The sample proportion p̂ (p-hat) is the proportion in a sample: p̂ = x ÷ n, where x is the number of successes and n is the sample size. It estimates the population proportion p. Use this calculator to find p̂, its standard error and a 95% confidence interval, and to check the conditions for using the normal approximation.
Sample Proportion Calculator
p̂ = x ÷ n — with standard error and a 95% interval.
Example: 60 successes in 100 trials gives p̂ = 0.60, or 60%.
What Is a Sample Proportion?
The sample proportion p̂ is the fraction of a sample with a certain characteristic: p̂ = x ÷ n. If 60 of 100 surveyed people say yes, p̂ = 0.60. It is a statistic — a number computed from a sample — and it estimates the parameter p, the proportion in the whole population. Different random samples give different p̂ values; that spread is called sampling variability.
The Sampling Distribution of p̂
For a large enough sample, the sample proportion is approximately normally distributed, centred on the true proportion p, with standard deviation (standard error) σ_p̂ = √(p(1 − p) ÷ n). In practice p is unknown, so we use p̂ in the formula: SE = √(p̂(1 − p̂) ÷ n). Larger samples give smaller standard errors and more precise estimates — quadrupling n halves the standard error.
Conditions for the Normal Approximation
The normal approximation works when the sample is random, when np̂ ≥ 10 and n(1 − p̂) ≥ 10, and when the sample is less than 10% of the population. These conditions make sure the sampling distribution is close enough to normal for z-procedures — the z-test and the confidence interval — to be valid.
Worked Examples
Example 1: In a poll of 400 voters, 220 support a candidate. p̂ = 220 ÷ 400 = 0.55, SE = √(0.55 × 0.45 ÷ 400) ≈ 0.0249, and the 95% interval is 0.55 ± 1.96 × 0.0249 = (0.501, 0.599). Example 2: A factory checks 200 items and finds 8 defective. p̂ = 0.04, SE = √(0.04 × 0.96 ÷ 200) ≈ 0.0139.
Sample Proportion FAQs
What is the difference between p and p̂?
p is the true population proportion — a fixed, usually unknown parameter. p̂ is the sample proportion — a statistic computed from data that estimates p.
What is the standard error of a sample proportion?
SE = √(p̂(1 − p̂) ÷ n). It measures how much p̂ typically varies from sample to sample, and it shrinks as the sample size grows.
Why do we need np̂ ≥ 10 and n(1 − p̂) ≥ 10?
These conditions make sure there are enough successes and failures for the sampling distribution of p̂ to be close to normal, so z-procedures are valid.
Related tools: Proportion Confidence Interval · One-Proportion Z-Test · Sample Size for Proportion · Statistics