Sampling Distribution of Proportion Calculator

Enter your population proportion (p), sample size (n), and choose a probability type (above, below, or between) with the corresponding proportion value(s) to get the probability P(p̂) for your sampling distribution. The calculator also returns the mean (μp̂) and standard error (σp̂) of the sample proportion, using the normal approximation via the Central Limit Theorem.

The true proportion of successes in the population (between 0 and 1).

The number of observations in your sample.

Choose the type of probability you want to compute.

For 'above' or 'below', enter x. For 'between' or 'tails', enter the lower limit x₁.

Required only for 'between' and 'two tails' probability types.

Results

Probability P(p̂)

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Mean of Sampling Distribution (μp̂)

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Standard Error (σp̂)

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Z-Score (x or x₁)

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Z-Score (x₂)

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Normal Approximation Valid?

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Probability Distribution of Sample Proportion

Frequently Asked Questions

What is the sampling distribution of the sample proportion?

The sampling distribution of the sample proportion describes how the sample proportion p̂ = X/n varies across all possible samples of size n drawn from a population with true proportion p. When the sample size is large enough (np ≥ 10 and n(1−p) ≥ 10), this distribution is approximately normal with mean μp̂ = p and standard error σp̂ = √(p(1−p)/n) by the Central Limit Theorem.

How do I find the sample proportion?

The sample proportion p̂ is calculated as p̂ = X / n, where X is the number of successes (favorable outcomes) observed in the sample and n is the total sample size. For example, if 40 out of 100 surveyed people prefer a product, then p̂ = 40/100 = 0.40.

How do I find the standard deviation (standard error) of the sample proportion?

The standard error of the sample proportion is σp̂ = √(p(1−p)/n), where p is the population proportion and n is the sample size. It measures how much the sample proportion p̂ is expected to vary from sample to sample. As the sample size increases, the standard error decreases.

What is the difference between population proportion and sample proportion?

The population proportion (p) is the true proportion of successes in the entire population — it is a fixed but often unknown parameter. The sample proportion (p̂) is the proportion of successes observed in a specific random sample drawn from that population. Because different samples yield different values of p̂, it is a random variable, while p is a constant.

What is the probability of getting a sample proportion higher than the population proportion?

By symmetry of the normal distribution, the probability of observing a sample proportion higher than the population proportion p is exactly 0.5 (50%), assuming the normal approximation holds. This reflects that p̂ is equally likely to be above or below p, since the sampling distribution is centered at p.

When is the normal approximation valid for the sampling distribution of proportion?

The normal approximation is considered valid when both np ≥ 10 and n(1−p) ≥ 10. These conditions ensure the binomial distribution of the count X is well-approximated by a normal distribution. If these conditions are not met, the approximation may be inaccurate and exact binomial methods should be preferred.

How do you calculate the z-score for a sample proportion?

The z-score for a sample proportion value x is calculated as z = (x − p) / σp̂, where p is the population proportion and σp̂ = √(p(1−p)/n) is the standard error. This z-score converts the proportion to a standard normal value, allowing you to look up (or compute) the corresponding probability.

What does P(p̂ between x₁ and x₂) mean?

P(x₁ < p̂ < x₂) is the probability that a randomly drawn sample of size n produces a sample proportion that falls strictly between x₁ and x₂. It is computed as the area under the normal curve between the two corresponding z-scores: P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁), where Φ is the standard normal CDF.

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