Sampling Distribution of the Sample Proportion Calculator

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The true proportion in the population, between 0 and 1. For example, 0.40 means 40% of the population has the characteristic of interest.
The number of independent observations drawn from the population. Larger samples produce a narrower sampling distribution.
Choose whether to find a two-sided probability (between two proportions), a left-tail probability (at most), or a right-tail probability (at least).
The lower boundary for the probability event. In "between" mode this is the lower limit; in "left tail" or "right tail" mode this is the single threshold.
The upper boundary used only in "between two values" mode.
Confidence level for the interval around the population proportion. 95% is most common in practice.
ProbabilityModerate probability
0.8257

Probability that the sample proportion falls in the chosen region.

Mean of p̂ (μₚ̂)0.4
Standard error (σₚ̂)0.049
Z-score of p₁-1.0206
Z-score of p₂2.0412
CI lower bound0.2498
CI upper bound0.5502
CLT approximation valid?Yes (np = 40.0, n(1-p) = 60.0)
-1.020615.4% below · Z-score

Probability = 82.57% under the normal approximation.

  • The sampling distribution of p-hat is centered at 0.4000 (the population proportion) with a standard error of 0.0490.
  • There is a 82.57% chance that a random sample of size 100 will produce a sample proportion between 0.350 and 0.500.
  • The 95% confidence interval for p is (0.2498, 0.5502).

Next stepThis result assumes independent random sampling and a large enough population (at least 10 times the sample size) so that sampling without replacement does not appreciably change the standard error.

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