Defining the Population Proportion and Binary Outcome

Why Define the Outcome First?

A percentage has meaning only after we know what is being counted and which units are included. Saying that 35% of a group “use the shuttle,” for example, leaves important questions: Which group? What counts as using the shuttle? Is the percentage based on everyone in that group or only on those who answered a survey? A precise definition answers these questions before a proportion is calculated.

In a test for one proportion, the quantity of interest is a population proportion, conventionally denoted by \(p\). The data produce an observed sample proportion, denoted by \(\hat{p}\). These quantities are related, but they are not the same: \(p\) describes the defined population, while \(\hat{p}\) describes the observed sample.

Definition: A binary outcome assigns each unit to exactly one of two categories, such as “uses the shuttle” or “does not use the shuttle.” After designating one category as the outcome of interest, call it a success. If a target population contains \(N\) units and \(K\) of them have that outcome, its population proportion is \(p=K/N\). In a sample of \(n\) units, if \(x\) have the outcome, the sample proportion is \(\hat{p}=x/n\).

“Success” is a counting label, not a judgment that an outcome is desirable. In a health study, a positive diagnosis might be designated a success for counting purposes even if it is an undesirable health outcome. The other category must be defined as the complement: every unit is either counted as a success or is not.

For a finite population, \(N\) is the number of units in the population being described, and \(K\) is the number of those units with the specified outcome. Thus \(p\) is a fixed proportion for that population, even if its value is not known. In a probability model for units generated from a larger process, \(p\) can also denote the probability that a unit has the specified outcome. In either use, \(p\) refers to the population-level quantity, not the fraction observed in one particular sample.

Population Proportion and Sample Proportion

The definition of \(p\) requires more than choosing a numerator. First specify the population: the people, items, or other units to which the question refers. Then define the binary outcome in terms that can be applied to each unit. The population denominator is the total number of units in that defined population; the population numerator is the number among them with the outcome of interest.

For a sample, use the same outcome rule. The sample denominator \(n\) is the number of sampled units with an observed, usable outcome, and \(x\) is the number of those units classified as successes. The observed proportion is therefore \(\hat{p}=x/n\). It is a summary of those sample observations. If the sample was selected in a way that represents the target population, \(\hat{p}\) may provide information about \(p\), but it does not become the population proportion merely because it is calculated from data.

The denominator must match the units and outcome in the question. A percentage of students is not a percentage of survey responses if some students did not respond. A percentage of inspected products is not automatically a percentage of all products made by a factory. State what was counted and what group the count came from.

Worked Example: Shuttle Use Among Campus Students

Hypothetical setting. A campus wants to describe shuttle use among all 8,000 undergraduate students enrolled on October 1. Define the binary outcome as “used a campus shuttle at least once during the past seven days.” A random sample of 240 undergraduates answers the question: 84 report shuttle use and 156 do not. Identify the population proportion and calculate the observed sample proportion.

Define the population quantity. The population consists of the 8,000 undergraduates enrolled on October 1. Let \(K\) be the number of those students who used a campus shuttle at least once during the specified seven-day period. The population proportion is \(p=K/8{,}000\). The sample alone does not tell us the value of \(K\), so it does not give the exact value of \(p\).

Identify the sample count and denominator. The success category is “used a campus shuttle at least once.” Thus \(x=84\). All 240 sampled students supplied an outcome, so \(n=240\).

Calculate the sample proportion.

\hat{p}=\frac{x}{n}=\frac{84}{240}=0.350

The observed sample proportion is 0.350, or 35.0%. In context, 35.0% of the 240 sampled undergraduates reported using a campus shuttle at least once in the specified week. It is not automatically correct to say that exactly 35.0% of all 8,000 undergraduates did so; \(p\) is the proportion for all 8,000, while \(\hat{p}\) is calculated from the sample.

Worked Examples Across Different Settings

The same counting structure applies whether the units are people, patients, or manufactured items. The details that change are the target population, the outcome rule, and therefore which observations belong in the numerator and denominator.

Worked Example: Readmission After a Clinic Discharge

Hypothetical setting. A health service defines its target population as all 1,200 adult patients discharged from a specified clinic during a three-month period. The outcome is readmission to that clinic within 30 days of discharge. A random sample of 150 eligible discharges has complete 30-day follow-up; 18 patients were readmitted and 132 were not. Define the population proportion and find the sample proportion.

Population quantity. The unit is one eligible patient discharge, and the population is the 1,200 specified adult discharges. If \(K\) of them are followed by a readmission within 30 days, then \(p=K/1{,}200\). The outcome is tied to the stated 30-day period, so a readmission after that period does not meet the outcome definition.

Sample calculation. The numerator is the number of sampled discharges followed by a readmission within 30 days, \(x=18\). The denominator is the 150 sampled eligible discharges with complete follow-up, \(n=150\). Therefore,

\hat{p}=\frac{x}{n}=\frac{18}{150}=0.120

The observed sample proportion is 0.120, or 12.0%. It describes the sampled eligible discharges. The denominator is not the number of all clinic visits, all patients ever treated by the service, or only the 18 readmitted patients; it is the 150 sampled discharges for which the defined outcome was determined.

