What Is The 10 Condition In Statistics | Sample Safety Rule

It’s the rule that a sample should stay at or under 10% of a finite population so the draws act close to independent.

You’ll see the “10% condition” once you start doing inference. It’s a small checkpoint with a big job: it tells you when you can treat your observations as if they don’t tug on each other.

That matters because most intro-stat formulas assume one pick doesn’t change the next. In real sampling, you usually don’t put items back. You take people off a roster, parts out of a batch, or rows out of a file. Each draw changes what’s left.

What the 10% condition means in plain language

The 10% condition says your sample size n must be no more than 10% of the population size N when you sample without replacement from a finite population. The quick check is:

• n ≤ 0.10 × N

If that inequality holds, many standard procedures treat the observations as “close enough” to independent for coursework and most basic reports.

What counts as the population here

For this condition, “population” means the finite set you could have drawn from at the moment you sampled. Think: all students enrolled this term, all units in a shipping crate, all registered members in a club, all invoices in a month’s folder.

If you can’t name a finite set with a count, this condition may not be the right tool.

Where the 10% condition shows up in statistics

Teachers often bundle the 10% condition under “independent observations.” You’ll see it paired with confidence intervals and hypothesis tests for means and proportions, and with normal approximations for counts.

AP Statistics materials spell it out directly: the “10 Percent Condition” is that the sample is less than 10 percent of the population, used to justify treating trials as close to independent when sampling from a finite group. College Board’s condition list puts it alongside the random condition in model checks.

What it does not fix

Passing the 10% condition doesn’t rescue a biased sample. If the sample was chosen in a skewed way, the math can be tidy while the conclusion misses the mark. This condition only targets dependence created by sampling without replacement.

Understanding the 10% condition in statistics for independence checks

Independence keeps your standard errors honest. When observations are linked, the usual “plug-in” formulas can misstate the spread of a statistic. That can make intervals too narrow and tests too eager.

Sampling without replacement creates that link because the pool shrinks. The more you take, the more the later draws feel the earlier draws. The 10% condition is the classroom-friendly line where that effect is treated as small.

Sampling with replacement vs. without replacement

With replacement: you draw, record, then return the item. Each draw faces the same pool, so independence is a fair assumption.

Without replacement: you draw and keep the item out. The pool changes, so strict independence fails.

Most real surveys are without replacement. You don’t interview the same person twice in one sample, and you don’t inspect the same part twice.

Why “fraction” beats “number”

A sample of 80 can be tiny in a population of 50,000, and huge in a population of 600. The 10% condition forces you to think in ratios, not raw counts.

How to check the 10% condition step by step

1. State the population and its size (N). Name the finite group you sampled from.
2. State the sample size (n). Use the number of observations collected.
3. Compute 10% of the population. Calculate 0.10 × N.
4. Compare. If n ≤ 0.10 × N, the condition is met.

When N is unknown

If you can’t get an exact N, use a defensible count or bound. A company headcount, an enrollment figure, an inventory count, or a database total often works. If the population isn’t a fixed finite list (like “all future visits”), don’t force this condition into the write-up.

A note for database samples

When you sample rows from a database extract, the population is the extract itself, not each record the company has stored. If your extract contains 120,000 transactions from January and you sample 1,000 of them, the 10% condition passes easily. If your extract contains 6,000 records and you sample 1,000, it fails. Same sample size, different context.

Common mistakes and quick fixes

Using the wrong population

If you sampled from a roster, the roster is the population for this check, even if you wish you could talk about a wider group. The condition follows the sampling frame you actually used.

Using it for repeated measurements

If you measure the same person twice, dependence comes from pairing or time effects, not from a shrinking pool. The 10% condition doesn’t handle that setup.

Checking it in situations that are not sampling

If you’re modeling a process with independent trials (coin flips, defect checks on a long production line where each item is new), independence comes from the process model, not from a roster being depleted.

