The 10% Condition for Sampling Without Replacement

Why the Sampling Fraction Matters

When a sample is selected without replacement, choosing one individual changes which individuals remain available for later selections. The observations are therefore not literally independent. For a small sample drawn from a much larger population, however, the effect of each selection on the next is slight. The 10% condition is a practical rule for deciding when that dependence is small enough to treat the observations as approximately independent.

This condition is one part of assessing a sampling procedure. It does not establish that the sample was randomly selected, nor does it address the shape of the data. Those are separate considerations. Here the question is narrower: if sampling is without replacement from a finite population, is the sample small enough relative to that population for the usual independence approximation to be reasonable?

Definition: For a sample of size \(n\) drawn without replacement from a finite population of size \(N\), the 10% condition is satisfied when \(n \leq 0.10N\). When the condition holds, the dependence among sample observations is usually small enough to treat them as approximately independent for standard inference calculations. It does not make them exactly independent.

The condition can also be expressed using the sampling fraction, \(n/N\). If this fraction is at most 0.10, no more than one tenth of the population is sampled. For example, selecting 30 people from a population of 500 gives a sampling fraction of \(30/500=0.06\), or 6%, so the condition is met.

Checking the Condition in Practice

To apply the rule, identify the population relevant to the parameter being estimated, determine its size \(N\), and compare the sample size \(n\) with 10% of that population. Be clear about what the population counts: if the target is the mean for all patients on a particular hospital's registry, \(N\) is the number of patients on that registry, not the number of hospitals in the region.

Identify the finite population.
State which individuals the inference concerns and find the population size \(N\), if it is known or can be reasonably estimated.

Find the sample size.
Use the number \(n\) of individuals actually sampled without replacement from that population.

Compare the sizes.
Check whether \(n \leq 0.10N\), or equivalently whether \(n/N \leq 0.10\). State the comparison in context.

Describe the implication accurately.
If the condition holds, approximate independence is reasonable for this issue. If it fails, the dependence may matter, so do not automatically use the independence approximation.

Worked Example: A Small Sampling Fraction

A hypothetical college has 1,200 students enrolled in a particular introductory course. An administrator selects 72 students without replacement to estimate the mean number of hours they studied for a recent exam. Assess the 10% condition.

The population is the 1,200 students enrolled in that course, so \(N=1{,}200\), and the sample size is \(n=72\). Ten percent of the population is

$$ 0.10N=0.10(1{,}200)=120. $$

Since \(72 \leq 120\), the 10% condition is satisfied. Equivalently, the sampling fraction is

\frac{n}{N}=\frac{72}{1{,}200}=0.06=6\%.

Because the sample was drawn without replacement from a population much larger than the sample, the dependence caused by each selection is small enough to treat observations as approximately independent for standard inference calculations. This conclusion concerns only the independence approximation; it does not establish that the sample is representative or satisfy other conditions needed for an interval.

Why the Rule Works: The Finite-Population Correction

A useful way to understand the condition is to examine the exact variability of a sample mean under simple random sampling without replacement. Suppose a finite population has \(N\) fixed values \(y_1,\ldots,y_N\), population mean \(\mu\), and population variance \(\sigma^2\), where the variance is defined using divisor \(N\). For a simple random sample of size \(n\), every subset of \(n\) population members is equally likely to be selected.

The exact variance of the sample mean under this sampling design is

\operatorname{Var}(\bar{x})= \frac{\sigma^2}{n}\left(\frac{N-n}{N-1}\right).

If the sample observations were independent draws from a population with variance \(\sigma^2\), the variance of their mean would be \(\sigma^2/n\). Sampling without replacement reduces that variance by the factor \((N-n)/(N-1)\). The corresponding multiplier for the standard error is the square root of that factor, called the finite-population correction.

Key result: Under simple random sampling without replacement from a finite population, the standard deviation of the sample mean's sampling distribution is multiplied by \(\sqrt{(N-n)/(N-1)}\) relative to the independent-draw standard error \(\sigma/\sqrt{n}\). The 10% condition is a practical rule for deciding when it is usually reasonable to ignore this reduction and use an independence approximation.

The result can be justified by considering two successive draws. Let \(Y_1\) and \(Y_2\) be the values selected on the first and second draws. Each draw has mean \(\mu\) and variance \(\sigma^2\). Center the population values by writing \(z_i=y_i-\mu\), so that \(\sum_{i=1}^N z_i=0\) and \(\sum_{i=1}^N z_i^2=N\sigma^2\). Every ordered pair of distinct population members is equally likely on the two draws. Thus

\operatorname{Cov}(Y_1,Y_2) =E[(Y_1-\mu)(Y_2-\mu)] =\frac{\sum_{i\ne j}z_i z_j}{N(N-1)} =-\frac{\sum_{i=1}^N z_i^2}{N(N-1)} =-\frac{\sigma^2}{N-1}.

The covariance is negative: after one value is selected, it cannot be selected again, which slightly changes the distribution of the next draw. The sample mean is the average of \(n\) draws, so its variance includes both the \(n\) individual variances and the covariances between all distinct pairs:

\operatorname{Var}(\bar{x}) =\frac{1}{n^2}\left[n\sigma^2+n(n-1)\left(-\frac{\sigma^2}{N-1}\right)\right] =\frac{\sigma^2}{n}\left(\frac{N-n}{N-1}\right).

