, such as the Wilson interval for a proportion and the Newcombe-Wilson interval for two proportions, calculate the lower and upper limits directly rather than using a single symmetric margin of error.

Which Confidence Interval Should I Use?

The correct confidence interval depends on the parameter you want to estimate and whether your samples are independent or related.

The table below provides a quick summary of the confidence intervals you can calculate with this calculator and when to use each of them.

Confidence IntervalUse It WhenDetailed Calculator
One MeanYou want to estimate a CI for one population mean either when the population standard deviation is unknown or known.Confidence Interval for Single Population Mean Calculator
One ProportionYou want to estimate a CI for a one-population proportion.Proportion Confidence Interval Calculator
Two MeansYou want to estimate the CI for the difference between two independent population means, when population standard deviation is known or unknown. Confidence Interval for Difference in Means Calculator
Paired Mean DifferenceYou want to estimate the CI for the mean difference between two related measurements.Paired T-Test Confidence Interval Calculator
Two ProportionsYou want to estimate the CI for the difference between two independent population proportionsTwo Proportion Confidence Interval Calculator

The sections below show the formula used for each interval and a brief example of how to calculate it using the calculator.

Confidence Interval for One Mean

Choose the One Mean option when you have one quantitative sample and want to estimate the population mean, μ. The calculator automatically uses the appropriate distribution based on whether the population standard deviation, σ, is known.

When σ is unknown, the sample standard deviation is used, and the confidence interval is based on Student’s t distribution. In this case, the confidence interval formula is:

CI=x‾±t1−α/2,;n−1(sn)\text {CI} = \bar{x} \pm t_{1-\alpha/2,;n-1} \left( \frac{s}{\sqrt{n}} \right)

where:

  • x̄ is the sample mean
  • s is the sample standard deviation
  • n is the sample size
  • t1-α/2, n-1 is the t critical value

However, if the population standard deviation (σ) is known, we estimate the confidence interval for the mean using the z distribution. In this case, the formula is:

CI=x‾±z1−α/2(σn)\text{CI} = \bar{x} \pm z_{1-\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)

where:

  • x̄ is the sample mean
  • σ is the population standard deviation
  • n is the sample size
  • z1-α/2 is the z critical value

Example 1. Confidence Interval for Population Mean (σ Unknown)

Suppose the following weights were obtained from a sample of 15 children

26.2, 24.0, 26.8, 29.8, 23.7, 23.7, 30.0, 27.2, 22.9, 26.4, 22.9, 22.9, 25.3, 17.8, 18.5.

The population standard deviation is unknown. Find the 95% confidence interval for the population mean.

Solution

To find the 95% confidence interval for a single mean using the calculator:

  1. Select One Mean.
  2. Choose Raw Data.
  3. Enter the Weight as the variable name and select Unknown — use t interval for the population standard deviation.
  4. Copy and paste the values into the calculator input field
  5. Enter 95 as the confidence level.
  6. Select 4 decimal places
  7. Click Calculate.

The calculator returns: 95% CI for μ = (22.6209, 26.4591). Therefore, we are 95% confident that the population mean weight for the children lies between 22.62 and 26.46.

Example 2. Confidence Interval for Population Mean (σ Is Known)

Suppose a sample of 36 employees has a mean weekly working time of 84.7 hours. From previous records, the population standard deviation is known to be 12 hours.

Find the 95% confidence interval for the population mean weekly working time.

Solution

To find the 95% confidence interval for a single mean when the population standard deviation is known using the calculator:

  1. Select One Mean.
  2. Choose Summary Statistics.
  3. Enter Weekly working time as the variable name and select Known — use z interval for the population standard deviation.
  4. Enter 36 as the sample size, 84.7 as the sample mean, and 12 as the population standard deviation.
  5. Enter 95 as the confidence level.
  6. Select 4 decimal places.
  7. Click Calculate.

The calculator returns: 95% CI for μ = (80.7801, 88.6199). Therefore, we are 95% confident that the population mean weekly working time lies between 80.78 and 88.62 hours.

Need to calculate a confidence interval for a mean when σ is known or unknown and see the complete z or t calculation? Use the Confidence Interval for Mean Calculator.

Confidence Interval for One Proportion

Choose the One Proportion option when you want to estimate a single population proportion, p. For example, you may want to estimate the proportion of students who pass an examination, customers who prefer a particular product, or patients who respond to a treatment.

