Population Proportion

1) Meaning

A population proportion is a type of parameter that describes the fraction (or percentage) of individuals in a population who have a certain characteristic.

It’s expressed as:

$p = \frac{X}{N}$

Where:

• $X$ = number of individuals in the population with the characteristic
• $N$ = total number of individuals in the population

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2) Example (Simple)

• Suppose a university has 10,000 students.
• Out of these, 6,000 are undergraduates.

$p = \frac{6000}{10000} = 0.6 \; (60\%)$

The population proportion of undergraduates = 0.6 (60%).

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3) In Statistics (Sampling Context)

When the entire population cannot be studied, we estimate the population proportion (p) using a sample proportion ($\hat{p}$​):

$\hat{p} = \frac{x}{n}$

Where:

• $x$ = number of individuals in the sample with the characteristic
• $n$ = sample size

Example:

• Survey 200 students, 120 are undergraduates.

$\hat{p} = \frac{120}{200} = 0.6$

This sample proportion is used to estimate the true population proportion.

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4) Confidence Intervals for Proportions

Because sample proportion $\hat{p}$ is only an estimate, we often construct a confidence interval for the population proportion $p$:

$CI = \hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Example:

• $\hat{p} = 0.6$, $n = 200$, $Z = 1.96$ (95% CI).
• Margin of error = $1.96 \times \sqrt{\frac{0.6 \times 0.4}{200}} \approx 0.068$.
• CI = $0.6 \pm 0.068 = (0.532, 0.668)$.

We’re 95% confident the true population proportion lies between 53.2% and 66.8%.

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5) Applications

• Elections: proportion of voters supporting a candidate.
• Healthcare: proportion of patients responding to a treatment.
• Business: proportion of customers buying a product.
• Uplift modeling/Qini curve: population proportion often appears on the X-axis, showing the share of customers targeted.

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Bottom line:
A population proportion is the percentage of individuals in a population with a certain trait. In practice, it’s often estimated using a sample proportion and evaluated with confidence intervals.