Confidence Interval for Single Proportion

Use this calculator to compute the confidence interval for population proportion when you have binary outcome data (success/failure, yes/no, defective/non-defective).

When to Use This Calculator

  • Binary outcome data (only two possible outcomes: success or failure)
  • Large sample size where $n\hat{p} \geq 5$ AND $n(1-\hat{p}) \geq 5$
  • Population proportion is unknown but estimated from sample
  • Computing range estimates for percentages or success rates
  • You want 90%, 95%, or 99% confidence level

Common Applications:

  • Estimating customer conversion rates from sample of visitors
  • Estimating defect rates in manufacturing
  • Estimating proportion of voters supporting a candidate
  • Estimating disease prevalence in a population sample
  • Estimating customer satisfaction proportion (satisfied vs unsatisfied)

Note: For small samples or extreme proportions (p near 0 or 1), use the Plus-Four method instead.

How to Use This Calculator

Step 1: Enter the sample size (n) - total number of observations

Step 2: Enter the number of successes (k) - how many had the desired outcome

Step 3: Select the confidence level (typically 95%)

Step 4: Click the “Calculate” button

Step 5: The calculator displays:

  • Sample proportion (p̂) - your point estimate
  • Standard Error (SE) - precision of your estimate
  • Z-critical value - from standard normal distribution
  • Margin of Error (E) - range around your estimate
  • Lower and Upper Confidence Limits - your final CI
Confidence Interval Calculator for proportion
Sample Size ($n$)
Number of successes ($k$)
Confidence Level ($1-\alpha$)
Results
Estimate of proportion: ($\hat{p}$)
Standard Error of proportion: ($SE$)
Z-critical value: ($Z_{\alpha/2}$)
Margin of Error: ($E$)
Lower Confidence Limits:
Upper Confidence Limits:

Theory & Formula

Confidence Interval for Single Proportion

When estimating a population proportion $p$ from sample data, use the standard normal (z) distribution.

The $100(1-\alpha)%$ confidence interval for population proportion is:

$$CI = \hat{p} \pm z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Where:

  • $\hat{p} = \frac{k}{n}$ = sample proportion (proportion of successes)
  • $z_{\alpha/2}$ = z-critical value for chosen confidence level
  • $n$ = sample size
  • $SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ = standard error of proportion

Margin of Error: $E = z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Assumptions & Requirements

  1. Binary outcome data - Each observation has exactly two possible outcomes
  2. Random sample - Data collected without systematic bias
  3. Independent observations - Each observation independent of others
  4. Sample size adequate - $n\hat{p} \geq 5$ AND $n(1-\hat{p}) \geq 5$
    • If not satisfied, use the Plus-Four method instead

Step-by-Step Procedure

Step 1: Verify Requirements

Check that:

  • $n\hat{p} \geq 5$ (enough successes)
  • $n(1-\hat{p}) \geq 5$ (enough failures)

If either is violated, use the Plus-Four method.

Step 2: Calculate Sample Proportion

$$\hat{p} = \frac{\text{Number of successes}}{n} = \frac{k}{n}$$

Step 3: Calculate Standard Error

$$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Step 4: Choose Confidence Level and Find z-Critical Value

Confidence α z-critical
90% 0.10 1.645
95% 0.05 1.96
99% 0.01 2.576

Step 5: Calculate Margin of Error

$$E = z_{\alpha/2} \times SE$$

Step 6: Construct Confidence Interval

$$CI = [\hat{p} - E, \hat{p} + E]$$

Worked Examples

Example 1: Online Store Conversion Rate

Scenario: An online store wants to estimate the conversion rate (proportion of visitors who make a purchase). From 500 recent visitors, 45 made a purchase.

Solution:

Step 1: Verify requirements

  • $n\hat{p} = 500 \times (45/500) = 45 \geq 5$ ✓
  • $n(1-\hat{p}) = 500 \times (455/500) = 455 \geq 5$ ✓

Step 2: Calculate sample proportion $$\hat{p} = \frac{45}{500} = 0.09 \text{ or } 9%$$

Step 3: Calculate standard error $$SE = \sqrt{\frac{0.09 \times 0.91}{500}} = \sqrt{\frac{0.0819}{500}} = 0.0128$$

Step 4: Find z-critical value (95% confidence) $$z_{0.025} = 1.96$$

Step 5: Calculate margin of error $$E = 1.96 \times 0.0128 = 0.0251 \text{ or } 2.51%$$

Step 6: Confidence interval $$CI = 0.09 \pm 0.0251 = [0.0649, 0.1151] \text{ or } [6.49%, 11.51%]$$

Interpretation: We’re 95% confident that the true conversion rate is between 6.49% and 11.51%.

Example 2: Product Defect Rate

Scenario: A manufacturer inspects 200 units and finds 8 defective units. What proportion of production is defective?

Solution:

Step 1: Verify requirements

  • $n\hat{p} = 200 \times (8/200) = 8 \geq 5$ ✓
  • $n(1-\hat{p}) = 200 \times (192/200) = 192 \geq 5$ ✓

Step 2: Calculate sample proportion $$\hat{p} = \frac{8}{200} = 0.04 \text{ or } 4%$$

Step 3: Calculate standard error $$SE = \sqrt{\frac{0.04 \times 0.96}{200}} = \sqrt{\frac{0.0384}{200}} = 0.0139$$

Step 4: Find z-critical value (99% confidence) $$z_{0.005} = 2.576$$

Step 5: Calculate margin of error $$E = 2.576 \times 0.0139 = 0.0358 \text{ or } 3.58%$$

Step 6: Confidence interval $$CI = 0.04 \pm 0.0358 = [0.0042, 0.0758] \text{ or } [0.42%, 7.58%]$$

Interpretation: With 99% confidence, the true defect rate is between 0.42% and 7.58%.

How to Interpret Results

Understanding the Confidence Interval

A 95% CI of [0.50, 0.60] means:

  • CORRECT: “If we repeated this study many times, 95% of the resulting intervals would contain the true population proportion”
  • INCORRECT: “There’s a 95% probability the true proportion is in this interval”
  • CORRECT: “We have 95% confidence this interval contains the true proportion”

Decision Guides

Q: “What does the margin of error tell me?”

Margin of Error = Radius of CI

  • Margin of Error = 3% means the point estimate can be ±3% from the true value
  • Smaller margin = more precise estimate
  • Larger margin = more uncertainty

Q: “How do I reduce margin of error?”

Margin of Error = z * √(p(1-p)/n)

Three ways to reduce it:
1. Increase sample size (n) - more data = narrower interval
2. Lower confidence level - 90% CI narrower than 95% or 99%
3. Choose p closer to 0.5 - proportions near 0/1 have different variability

Q: “Is this proportion equal to a target value?”

Is target (e.g., 5%) within the CI?

If YES → Cannot conclude it differs from target
If NO → Target proportion likely incorrect

When Sample Size is Small: Use Plus-Four Method if:

  • $n < 40$ OR
  • $\hat{p} < 0.1$ (less than 10% successes) OR
  • $\hat{p} > 0.9$ (more than 90% successes)

Comparing Two Proportions: Use CI for Two Proportions calculator

Tutorial: Complete guide to CI for proportions Examples: Worked examples with solutions