This also explains why sampling fraction matters. When \(n\) is small relative to \(N\), the correction factor is close to 1. At a sampling fraction near 10%, the factor is about \(\sqrt{0.9}\), or 0.949: the finite-population standard error is roughly 5% smaller than the independent-draw version. This is why the 10% rule is a useful approximation, rather than a claim that dependence vanishes.

Worked Example: A Sample Just Over 10%

A hypothetical clinic has 500 patients on a registry. A simple random sample of 64 patients is selected without replacement to estimate the mean number of days between appointments. The sample standard deviation is 19.2 days. Determine whether the 10% condition holds and compare the usual standard error with the finite-population adjustment.

Ten percent of the registry is \(0.10(500)=50\) patients. Since \(64>50\), the condition is not met. The sampling fraction is

\frac{n}{N}=\frac{64}{500}=0.128=12.8\%.

Using the usual independent-observation formula gives

SE_{\text{usual}}=\frac{s}{\sqrt{n}} =\frac{19.2}{\sqrt{64}} =\frac{19.2}{8} =2.40\text{ days}.

For illustration, the finite-population multiplier is

\sqrt{\frac{N-n}{N-1}} =\sqrt{\frac{500-64}{499}} =\sqrt{\frac{436}{499}} \approx 0.9347.

Applying this factor to the usual standard error gives \(2.40(0.9347)\approx 2.243\) days. Thus, for this sampling design, the usual formula is somewhat larger than the adjusted value. Because the 10% condition fails, one should not simply claim that approximate independence is justified by that rule. The example shows the size of the finite-population adjustment; whether and how to use it in a particular inference procedure depends on the procedure and its assumptions.

When the Population Is Small

A small population requires particular care because even a moderate sample can represent a substantial fraction of all its members. Suppose a survey samples 18 of 120 employees without replacement. The sampling fraction is \(18/120=0.15\), or 15%, so the 10% condition fails. Treating the observations as independent may overstate the sampling variability of the sample mean when the target is specifically the mean of those 120 employees.

At the extreme, if every member of a finite population is measured, \(n=N\), then the sample mean equals the population mean exactly for that finite population. The correction factor is zero, reflecting no sampling uncertainty about that finite-population mean. This does not mean that other sources of uncertainty, such as measurement error or a mismatch between the target population and the observed group, disappear.

Worked Example: A Large Fraction of a Small Population

A hypothetical research team wants to estimate the mean daily water use among the 80 households in a small housing complex. It randomly selects 20 households without replacement. Assess the condition and explain what the team should report about independence.

Here \(N=80\) and \(n=20\). Ten percent of the population is

$$ 0.10N=0.10(80)=8. $$

The sample of 20 exceeds 8, and its sampling fraction is

\frac{n}{N}=\frac{20}{80}=0.25=25\%.

The 10% condition is not satisfied. Each household selected is a noticeable share of this small target population, so the usual approximate-independence justification is not available from the 10% rule. The team should identify the substantial sampling fraction and consider the finite-population aspect when choosing and explaining its standard error calculation. It should not describe the 20 household observations as literally independent.

Common Mistakes and Practical Limits

Calling the observations independent because the condition holds. Sampling without replacement still creates dependence. The condition says that the dependence is small enough for a common approximation, not that it is exactly absent.
Checking the sample size without identifying the population. The comparison must use the size of the population the inference targets, not a larger or unrelated group.
Treating a failed condition as proof that an analysis is impossible. Failure means the independence approximation needs more care; it does not by itself invalidate the data. A finite-population correction may be relevant when the target is a finite population sampled without replacement.
Assuming the 10% condition addresses every assumption. It addresses dependence arising from sampling without replacement. It does not establish random selection, an appropriate measurement process, or other conditions for an inference procedure.
Using the rule for a different sampling design. The condition is intended for sampling without replacement from a finite population. Sampling with replacement has a different dependence structure, and a complicated design may require a design-specific analysis.

The 10% condition matters because it links the sampling plan to the accuracy of the usual standard-error approximation. When it holds, approximate independence is generally a reasonable simplification for a modest sampling fraction. When it does not, especially when the target population is small, report the sampling fraction and avoid relying on the rule as justification for independence. The distinction helps keep the uncertainty calculation aligned with how the data were actually collected.

Check Your Understanding

For each question, focus on the population being sampled and the role of the sampling fraction.

A simple random sample of 45 items is drawn without replacement from a batch of 600. Calculate the sampling fraction and decide whether the 10% condition holds.
A sample of 30 households is taken without replacement from a target population of 250 households. What is the largest sample size that would satisfy the 10% condition?
In your own words, explain why the 10% condition supports approximate independence but does not make observations exactly independent.
A sample includes 90 people from a registry of 700. Does the 10% condition hold? State the comparison that supports your answer.
What does the finite-population correction do to the standard error of a sample mean, and why is its value close to 1 when the sample is small relative to the population?