Often in statistics we’re interested in estimating the proportion of individuals in a population with a certain characteristic.

For example, we might be interested in estimating the proportion of residents in a certain city who support a new law.

Instead of going around and asking every individual resident if they support the law, we would instead collect a simple random sample and find out how many residents in the sample support the law.

We would then calculate the **sample proportion (p̂)** as:

Sample Proportion Formula:

p̂ = x / n

where:

**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 47 of the 300 residents in the sample supported the new law, the sample proportion would be calculated as 47 / 300 = **0.157**.

This means our best estimate for the proportion of residents in the population who supported the law would be **0.157**.

However, there’s no guarantee that this estimate will exactly match the true population proportion so we typically calculate the **standard error of the proportion** as well.

This is calculated as:

Standard Error of the Proportion Formula:

Standard Error = √p̂(1-p̂) / n

For example, if p̂ = 0.157 and n = 300, then we would calculate the standard error of the proportion as:

Standard error of the proportion = √.157(1-.157) / 300 = **0.021**

We then typically use this standard error to calculate a confidence interval for the true proportion of residents who support the law.

This is calculated as:

Confidence Interval for a Population Proportion Formula:

Confidence Interval = p̂+/- z*√p̂(1-p̂) / n

Looking at this formula, it’s easy to see that **the larger the standard error of the proportion, the wider the confidence interval**.

Note that the **z** in the formula is 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 |

For example, here’s how to calculate a 95% confidence interval for the true proportion of residents in the city who support the new law:

- 95% C.I. = p̂ +/- z*√p̂(1-p̂) / n
- 95% C.I. = .157 +/- 1.96*√.157(1-.157) / 300
- 95% C.I. = .157 +/- 1.96*(.021)
- 95% C.I. = [ .10884 , .19816]

Thus, we would say with 95% confidence that the true proportion of residents in the city who support the new law is between 10.884% and 19.816%.

**Additional Resources**

Standard Error of the Proportion Calculator

Confidence Interval for Proportion Calculator

What is a Population Proportion?