When you survey n people and count how many say yes, the share p̂ = successes/n varies from sample to sample. Its sampling distribution is centred on the true population proportion p, and its spread – the standard error – is √(p(1 − p)/n). Because p̂ is an average of 0/1 values, the central limit theorem makes it approximately normal when the sample has enough successes and failures.
The calculator uses that normal model to answer questions like “if 70% of customers are satisfied, how likely is a survey of 500 to show more than 65% satisfied?”. It reports the standard error and z-scores and shades the probability on the curve.
Because the count of successes is really binomial, the calculator also gives the exact binomial probability for the same event. When n·p and n·(1 − p) are large the two agree closely; when they are small the exact figure is the one to trust, and the conditions check tells you which situation you are in.
How to use the Sampling Distribution of the Sample Proportion Calculator
- Population proportion (p)
- Sample size (n)
- What probability do you want?
- p₁ (sample proportion bound)
- p₂ (upper bound)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The standard error √(p(1 − p)/n) converts each bound to a z-score and the normal CDF gives the probability. The exact value counts the successes X = n·p̂ with the binomial distribution, using strict inequalities.
Worked example
Suppose 70% of a company’s customers are satisfied and you survey 500 at random. The standard error is √(0.7 × 0.3/500) = 0.02049. The probability that more than 65% of the sample is satisfied is P(z > (0.65 − 0.70)/0.02049) = P(z > −2.44) = 99.27%. The exact binomial probability of more than 325 satisfied customers is 99.09% – close, as expected with 350 expected successes and 150 failures.
How the sample size changes the spread (p = 50%)
| n | Standard error | 95% of p̂ within |
|---|---|---|
| 100 | 0.0500 | ± 9.8 points |
| 400 | 0.0250 | ± 4.9 points |
| 1,000 | 0.0158 | ± 3.1 points |
| 2,500 | 0.0100 | ± 2.0 points |
Formula
μp̂ = p · σp̂ = √(p(1 − p) ÷ n) · z = (p̂ − p) ÷ σp̂ · valid when n·p ≥ 10 and n·(1 − p) ≥ 10
Worked example
Inputs — Population proportion (p): 70 %; Sample size (n): 500; What probability do you want?: P(p̂ > p₁) – above; p₁ (sample proportion bound): 65 %.
Results — Normal probability = 99.2651 %; Result = P(p̂ > 0.65) = 0.992651 (99.2651%); Mean of p̂ = 0.7; Standard error σp̂ = 0.020494; z-score of p₁ = -2.4398; Exact binomial probability (for comparison) = 99.0942 %.
Key terms
- Sample proportion (p̂)
- The share of the sample with the attribute: successes ÷ n.
- Population proportion (p)
- The true share in the whole population.
- Standard error (σp̂)
- √(p(1 − p)/n), the standard deviation of p̂.
Frequently asked questions
What is the standard error of a sample proportion?
√(p(1 − p)/n). For p = 0.7 and n = 500 it is √(0.21/500) = 0.0205, about 2 percentage points.
When is the normal approximation valid?
When the expected numbers of successes and failures, n·p and n·(1 − p), are both at least 10 (some books ask for 15).
Is the sampling distribution of p̂ symmetric?
Approximately, when the conditions hold. With p close to 0 or 1 and small n it is skewed, and the exact binomial value is better.
What is the mean of p̂?
The population proportion p – the sample proportion is unbiased.
How does the sample size affect it?
The standard error falls with √n: four times the sample halves the spread.
Why does the exact binomial result differ slightly?
The normal curve is continuous while counts are whole numbers; a continuity correction (±0.5/n) brings the two closer.