This calculator uses the Wilson score interval by default. The Wilson confidence interval is:

CI=p^+z22n±zp^(1−p^)n+z24n21+z2n\text{CI} = \frac{ \hat{p}+\frac{z^2}{2n} \pm z\sqrt{ \frac{\hat{p}(1-\hat{p})}{n} + \frac{z^2}{4n^2} } }{ 1+\frac{z^2}{n} }

where:

  • p̂ is the sample proportion
  • n is the sample size
  • z is the z critical value for the selected confidence level

The Wilson interval is generally preferred over the simple normal approximation because it performs better when the sample size is relatively small or when the sample proportion is close to 0 or 1.

However, if you select Normal Approximation, the calculator uses:

CI=p^±z1−α/2p^(1−p^)n\text{CI} =\hat{p} \pm z_{1-\alpha/2} \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }

Example 3. Confidence Interval for Population Proportion

Suppose 84 of 130 students who enrolled in an online course completed the course. Find the 95% confidence interval for the population proportion of students who complete the course.

Solution

To find the 95% confidence interval for a single proportion using the calculator:

  1. Select One Proportion.
  2. Choose Counts.
  3. Enter Course completion as the outcome, 84 as the number of successes, and 130 as the sample size.
  4. Leave the method as Wilson Score.
  5. Enter 95 as the confidence level.
  6. Select 4 decimal places.
  7. Click Calculate.

The calculator returns: 95% CI for p = (0.5608, 0.7231). Therefore, we are 95% confident that the population proportion of students who complete the online course lies between 0.5608 and 0.7231, or approximately 56.08% and 72.31%.

Need to calculate a confidence interval for a proportion and see the complete calculation? Use the One Proportion Confidence Interval Calculator.

Confidence Interval for Two Means

Choose the Two Means option when you have two independent samples and want to estimate the difference between two population means, μ1−μ2\mu_1-\mu_2. The method used depends on whether the population standard deviations are known.

When the population standard deviations are unknown, the calculator uses a t interval. Welch’s method is selected by default and uses:

CI=(x‾1−x‾2)±t1−α/2,νs12n1+s22n2\text{CI} = (\bar{x}_1-\bar{x}2) \pm t{1-\alpha/2,\nu} \sqrt{ \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} }

where:

  • x̄₁ and x̄₂ are the sample means
  • s₁ and s₂ are the sample standard deviations
  • n₁ and n₂ are the sample sizes
  • t₁−α/2, ν is the t critical value based on the Welch degrees of freedom

However, if equal population variances can reasonably be assumed, you can select the Pooled Variance method. The confidence interval is then:

CI=(x‾1−x‾2)±t1−α/2,n1+n2−2sp1n1+1n2\text{CI} = (\bar{x}_1-\bar{x}2) \pm t{1-\alpha/2,n_1+n_2-2} s_p \sqrt{ \frac{1}{n_1} + \frac{1}{n_2} }

where sp is the pooled sample standard deviation.

However, when both population standard deviations, σ1 and σ2, are known, the calculator uses the z distribution. The confidence interval formula is

CI=(x‾1−x‾2)±z1−α/2σ12n1+σ22n2\text{CI} = (\bar{x}_1-\bar{x}2) \pm z{1-\alpha/2} \sqrt{ \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} }

Example 4. Confidence Interval for Two Means When Population Standard Deviations Are Unknown

Suppose a researcher compares reading scores for students taught using two different teaching methods.

The scores are:

Method A: 78, 82, 75, 88, 84, 79, 81, 85, 77, 83, 86, 80

Method B: 72, 76, 74, 79, 71, 75, 78, 73, 77, 70, 76

The population standard deviations are unknown. Find the 95% confidence interval for the difference between the population mean reading scores.

Solution

To find the 95% confidence interval for the difference between two independent means using the calculator:

  1. Select Two Means.
  2. Choose Raw Data.
  3. Enter Reading scores as the variable name, Method A as Group 1, and Method B as Group 2.
  4. Select Unknown — use t interval for the population standard deviations and leave the method as Welch.
  5. Copy and paste the Method A and Method B scores into their corresponding input fields.
  6. Enter 95 as the confidence level.
  7. Select 4 decimal places.
  8. Click Calculate.

The calculator returns: 95% CI for μ₁ − μ₂ = (3.8909, 9.8364). Therefore, we are 95% confident that the population mean reading score for students taught using Method A is between 3.89 and 9.84 points higher than the population mean score for students taught using Method B.

Example 5. Confidence Interval for Two Means When Population Standard Deviations Are Known

Suppose a researcher compares weekly transport costs for employees from two locations. For Location A, a sample of 50 employees has a mean weekly transport cost of $102.40. The population standard deviation is known to be $15.00. For Location B, a sample of 45 employees has a mean weekly transport cost of $96.10. The population standard deviation is known to be $13.20.

