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3.2 Sampling Distributions for Sample Proportions

One Population, Many Possible Samples

Imagine a library with 10,000 registered members. Its complete records show that exactly 60% have borrowed an audiobook this year. In this teaching model, the population proportion is known: p=0.60p=0.60. A simple random sample of 100 members might contain 58 audiobook borrowers, producing p^=0.58\hat p=0.58. Another sample might contain 64. The observed sample proportions vary; the population proportion remains fixed.

The sampling distribution of p^\hat p describes the proportions from all possible random samples of the same size from this population. Its mean is the long-run center of those proportions, not the result of one particular sample. When sampled outcomes are independent, its mean and standard deviation are

μp^=p,σp^=p(1−p)n.\mu_{\hat p}=p,\qquad \sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}.

The standard deviation is measured in proportion units. A value of 0.05 means about five percentage points of typical sample-to-sample deviation from the center. It is not five people and does not predict the difference between two consecutive samples. Increasing n reduces this spread: larger samples tend to estimate the same fixed p more precisely.

For sampling without replacement, the observations are not literally independent. The usual 10% condition makes dependence small enough for the standard independent-sampling formula to serve as a close approximation. A finite-population correction is outside the calculations used here. We will identify when we use this approximation rather than calling every finite-population calculation exact.

Check the Design Before the Curve

Two design conditions matter when sampling without replacement. First, the data must come from a random sample. A large voluntary poll does not become a random sample just because many people respond. Second, the population must be at least ten times the sample size:

n≤0.10N.n\leq 0.10N.

Here N is the number of members in the population and n is the number sampled. For the library, 100≤0.10(10,000)=1,000100\leq 0.10(10{,}000)=1{,}000. Sampling with replacement or an explicitly independent trial model does not require this without-replacement check.

The distribution of a sample proportion is approximately normal when both expected counts are at least 10:

np≥10andn(1−p)≥10.np\geq 10\quad\text{and}\quad n(1-p)\geq 10.

Use the population p in this theoretical calculation. The library’s expected counts are 60 audiobook borrowers and 40 nonborrowers, so both meet the condition. These are expected counts over repeated samples; an actual sample need not have exactly 60 and 40.

SettingDesign and countsIs a normal probability calculation justified?
Library SRS of 100Random; 100 is at most 10% of 10,000; expected counts 60 and 40Yes, using the usual sampling approximation
Rare borrowing category with p = 0.02 and n = 100Expected counts 2 and 98No: the expected-success count is too small
First 100 people who answer an open surveyNo random selectionThe random-sampling model is unsupported

Worked Example: An Unusually High Sample Proportion

Under the library model,

μp^=0.60,σp^≈0.60(0.40)100=0.04899.\mu_{\hat p}=0.60,\qquad \sigma_{\hat p}\approx\sqrt{\frac{0.60(0.40)}{100}}=0.04899.

To estimate the chance that an SRS of 100 has more than 70% audiobook borrowers, compare the cutoff 0.70 with the sampling distribution, rather than with the distribution of individual yes/no responses:

z=0.70−0.600.04899≈2.0412,P(p^>0.70)≈P(Z>2.0412)=0.0206.z=\frac{0.70-0.60}{0.04899}\approx 2.0412,\qquad P(\hat p>0.70)\approx P(Z>2.0412)=0.0206.

This is a normal approximation without a continuity correction. A discrete sample proportion can take only values 0, 0.01, 0.02 and so on; the smooth normal curve estimates its probability. The result says that about 2.1% of repeated random samples of 100 from this library would have a proportion above 0.70 under the stated model. It does not say that 2.1% of individual members borrow audiobooks.

Keep the Interpretation Attached to the Sample Size

The mean 0.60 is the long-run average sample proportion for repeated size-100 samples. The SD of about 0.049 is the typical deviation of those sample proportions from 0.60. The calculated probability concerns the proportion in an entire random sample of 100. All three statements name the same population and the same statistic.

If an actual sample gives p^=0.58\hat p=0.58, that observed number does not replace the known population p in this theoretical model. In a real study where p is unknown, a sample proportion estimates p and can be used in an estimated standard error for inference. That is a different calculation with a different source of information. Always ask whether the problem supplied a known parameter or only an observed summary.

Try It Yourself

The following original practice applies the ideas above to a fresh setting. Show the reasoning requested in each part. Answers and task-specific rubrics are on the separate solutions page.

Original generated practice for the Fall 2026 course.

Repeated Library Proportions

A city library has 20,000 registered members. Complete membership records establish that exactly 40% renewed online this year, so the population proportion is known to be p=0.40. Select an SRS of 200 distinct members without replacement. Independently repeat this whole sampling procedure when discussing its sampling distribution. One actual sample happens to contain 79 online renewals. Use the usual 10% approximation and, where justified, the normal approximation without a continuity correction.