When the 10% condition matters most

This condition becomes a real decision point when your sample is a noticeable slice of a finite group. It comes up a lot in:

• Small schools and classes
• Internal staff surveys
• Small customer lists
• Batch and lot inspection
• Audits of a fixed file or folder

In these settings, a “normal” sample size can cross 10% fast. That’s not a disaster. It’s a signal to be precise about assumptions and method.

Decision table for the 10% condition in real tasks

This table helps you spot when the condition is the right check.

Situation	Check 10% condition?	Reason
Survey a fixed roster (school, club, staff list)	Yes	Finite list; sampling without replacement creates dependence
Audit invoices from one month’s file	Yes	Each draw changes what remains in the file
Pick students for a committee from a small class	Yes	Finite class list; sample fraction can be large
Inspect parts from one batch or lot	Yes	Batch is finite; later picks depend on earlier picks
Randomized experiment with distinct subjects	Usually no	Independence comes from separate units, not a shrinking pool
Repeated measurements on the same unit	No	Dependence comes from pairing or time effects
Process model like coin flips or new items on a line	No	Independence is built into the trial model
Sampling “future events” with no fixed list	No	No finite population size to compare against

What happens if the 10% condition fails

If n is more than 10% of N, dependence from sampling without replacement is no longer small. Standard intro procedures may still be used in some settings, but you should explain why the method fits, or adjust the method to match the setup.

Finite population correction in one sentence

In survey sampling, the finite population correction (FPC) is a standard-error adjustment that accounts for a large sampling fraction, since a big slice of a small population leaves less uncertainty than “infinite population” formulas assume.

When it turns into a census

If you measured almost all members of the population, you might not need inference. You can report the parameter for that group directly, then state limits on generalizing beyond that group.

Worked checks you can copy into homework and reports

These are short write-ups you can adapt without rewriting your whole solution.

Example: Student survey in one grade

Population: all 480 tenth-grade students. Sample: 32 students chosen at random from the grade roster.

• 0.10 × 480 = 48
• 32 ≤ 48, so the 10% condition is met

Example: Batch inspection in a small lot

Population: 260 parts in one lot. Sample: 40 parts selected without replacement for inspection.

• 0.10 × 260 = 26
• 40 > 26, so the 10% condition is not met

Table of pass/fail checks with report-ready wording

Use this as a template for clean justification sentences.

Population N and sample n	Pass?	Sentence to use
N = 2,500; n = 120	Yes	120 is within 10% of 2,500, so draws act close to independent
N = 900; n = 70	Yes	70 is within 10% of 900, so near-independence is reasonable
N = 600; n = 85	No	85 is over 10% of 600, so dependence from sampling without replacement is not small
N = 1,200; n = 140	No	140 is over 10% of 1,200, so an FPC-style adjustment may fit better
N = 50,000; n = 800	Yes	800 is a small slice of 50,000, so the shrinking pool effect is tiny
N = 320; n = 20	Yes	20 is within 10% of 320, so standard independence checks can proceed

How it fits with other condition checks

Students often mix conditions together. A clean split is:

• Random selection: guards against selection bias.
• 10% condition: guards against dependence from sampling without replacement.
• Shape checks: guard the model form you’re about to use (normal, t, or count-based approximations).

Khan Academy’s reference page for inference on a mean lists the 10% rule in this role: it’s the independence check used when sampling without replacement from a finite population. Khan Academy’s conditions for inference on a mean includes that checklist language.

Practical tips for exams and project write-ups

If you’re writing a full solution, don’t bury the 10% check in a paragraph. Put it in a short, visible block: state N, state n, compute 0.10 × N, then write the one-line conclusion. Graders love that layout because they can scan it.

If you’re building your own study notes, keep one reminder next to the formula sheet: “Only check this when sampling without replacement from a finite group.” That single line keeps you from wasting time on problems that don’t even have an N to compare against.

A short checklist for clean write-ups

• Name the population and state N.
• State the sample size n.
• Show 0.10 × N.
• Write one sentence connecting the check to near-independence.