Find the 95% confidence interval for the difference between the two population means.

Solution

To find the 95% confidence interval for the difference between two means when the population standard deviations are known using the calculator:

  1. Select Two Means.
  2. Choose Summary Statistics.
  3. Enter Weekly transport cost as the variable name, Location A as Group 1, and Location B as Group 2.
  4. Select Known — use z interval for the population standard deviations.
  5. For Location A, enter 50 as the sample size, 102.4 as the mean, and 15 as the population standard deviation.
  6. For Location B, enter 45 as the sample size, 96.1 as the mean, and 13.2 as the population standard deviation.
  7. Enter 95 as the confidence level.
  8. Select 4 decimal places.
  9. Click Calculate.

The calculator returns: 95% CI for μ₁ − μ₂ = (0.6290, 11.9710). Therefore, we are 95% confident that the population mean weekly transport cost for Location A is between $0.63 and $11.97 higher than the population mean weekly transport cost for Location B.

Need to calculate a confidence interval for the difference between two means and see the complete steps? Use the Confidence Interval for Difference in Means Calculator.

Confidence Interval for a Paired Mean Difference

Choose the Paired Mean Difference option when the two sets of observations are related. This commonly occurs when the same participants are measured twice, such as before and after an intervention, or when observations are naturally matched in pairs.

The calculator first finds the difference within each pair and then estimates the population mean difference (μd).

The confidence interval formula is:

CI=d‾±t1−α/2,n−1(sdn)\text{CI} = \bar{d} \pm t_{1-\alpha/2,n-1} \left( \frac{s_d}{\sqrt{n}} \right)

where:

  • d̄ is the sample mean of the paired differences
  • sd is the sample standard deviation of the differences
  • n is the number of pairs
  • t₁−α/2, n−1 is the t critical value

You can enter the original paired observations, enter the differences directly, or use summary statistics for the differences.

Example 6. Confidence Interval for a Paired Mean Difference

Suppose a researcher measures reaction time in milliseconds for 12 participants before and after a training program.

Before: 315, 298, 322, 305, 310, 287, 301, 319, 294, 308, 312, 296

After: 302, 290, 310, 297, 300, 281, 294, 306, 288, 299, 304, 289

Find the 95% confidence interval for the population mean difference in reaction time.

Solution

To find the 95% confidence interval for a paired mean difference using the calculator:

  1. Select Paired Mean Difference.
  2. Choose Paired Observations.
  3. Enter Reaction time as the variable name, Before as Condition 1, and After as Condition 2.
  4. Copy and paste the Before and After values into their corresponding input fields.
  5. Enter 95 as the confidence level.
  6. Select 4 decimal places.
  7. Click Calculate.

The calculator calculates the differences as Before − After and returns: 95% CI for μd = (7.3034, 10.5299). Therefore, we are 95% confident that the population mean reaction time before the training program is between 7.30 and 10.53 milliseconds higher than after the training program.

Need to calculate a confidence interval for paired data and see the complete calculation? Use the Paired T-Test Confidence Interval Calculator.

Confidence Interval for Two Proportions

Choose the Two Proportions option when you want to estimate the difference between two independent population proportions (p1-p2). For example, you may want to compare the proportion of patients who recover under two treatments or the proportion of customers who purchase under two marketing strategies.

The calculator uses the Newcombe-Wilson method by default.

Suppose ((L1, U1)) and ((L2, U2)) are the Wilson confidence intervals for the two population proportions. The lower limit is:

(p^1−p^2)(p^1−L1)2+(U2−p^2)2(\hat{p}_1-\hat{p}_2) \sqrt{ (\hat{p}_1-L_1)^2 + (U_2-\hat{p}_2)^2 }

and the upper limit is:

(p^1−p^2)+(U1−p^1)2+(p^2−L2)2(\hat{p}_1-\hat{p}_2) + \sqrt{ (U_1-\hat{p}_1)^2 + (\hat{p}_2-L_2)^2 }

You can also select the normal approximation method. In this case, the confidence interval formula is:

CI=(p^1−p^2)±z1−α/2p^1(1−p^1)n1+p^2(1−p^2)n2\text{CI} = (\hat{p}_1-\hat{p}2) \pm z{1-\alpha/2} \sqrt{ \frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2} }

where:

  • p̂₁ and p̂₂ are the sample proportions
  • n₁ and n₂ are the sample sizes
  • z₁−α/2 is the z critical value

Example 7. Confidence Interval for the Difference Between Two Proportions

Suppose 67 of 110 customers who saw Advertisement A purchased a product, compared with 49 of 105 customers who saw Advertisement B.