Worked Example: Defects in a Production Lot

Hypothetical setting. A quality-control team is interested in a lot of 500 sealed components. A component is classified as defective if it fails a specified inspection criterion. The team inspects a random sample of 80 components and finds 3 defective and 77 not defective. Identify the population proportion and calculate the observed sample proportion.

Population quantity. Each component is one unit, and the population is all 500 components in this lot. If \(K\) components in the lot are defective under the stated criterion, then \(p=K/500\). The sample does not reveal the exact value of \(K\) unless the entire lot is inspected.

Sample calculation. Three inspected components meet the defect definition, so \(x=3\). The sample contains 80 inspected components with a recorded outcome, so \(n=80\). The calculation is

\hat{p}=\frac{x}{n}=\frac{3}{80}=0.0375

The observed sample proportion is 0.0375, or 3.75% of the inspected components. It is the sample fraction classified as defective, not a direct count of defective components and not necessarily the exact fraction defective in the whole lot.

Why the Sample Fraction Has This Form

The numerator-and-denominator rule can be expressed using an indicator for each unit. For sampled unit \(i\), define \(Y_i=1\) if it has the outcome of interest and \(Y_i=0\) otherwise. Adding the indicators counts the successes: each success contributes one, and each non-success contributes zero. Dividing by the number of sampled units gives the fraction that are successes.

\hat{p}=\frac{Y_1+Y_2+\cdots+Y_n}{n} =\frac{x}{n}

For a finite population of \(N\) units, the same representation gives its proportion:

p=\frac{Y_1+Y_2+\cdots+Y_N}{N} =\frac{K}{N}

This also explains why the outcome rule and denominator must be fixed before calculating a proportion. Changing what counts as \(Y_i=1\), or changing which units are included, changes the count and may change the quantity being described.

As a brief justification for why a sample proportion can represent a population proportion, suppose a sample of size \(n\) is selected by simple random sampling without replacement from the \(N\) population units. Each population unit then has probability \(n/N\) of being included. Write \(I_j=1\) if population unit \(j\) is selected and \(I_j=0\) otherwise. The sample proportion can be written as

\hat{p}=\frac{1}{n}\sum_{j=1}^{N} I_jY_j

Here, \(Y_j\) is the fixed zero-or-one outcome for population unit \(j\). Since \(P(I_j=1)=n/N\), the expected sample proportion under this sampling design is

E(\hat{p}) =\frac{1}{n}\sum_{j=1}^{N} E(I_j)Y_j =\frac{1}{n}\sum_{j=1}^{N}\frac{n}{N}Y_j =\frac{1}{N}\sum_{j=1}^{N}Y_j =\frac{K}{N} =p

Thus, across repeated simple random samples, the sample proportion has average value \(p\). This does not mean every sample proportion equals \(p\). A particular sample can contain more or fewer successes than the population fraction would suggest.

Common Mistakes and Why the Definition Matters

Calling the sample proportion the population proportion. From the campus example, \(84/240=0.350\) is \(\hat{p}\), the observed sample proportion. The population proportion is \(p=K/8{,}000\); the value of \(K\) was not measured for every student.
Using the wrong denominator. A proportion of readmissions among eligible discharges uses eligible discharges as its denominator, not all clinic visits or only the patients who were readmitted. Name the unit and target group before counting.
Treating a missing outcome as a non-success. If a sampled person has no recorded response, that alone does not establish that the person belongs in the non-success category. Do not silently count an unknown outcome as “no.” State which units have usable outcomes when reporting \(\hat{p}\).
Assuming success means a good result. “Success” is simply the category whose count is in the numerator. A defective item or a positive diagnosis can be designated as a success for the calculation, even though it is undesirable.
Leaving the outcome vague. “Shuttle user” could mean used a shuttle today, in the past week, or at least once this term. The time window and classification rule need to be explicit so that each unit can be assigned consistently to one of the two categories.

A clear definition makes percentages interpretable and makes later statistical questions possible. Before using a one-proportion method, be able to state the population, the unit, the two outcome categories, the success category, the population quantity \(p\), and the sample counts \(x\) and \(n\). The procedure for writing hypotheses about \(p\) is a separate step; it depends on having first defined what \(p\) means.

Check Your Understanding

For each setting, identify the population, define the binary outcome, and state the correct numerator and denominator for the sample proportion.

A campus surveys 300 randomly selected students about whether they have a meal plan. Of these, 174 report having one. What are \(x\), \(n\), and \(\hat{p}\), and what group does that proportion describe?
A clinic is interested in the fraction of its eligible appointments during May that were missed. What must be specified to define the population proportion precisely?
A factory inspects 120 items from a production lot and finds 6 that fail inspection. What is the observed sample proportion of items that fail, and what is its denominator?
Explain why a sample proportion of 0.20 does not necessarily mean that exactly 20% of the target population has the outcome.
A survey records “Yes,” “No,” or “No response.” Why should “No response” not automatically be counted as a non-success when calculating the proportion from observed outcomes?