Why Check the Population Size?
In Recognizing Random Sampling Methods in Study Descriptions, you learned to identify how a sample was selected. Now consider another detail: how large was the population it came from? When a random sample is taken without replacement, each selection slightly changes the group left to be sampled. The 10% condition checks whether that dependence is small enough to treat the observations as approximately independent for the usual one-proportion inference procedures.
The condition is not a test of whether the sample is random. A random sample can be too large relative to its population to meet the condition, and a small sample does not become random simply because it is less than 10% of a population. These are separate checks, as discussed in Why Inference Procedures Need Conditions.
If the condition is met, the observations can be treated as approximately independent for the standard sampling model. They are not literally independent: selecting one individual still changes who remains. The condition says that this effect is small enough for the usual AP Statistics approach. As in When the 10 Percent Condition Applies, the condition is relevant to sampling without replacement from a finite population.
Two Equivalent Ways to Check
You can check the condition by calculating 10% of the population and comparing it with the sample size. Alternatively, calculate the sample fraction \(n/N\) and see whether it is at most 0.10. These methods give the same decision. The first is often easier when the population size is a whole number; the second makes the sample’s share of the population especially clear.
You can also rearrange the inequality to find the minimum population size for a given sample size:
This rearrangement is useful when a study description gives \(n\) but asks how large the population must be. For instance, a sample of 150 requires a population of at least \(10(150)=1{,}500\) to meet the condition.
The boundary counts: a sample exactly equal to 10% of the population meets the condition. A sample even slightly larger than 10% does not. Use the actual sample size and population size rather than rounding a percentage in a way that changes the decision.
Choose the Right Population Size
The \(N\) in the condition is the number of individuals in the population from which the sample was selected—not automatically the population someone hopes to describe. Begin with the sampling frame: the list or group from which the random selection was made. Then check that this frame corresponds to the target population for the inference. The random condition concerns how well the sampling process supports that target; the 10% condition concerns the sample’s size relative to its source population.
For example, if a researcher randomly selects customers from a list of active account holders, use the number of active account holders on that list as \(N\). Do not use the much larger number of all residents in the area just because the researcher hopes to describe those residents. If the sampling frame does not represent the intended target, the 10% condition cannot fix that problem.
A one-proportion inference may involve several conditions. The 10% condition addresses the independence approximation for sampling without replacement. It does not establish that the sample was randomly selected, and it does not replace the Large Counts condition for a Normal-based procedure. Check each applicable condition for its own reason.
Worked Examples
Worked Example: A Random Sample of Library Members
A fictional library system has 2,400 active members. A researcher uses a random-number generator to select 180 members without replacement and asks whether they have borrowed an audiobook in the past month. Does the sample meet the 10% condition?
State. The sample size is \(n=180\), and the population size from which the members were selected is \(N=2{,}400\). The question is whether \(n \leq 0.10N\).
Plan. Because the members were selected without replacement, compare the sample size with 10% of the 2,400 active members. As a second check, calculate the sample fraction.
Do. Ten percent of the population is \(0.10(2{,}400)=240\). Since \(180 \leq 240\), the condition is met. Checking the fraction gives \(180/2{,}400=0.075\), or 7.5%, also no more than 10%.
Conclude. The sample uses 7.5% of the active members, so the 10% condition is met. The observations may be treated as approximately independent for an appropriate one-proportion inference procedure. This conclusion addresses only the 10% condition; the random condition and any other requirements still need their own checks.
Worked Example: Equality at the Boundary
A fictional school has exactly 840 students. A researcher randomly selects 84 students without replacement to ask whether they used the school’s tutoring center this term. Does this sample meet the condition? What if 85 students had been selected instead?
Check the sample of 84. Here \(N=840\) and \(n=84\). Ten percent of the population is \(0.10(840)=84\). Therefore, \(84 \leq 84\), so the condition is met exactly at the boundary. The fraction check agrees: \(84/840=0.10\), or 10%.
Check the sample of 85. Ten percent of the population is still 84, and \(85 \not\leq 84\), so a sample of 85 does not meet the condition. The sample fraction confirms the decision: \(85/840 \approx 0.1012\), or about 10.12%.
Conclude. Equality counts: the sample of 84 meets the condition, but the sample of 85 does not. Do not interpret “no more than 10%” as “strictly less than 10%.” Also avoid rounding 10.12% to 10% and then declaring the second sample acceptable; the exact comparison shows it is over the limit.
Worked Example: Finding the Minimum Population Size
A fictional environmental team plans to select a random sample of 600 registered permit holders without replacement. What is the smallest population size that would meet the 10% condition? If the current list contains 5,600 permit holders, does the condition hold?
