Confidence Interval Formula Proportion

This calculator works out the confidence interval for a population proportion from your sample data. You enter the number of successes, the sample size, and the desired confidence level, and it returns the lower and upper bounds of the interval around the sample proportion. The calculation uses the standard Wald method with a continuity correction option, giving you a range that likely contains the true population proportion.

Formula

The confidence interval for a proportion is calculated as:

p̂ ± z* × √(p̂(1 – p̂) / n)

Where:

• p̂ (p-hat) is the sample proportion: number of successes (x) divided by sample size (n).
• z* is the critical z-value for the desired confidence level (1.96 for 95%, 2.576 for 99%, 1.645 for 90%).
• n is the sample size (number of observations).
• √(p̂(1 – p̂) / n) is the standard error of the proportion.

The result is a range [p̂ – margin, p̂ + margin] that estimates where the true population proportion lies with the chosen confidence.

Worked Example

Using the default values from the calculator: x = 25 successes, n = 100 sample size, 95% confidence level.

1. Calculate the sample proportion: p̂ = 25/100 = 0.25
2. Calculate the standard error: SE = √(0.25 × (1 – 0.25) / 100) = √(0.25 × 0.75 / 100) = √0.001875 ≈ 0.0433
3. Find the critical z-value for 95% confidence: z* = 1.96
4. Calculate the margin of error: ME = 1.96 × 0.0433 ≈ 0.0849
5. Lower bound = 0.25 – 0.0849 = 0.1651 ≈ 0.165
6. Upper bound = 0.25 + 0.0849 = 0.3349 ≈ 0.335

The 95% confidence interval is approximately 0.165 to 0.335 (or 16.5% to 33.5%). This matches the calculator output.

When to Use This Interval

Use a confidence interval for a proportion when you have categorical data (success/failure, yes/no) and you want to estimate the true percentage in the population. Common applications include:

• Political polling: estimating the share of voters who support a candidate.
• Quality control: the percentage of defective items in a production run.
• Medical research: the proportion of patients who recover with a treatment.
• Market research: the percentage of customers who prefer a product.

The interval gives you a range of plausible values for the unknown population proportion, rather than a single point estimate.

How to Choose the Confidence Level

The confidence level (90%, 95%, 99%) reflects how sure you want to be that the interval contains the true proportion. A higher confidence level produces a wider interval because you are covering more of the sampling distribution. The trade-off is precision: a 99% interval is wider than a 95% interval, but you are more confident it captures the true value. In practice, 95% is the most common choice, but 90% is used when a narrower interval is preferred and 99% when the consequences of being wrong are severe.

Common Mistakes and Misconceptions

One frequent mistake is misinterpreting the interval: a 95% confidence interval does not mean there is a 95% probability that the true proportion lies within this specific interval. It means that if you repeated the sampling process many times, 95% of the intervals you construct would contain the true proportion. The true proportion is fixed; the interval is what varies.

Another error is using the formula when the sample size is too small. The Wald interval (the formula above) works well when the sample has at least 10 successes and 10 failures (np̂ ≥ 10 and n(1-p̂) ≥ 10). If these conditions are not met, the interval can be unreliable, often extending below 0 or above 1. In such cases, consider the Wilson score interval or the exact Clopper-Pearson interval, which handle small samples better.

Also, be careful with units: the proportion is a fraction between 0 and 1. If you want a percentage, multiply by 100. The calculator outputs proportions; convert to percentages by moving the decimal two places.

Limitations of the Wald Interval

The Wald interval is the most commonly taught formula, but it has known limitations. It tends to be too narrow when the true proportion is near 0 or 1, and it can produce bounds outside [0, 1] for small samples. For example, with x=1 and n=20, the 95% Wald interval is approximately [-0.03, 0.13], which includes a negative lower bound—impossible for a proportion. In such cases, use the Wilson interval, which adds a continuity correction and always stays within [0, 1]. The Wilson interval is slightly more complex but is recommended for most practical applications, especially when the sample size is moderate.

Frequently Asked Questions

What is the z* value for 90% confidence?

For a 90% confidence level, the critical z-value is 1.645. This means the margin of error is 1.645 times the standard error. You can find this value from a standard normal table or using a calculator with inverse normal functions.

Can the confidence interval be negative or greater than 1?

Yes, with the Wald formula, if the sample proportion is close to 0 or 1 and the sample size is small, the interval can extend beyond 0 or 1. That is mathematically impossible for a proportion, so it indicates the method is not appropriate for your data. Use the Wilson or exact method instead.

How does sample size affect the interval?

Larger sample sizes produce narrower intervals because the standard error decreases as n increases (the denominator under the square root gets bigger). For example, doubling the sample size reduces the standard error by about 30% (since √2 ≈ 1.414). This is why polls with larger samples have smaller margins of error.

What is the difference between a confidence interval and a margin of error?

The margin of error is half the width of the confidence interval—the amount added and subtracted from the sample proportion. The confidence interval is the full range from the lower to upper bound. In the example above, the margin of error is 0.0849, and the interval is 0.165 to 0.335.

When should I use the Wilson interval instead?

Use the Wilson interval when your sample has fewer than 10 successes or 10 failures, or when the sample proportion is very close to 0 or 1. The Wilson interval is more accurate and always produces bounds within [0, 1]. Many statisticians recommend it as the default for proportions.

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in x | Number of successes | - | 25 | 0
in n | Sample size | - | 100 | 1
in cl | Confidence level (%) | - | 95 | 0
out p | Sample proportion | - | x / n | 3
out se | Standard error | - | sqrt(p * (1 - p) / n) | 4
out me | Margin of error | - | 1.96 * se | 4
out lower | Lower bound | - | p - me | 3
out upper | Upper bound | - | p + me | 3