Part a

Calculate the mean and standard deviation of the sampling distribution of the online-renewal sample proportion. Show the formula and substitution. Calculate the observed sample proportion from 79 renewals, and explain why it does not replace the known p in your theoretical mean or SD.

Part b

Justify whether this sample design and a normal model for its sample proportion are appropriate. Address randomization, the without-replacement 10% condition with N and n, and both expected-success and expected-failure counts. Explain what the 10% condition approximates.

Part c

Evaluate three alternative designs separately: (i) an SRS of 3,000 from the same 20,000 members; (ii) an SRS of 200 for a different known category with p=0.02; (iii) 200 responses to a voluntary website poll. Identify each failed condition with a calculation or design reason. Would an independently generated with-replacement sample require the 10% check? Explain.

Part d

Estimate P(p^≥0.45)P(\hat p\geq 0.45) for the original SRS of 200 using your justified normal model. Show the standardized cutoff and area direction. Interpret the distribution mean, its SD and this probability in the library context, with the repeated sample size explicit.

Worked answer and rubric

Glossary and deeper reading · 16 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Population proportion

    A population proportion is the fraction of the entire population with a specified characteristic. It is a parameter, commonly written p, that a sample proportion can estimate. For example, the fraction of all enrolled students who walk to school is the population proportion of walkers.

    Read more: Population proportion in the captured Statistics reading →

  • Simple random sample

    A simple random sample of a fixed size gives every possible subset of that size the same chance of selection. Giving individuals equal selection chances alone does not establish that the design is an SRS. For example, randomly choosing three identifiers without replacement from ten gives an SRS when every three-identifier set is equally likely.

    Read more: Simple random sample in the captured Statistics reading →

  • Sampling distribution

    A sampling distribution describes how a statistic varies across all possible samples drawn under a specified sampling design and sample size. It is a distribution of statistics rather than of individual observations. For example, the proportions from repeated random samples of 60 students form a sampling distribution.

    Read more: Sampling distribution in the captured Statistics reading →

  • Random sampling

    Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.

    Read more: Random sampling in the captured Statistics reading →

  • Standard deviation

    Standard deviation measures spread around the mean using squared deviations and a square root. A sample standard deviation uses the sum of squared deviations divided by one less than the sample size before taking the square root. For example, a larger standard deviation of travel times indicates greater variability, in the original time units.

    Read more: Standard deviation in the captured Statistics reading →

  • Sample

    A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.

    Read more: Sample in the captured Statistics reading →

  • Sampling without replacement

    Sampling without replacement does not return selected items before later selections. Selections from a finite population are dependent, although treating them as approximately independent can be reasonable when the sample is a small fraction of the population. For example, selecting students without replacement ensures that no student is chosen twice.

    Read more: Sampling without replacement in the captured Statistics reading →

  • Statistical independence

    Statistical independence means that learning one outcome does not change the probability distribution of another. For two events, it implies that the probability of both occurring equals the product of their probabilities. For example, independent draws with replacement keep the second draw’s color probabilities unchanged by the first.

    Read more: Statistical independence in the captured Statistics reading →

  • Population

    A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.

    Read more: Population in the captured Statistics reading →

  • Sampling with replacement

    Sampling with replacement returns a selected item to the pool before the next selection. With independent random selections, the available composition and selection probabilities remain unchanged. For example, replacing a drawn card before reshuffling allows that same card to be drawn again.

    Read more: Sampling with replacement in the captured Statistics reading →

  • Sample proportion

    A sample proportion is the number of sampled individuals with a specified characteristic divided by the sample size. It is commonly written p-hat and estimates the corresponding population proportion. For example, 18 walkers among 60 sampled students give a sample proportion of 0.30.

    Read more: Sample proportion in the captured Statistics reading →

  • Expected count

    An expected count is the count predicted for a category or cell under the null model. Goodness-of-fit expectations use the total count times the null proportion, while independence-table expectations use row total times column total divided by the grand total. For example, a null proportion of 0.2 in 50 trials gives an expected count of 10.

    Read more: Expected count in the current two-way chi-square lesson →

  • Mean

    The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.

    Read more: Mean in the captured Statistics reading →

  • Probability

    Probability quantifies how likely an event is under a specified chance model. It lies between 0 and 1, with larger values representing greater likelihood. For example, drawing a red card from a well-shuffled standard deck has probability one half.

    Read more: Probability in the captured Statistics reading →

  • Standard error

    A standard error describes the standard deviation of a statistic’s sampling distribution, usually estimated from data. It measures uncertainty in an estimate rather than the spread of individual observations. For example, the estimated standard error of a sample mean is the sample standard deviation divided by the square root of the sample size.

    Read more: Standard error in the captured Statistics reading →

  • Statistical inference

    Statistical inference uses data and a study design to draw conclusions beyond the observed values. Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions. For example, a random sample can estimate a school-wide preference without surveying every student.

    Read more: Statistical inference in the captured Statistics reading →

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