Sampling Distribution of p̂

Calculate probabilities and statistics for the sample proportion distribution.

Last updated: March 2026

Calculator

Population proportion (p)

Proportion in population (0 to 1)

Sample size (n)

Sample proportion (p̂)

Observed proportion in sample

P(p̂ ≤ 0.45)

0.846283

P(p̂ ≥ value)0.153717

Mean (μ_p̂)0.4000

SE(p̂)0.048990

z-score1.0206

np40.0

nq60.0

What is Sampling Distribution of p̂?

The sampling distribution of p̂ (p-hat) describes how sample proportions vary when you repeatedly sample from a population. If you took many samples from a population and measured the proportion of success in each sample, you'd get different values—this collection of values has the sampling distribution.

The sampling distribution is centered at the true population proportion (p) and has standard error SE(p̂) = √(p×(1−p)/n). As sample size increases, the distribution becomes narrower (less variation). By the Central Limit Theorem, this distribution approaches normal shape when np and nq are both at least 10.

This is fundamental to statistical inference: it lets us calculate how likely it is to observe a sample proportion (p̂) given a true population proportion (p), and conversely, estimate population proportions from sample data while quantifying uncertainty.

How to Use This Calculator

Step-by-Step Process

Step 1: Enter the true population proportion (p) between 0 and 1

Step 2: Enter your sample size (n) in number of observations

Step 3: Enter the observed sample proportion (p̂) you obtained

Step 4: View the probability, z-score, and whether normality assumptions are met

Key Formulas

Mean of p̂:

μ_p̂ = p

Standard Error:

SE(p̂) = √(p×(1−p)/n)

Z-Score:

z = (p̂ − p) / SE(p̂)

Normality Check:

np ≥ 10 AND nq ≥ 10 (where q = 1−p)

Real-World Example

Customer Satisfaction: Is this sample result unusual?

Scenario:

Your company reports 60% customer satisfaction. In a sample of 200 customers, 140 (70%) reported satisfaction. How unusual is this observation?

Given:

• p = 0.60 (claimed population proportion)

• n = 200 (sample size)

• p̂ = 140/200 = 0.70 (observed sample proportion)

Calculation:

SE = √(0.60×0.40/200) = √0.0012 = 0.0346

z = (0.70 − 0.60) / 0.0346 = 2.887

P(p̂ ≤ 0.70) ≈ 0.998 or 99.8%

Interpretation:

The 70% result is quite unusual.

If the true satisfaction is 60%, there's only a 0.2% chance of observing 70% or higher in this sample. This suggests the actual satisfaction may be higher than claimed.

Frequently Asked Questions

What does p̂ represent?

p̂ (p-hat) is the proportion observed in a sample. If 70 out of 100 surveys say 'yes', then p̂ = 0.70. It estimates the true population proportion p.

Why does sample size matter?

Larger samples have smaller standard errors, making the distribution tighter. SE ∝ 1/√n, so quadrupling sample size cuts SE in half. Larger samples = more precise estimates.

When can I use the normal approximation?

The Rule of 10: when both np ≥ 10 and nq ≥ 10, the sampling distribution is approximately normal. This lets you use z-scores and calculate probabilities accurately.

What does a z-score tell me?

A z-score measures how many standard errors away your sample proportion is from the population mean. z = 2 means 2 SEs away. Larger |z| means more unusual observations.

How do I interpret the probabilities?

P(p̂ ≤ value) is the cumulative probability—the chance of observing that proportion or lower. P(p̂ ≥ value) is 1 minus this. These help assess how unusual your sample is.

Can I use this with small samples?

Theoretically yes, but with caution. When np or nq < 10, the normal approximation fails. Use exact binomial methods instead for more accuracy with small samples.

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