In statistics, a population proportion refers to the fraction of individuals in a population with a certain characteristic.
For example, suppose 43.8% of individuals in a certain city support a new law. The value 0.438 represents a population proportion.
Formula for a Population Proportion
A population proportion always ranges between 0 and 1 (or 0% to 100% in percentage terms) and it is calculated as follows:
p = X / N
where:
- p: The population proportion
- X: The count of individuals in a population with a certain characteristic.
- N: The total number of individuals in a population.
How to Estimate a Population Proportion
Since it is usually too time-consuming and costly to collect data for every individual in a population, we often collect data for a sample instead.
For example, suppose we want to know what proportion of residents in a certain city support a new law. If the population consists of 50,000 total residents, we may take a simple random sample of 1,000 residents:
We would then calculate the sample proportion as follows:
p̂ = x / n
where:
- p̂: The sample proportion
- x: The count of individuals in the sample with a certain characteristic.
- n: The total number of individuals in the sample.
We would then use this sample proportion to estimate the population proportion. For example, if 367 of the 1,000 residents in the sample supported the new law, the sample proportion would be calculated as 367 / 1,000 = 0.367.
Thus, our best estimate for the proportion of residents in the population who supported the law would be 0.367.
Confidence Interval for a Population Proportion
Although the sample proportion provides us with an estimate of the true population proportion, there’s no guarantee that the sample proportion will exactly match the population proportion.
For this reason, we typically construct a confidence interval – a range of values that are likely to contain the true population proportion with a high degree of confidence.
The formula to calculate a confidence interval for a population proportion is:
Confidence Interval = p̂ +/- z*√p̂(1-p̂) / n
where:
- p̂: sample proportion
- z: the chosen z-value
- n: sample size
The z-value that you will use is dependent on the confidence level that you choose. The following table shows the z-value that corresponds to popular confidence level choices:
| Confidence Level | z-value |
|---|---|
| 0.90 | 1.645 |
| 0.95 | 1.96 |
| 0.99 | 2.58 |
Notice that higher confidence levels correspond to larger z-values, which leads to wider confidence intervals. This means that, for example, a 95% confidence interval will be wider than a 90% confidence interval for the same set of data.
Example: Confidence Interval for a Population Proportion
Suppose we want to estimate the proportion of residents in a city that are in favor of a certain law. We select a random sample of 100 residents and ask them about their stance on the law. Here are the results:
- Sample size n = 100
- Proportion in favor of law p̂ = 0.56
Here is how to find various confidence intervals for the population proportion:
90% Confidence Interval: 0.56 +/- 1.645*(√.56(1-.56) / 100) = [0.478, 0.642]
95% Confidence Interval: 0.56 +/- 1.96*(√.56(1-.56) / 100) = [0.463, 0.657]
99% Confidence Interval: 0.56 +/- 2.58*(√.56(1-.56) / 100) = [0.432, 0.688]
Note: You can also find these confidence intervals by using the Confidence Interval for Proportion Calculator.