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Large Counts Condition — Definition, Formula & Examples

The Large Counts Condition is a rule that checks whether a sample is large enough for the sampling distribution of a proportion to be approximately normal. It requires that both the expected number of successes and the expected number of failures are at least 10.

For inference about a population proportion pp based on a sample of size nn, the Large Counts Condition states that the sampling distribution of p^\hat{p} is approximately normal when np≥10np \geq 10 and n(1−p)≥10n(1 - p) \geq 10. When testing a hypothesis, use the hypothesized proportion p0p_0; when constructing a confidence interval, use the sample proportion p^\hat{p}.

Key Formula

np≥10andn(1−p)≥10np \geq 10 \quad \text{and} \quad n(1 - p) \geq 10
Where:
  • nn = Sample size
  • pp = Population proportion (use $p_0$ for hypothesis tests or $\hat{p}$ for confidence intervals)

How It Works

Before performing a z-test or z-interval for a proportion, you must verify this condition. Multiply the sample size by the proportion to get the expected successes, and multiply the sample size by one minus the proportion to get the expected failures. If both values are at least 10, the condition is satisfied and the normal approximation is reliable. If either value falls below 10, the sampling distribution may be too skewed for z-procedures to give accurate results.

Worked Example

Problem: A researcher surveys 200 voters and wants to test whether the true proportion who support a policy differs from 0.40. Check the Large Counts Condition using the null hypothesis proportion p0=0.40p_0 = 0.40.
Expected successes: Multiply the sample size by the hypothesized proportion.
np0=200×0.40=80np_0 = 200 \times 0.40 = 80
Expected failures: Multiply the sample size by one minus the hypothesized proportion.
n(1−p0)=200×0.60=120n(1 - p_0) = 200 \times 0.60 = 120
Check the condition: Both 80 and 120 are at least 10, so the condition is satisfied.
80≥10✓120≥10✓80 \geq 10 \quad \checkmark \qquad 120 \geq 10 \quad \checkmark
Answer: The Large Counts Condition is met. The normal approximation is appropriate for this hypothesis test.

Why It Matters

On the AP Statistics exam, failing to verify the Large Counts Condition before performing a one-proportion z-test or z-interval will cost you points, even if your calculations are correct. Beyond the exam, this check prevents you from drawing conclusions based on a normal model that does not actually fit the data.

Common Mistakes

Mistake: Using the sample proportion p^\hat{p} when checking the condition for a hypothesis test.
Correction: For hypothesis tests, always use the null hypothesis value p0p_0. The sample proportion p^\hat{p} is only used when checking this condition for confidence intervals, where no hypothesized value exists.

Related Terms