Find the minimum. Start with \(n \leq 0.10N\) and rearrange: \(N \geq 10n\). Substituting \(n=600\), the population must have at least \(10(600)=6{,}000\) permit holders.
Check the current list. With \(N=5{,}600\), 10% of the population is \(0.10(5{,}600)=560\). The proposed sample size of 600 is greater than 560, so the condition is not met. The fraction check gives \(600/5{,}600 \approx 0.1071\), or about 10.71%, which is also over 10%.
Conclude. A sample of 600 meets the condition only if it is drawn from a population of at least 6,000. From the current list of 5,600, it would be about 10.71% of the population, so the usual 10% condition fails. Do not change the denominator to a larger group unless that larger group is actually the population from which the sample was selected.
Worked Example: Use the Sampling Frame, Not a Larger Community
A fictional service randomly selects 100 people without replacement from its list of 900 active account holders. The service hopes to describe those account holders. The surrounding county has 40,000 residents. Which population size should be used, and does the sample meet the condition?
Identify the relevant population. The random selection was made from the 900 active account holders, so \(N=900\) for the 10% check. The county’s 40,000 residents were not the group from which these 100 people were selected.
Check the condition. Ten percent of the account-holder population is \(0.10(900)=90\). Since \(100>90\), the condition is not met. The sample fraction gives the same result: \(100/900 \approx 0.1111\), or about 11.11%. Using the county count would give \(100/40{,}000=0.0025\), or 0.25%, but that calculation uses the wrong population for this selection process.
Conclude. The sample is about 11.11% of the active account holders, so it does not meet the 10% condition. The county’s larger population size is irrelevant to this check. Separately, a sample drawn only from account holders would not automatically support conclusions about all county residents; that is a question about the target population and the random condition.
What the Condition Does—and Does Not—Tell You
When the condition is met, it supports treating observations as approximately independent in the standard model for \(\hat{p}\). In the earlier tutorials on the standard deviation and sampling distribution of \(\hat{p}\), independence was part of the setting for the usual spread formula. The 10% condition is a practical check for that independence assumption when sampling without replacement from a finite population.
When the condition is not met, do not claim that the observations are approximately independent under the usual 10% guideline. That does not mean the sample was necessarily selected incorrectly or that its descriptive results cannot be reported. It means this condition does not justify the usual independence approximation for inference. Follow the requirements of the specific procedure rather than quietly ignoring a failed check.
The condition applies to the sampling fraction, not to the number or percentage of successes. A sample could contain 20% successes and still meet the 10% condition if the sample is no more than 10% of its source population. Conversely, a sample with very few successes could fail the condition if it is too large relative to that population.
If individuals are sampled with replacement, the 10% condition for sampling without replacement is not the relevant check: the selection process does not shrink the pool after each draw. In typical survey settings, however, individuals are sampled without replacement, so identify that detail before deciding whether to apply the condition.
Common Mistakes and AP Exam Tip
- Using the target population instead of the source population. Use the number of individuals who were eligible to be selected from the stated frame. If the frame does not match the target, discuss that limitation separately.
- Treating 10% as a strict cutoff. The condition is \(n \leq 0.10N\), so exactly 10% meets it. A sample above 10% does not.
- Confusing the condition with the sample’s response rate. The comparison is \(n/N\), where \(n\) is the selected sample size and \(N\) is the source population size. It is not the percentage of invited people who answered.
- Assuming this check proves the whole procedure is appropriate. The 10% condition does not prove that the sample is random, representative of the target, or large enough for a Normal approximation.
- Rounding before making the decision. Compare \(n\) with \(0.10N\), or calculate \(n/N\) with enough precision to determine whether it exceeds 0.10. Report rounded percentages only after the decision is clear.
Key Takeaway
For sampling without replacement, compare the sample size with 10% of the population it was drawn from. Equality meets the condition, and the equivalent population-size check is \(N \geq 10n\). Keep this independence check separate from the Random condition, the target-population question, and the Large Counts condition.
Check Your Understanding
For each situation, identify the appropriate \(n\) and \(N\), check the 10% condition, and state your conclusion in context.
- A random sample of 250 households is drawn without replacement from a city list of 3,000 households. Does the condition hold? Show the population-threshold comparison.
- A researcher samples 72 students without replacement from a school with 720 students. Is the sample exactly at, below, or above the 10% boundary?
- A random sample of 410 members is drawn without replacement from a membership list of 4,000. Check the condition using both \(0.10N\) and the sample fraction.
- A company selects 120 people from a list of 1,000 active customers but wants to make a claim about all residents of the state. Which population count is used for the 10% check, and what separate issue should be considered?
- A student says, “The sample had only 8% successes, so it meets the 10% condition.” Explain what quantity the student should compare instead.