Find the 95% confidence interval for the difference between the two population purchase proportions.

Solution

To find the 95% confidence interval for the difference between two proportions using the calculator:

  1. Select Two Proportions.
  2. Choose Counts.
  3. Enter Purchased product as the outcome.
  4. For Group 1, enter Advertisement A, 67 successes, and a sample size of 110.
  5. For Group 2, enter Advertisement B, 49 successes, and a sample size of 105.
  6. Leave the method as Newcombe-Wilson.
  7. Enter 95 as the confidence level.
  8. Select 4 decimal places.
  9. Click Calculate.

The calculator returns: 95% CI for p₁ − p₂ = (0.0093, 0.2688). Therefore, we are 95% confident that the population purchase proportion for Advertisement A is between 0.0093 and 0.2688 higher than the population purchase proportion for Advertisement B.

In percentage-point terms, Advertisement A is estimated to produce a purchase rate between approximately 0.93 and 26.88 percentage points higher than Advertisement B.

Need to calculate a confidence interval for the difference between two proportions and see the complete calculation? Use the Two Proportion Confidence Interval Calculator.

How to Interpret a Confidence Interval

Suppose a 95% confidence interval for a population mean is (68.4, 75.7). We interpret it as follows: We are 95% confident that the population mean lies between 68.4 and 75.7.

For confidence intervals involving a difference, such as (μ1−μ2\mu_1-\mu_2), (μd), or (p1-p2), you should also check whether the interval contains 0.

  • If the entire confidence interval is greater than 0, the first population parameter is estimated to be larger than the second.
  • If the entire interval is less than 0, the first population parameter is estimated to be smaller than the second.
  • However, if the interval contains 0, a population difference of zero is still consistent with the data at the selected confidence level.

This is also why a confidence interval and a corresponding two-tailed hypothesis test usually give the same statistical conclusion when the confidence level and significance level correspond.

What Confidence Level Should You Use?

A 95% confidence level is the most commonly used, although 90% and 99% confidence intervals are also common.

The confidence level affects the critical value and, therefore, the width of the confidence interval.

For the same sample data:

  • a 90% confidence interval is generally narrower than a 95% confidence interval
  • a 99% confidence interval is generally wider than a 95% confidence interval

A higher confidence level gives greater confidence that the interval-producing procedure captures the population parameter, but this comes at the cost of a wider interval.

What Affects the Width of a Confidence Interval?

The width of a confidence interval mainly depends on the confidence level, sample size, and variability in the data.

  • A larger sample size reduces the standard error and usually produces a narrower confidence interval.
  • Greater variability increases the standard error and produces a wider confidence interval.
  • Increasing the confidence level also increases the critical value, which makes the interval wider.

For this reason, collecting more observations is one of the main ways to obtain a more precise estimate without reducing the confidence level.

Frequently Asked Questions

What is a confidence interval calculator?

A confidence interval calculator uses sample data to estimate a range of plausible values for an unknown population parameter. This calculator can find confidence intervals for one population mean, one population proportion, the difference between two independent means, a paired mean difference, and the difference between two independent proportions.

Can I calculate a confidence interval from raw data?

Yes. The calculator accepts raw data for one mean, two independent means, and paired observations. It automatically calculates the sample statistics needed to find the confidence interval. You can also enter summary statistics or upload a CSV, TXT, XLS, or XLSX dataset where supported.

What makes a confidence interval narrower?

A larger sample size generally produces a narrower confidence interval because it reduces the standard error. Lower variability also produces a narrower interval. In contrast, increasing the confidence level makes the interval wider.

What is the general formula for a confidence interval?

Most confidence intervals follow the general form: Point estimate±(critical value)(standard error)\text{Point estimate} \pm (\text{critical value})(\text{standard error}). However, some methods, such as the Wilson and Newcombe-Wilson intervals, calculate the lower and upper limits directly and may not be symmetric around the point estimate.

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StatCalc.net. (2026, October 17). Confidence Interval Calculator. Retrieved October 19, 2026, from https://statcalc.net/confidence-interval-calculator/

About This Calculator

This calculator was developed by the StatCalc.net Team and statistically reviewed by , Senior Statistician & Editor.

Formulas, calculation logic, worked examples, and explanatory content are reviewed for statistical accuracy and clarity.

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