Sampling Distribution of Sample Proportion

Analyze the properties and probabilities of a sample proportion's sampling distribution.

Enter the population proportion and sample size to understand the distribution's characteristics. You can also calculate probabilities for a specific sample proportion.

Probability Calculation

Examples

Use these examples to see how the calculator works.

Election Polling

election-poll

A political pollster wants to know the sampling distribution for a candidate who has 55% support in the population, based on a sample of 500 voters.

p: 0.55, n: 500

p̂: 0.58

Manufacturing Quality Control

quality-control

A factory produces light bulbs, and 5% are known to be defective. What is the probability that in a sample of 200 bulbs, more than 7% are defective?

p: 0.05, n: 200

p̂: 0.07

Market Research

market-research

A company believes that 30% of consumers prefer their product. They survey 150 people. What is the probability that the sample proportion is less than 0.25?

p: 0.30, n: 150

p̂: 0.25

Low Sample Size Example

low-sample-size

An example demonstrating a scenario where the normality conditions are not met. A researcher studies a rare disease (1% of population) with a small sample of 40.

p: 0.01, n: 40

p̂: 0.02

Other Titles
Understanding the Sampling Distribution of the Sample Proportion
A Comprehensive Guide to a Fundamental Statistical Concept

1. What is the Sampling Distribution of the Sample Proportion?

  • Core Concepts
  • Key Characteristics
  • Central Limit Theorem
The sampling distribution of the sample proportion is a theoretical distribution that describes the possible values of the sample proportion (p̂) from all possible random samples of a given size (n) drawn from a population. It's a cornerstone concept in inferential statistics, allowing us to make inferences about a population proportion based on a single sample.
Key Characteristics
Three main features define this distribution: its mean, its standard deviation (standard error), and its shape.
Mean (μp̂): The mean of all possible sample proportions is equal to the true population proportion (p). This means p̂ is an unbiased estimator of p.
Standard Error (σp̂): This measures the typical deviation of the sample proportions from the population proportion. The formula is σp̂ = √[p(1-p)/n]. A smaller standard error implies that sample proportions are likely to be close to the population proportion.
Shape: According to the Central Limit Theorem, the shape of the sampling distribution is approximately Normal, provided that certain conditions are met (np ≥ 10 and n(1-p) ≥ 10).

2. Step-by-Step Guide to Using the Calculator

  • Data Input
  • Interpreting Results
  • Normality Check
This calculator simplifies the process of analyzing the sampling distribution. Follow these steps for an accurate analysis:
Inputting the Data
Population Proportion (p): Enter the known or assumed proportion of the population. This must be a decimal between 0 and 1 (e.g., enter 0.65 for 65%).
Sample Size (n): Enter the total number of items in your sample. This must be a positive whole number.
Sample Proportion (p̂) (Optional): If you want to find the probability associated with a specific sample proportion, enter that value here. It must also be a decimal between 0 and 1.
Interpreting the Results
After clicking 'Calculate', you will receive:
Mean (μp̂): The expected value of the sample proportion.
Standard Error (σp̂): The standard deviation of the sampling distribution.
Normality Condition Check: The calculator verifies if np and n(1-p) are both at least 10. A 'Passed' status indicates the normal approximation is reliable.
Z-Score: If you entered a sample proportion (p̂), this is its standardized score, indicating how many standard errors it is from the mean.
Probabilities: The calculator provides the cumulative probability P(p̂ < X) and the survival probability P(p̂ > X) for your given sample proportion.

3. Real-World Applications

  • Political Polling
  • Quality Control
  • Medical Research
This concept is not just theoretical; it's applied across many fields to make data-driven decisions.
Political Polling
Pollsters survey a sample of voters to estimate the proportion of the entire population that supports a candidate. The standard error helps them construct a margin of error for their prediction.
Quality Control
A manufacturer tests a sample of products to see what proportion is defective. The sampling distribution helps determine if the defect rate in a particular batch is unusually high compared to the overall production standard.
Medical Research
Researchers test a new drug on a sample of patients to estimate the proportion of all patients who will be cured. This helps them determine the drug's effectiveness and whether the observed results are statistically significant.

4. Common Misconceptions

  • Standard Deviation vs. Standard Error
  • Normality Assumption
  • Finite Population Correction
Understanding common pitfalls can help ensure correct application of the concept.
Confusing Standard Deviation and Standard Error
The standard deviation of the population (σ) measures variability within the population. The standard error of the proportion (σp̂) measures the variability of sample proportions around the population proportion. It's a measure of sampling error.
Assuming Normality
The sampling distribution is only approximately normal if the conditions (np ≥ 10 and n(1-p) ≥ 10) are met. For small samples or proportions close to 0 or 1, the distribution is skewed, and using a normal approximation can lead to incorrect probabilities. In such cases, methods like the binomial distribution are more appropriate.
Ignoring the 'Finite Population Correction Factor'
The formula σp̂ = √[p(1-p)/n] assumes sampling with replacement or from an infinite population. If the sample size (n) is more than 5% of the total population size (N), a correction factor should be used. This calculator assumes the sample is less than 5% of the population.

5. Mathematical Derivation

  • Deriving the Mean
  • Deriving the Standard Error
  • Binomial Basis
The properties of the sampling distribution are derived from the binomial distribution.
Deriving the Mean
Let X be a binomial random variable representing the number of successes in a sample of size n. The mean of X is E[X] = np. The sample proportion is p̂ = X/n. The mean of p̂ is E[p̂] = E[X/n] = (1/n)E[X] = (1/n)(np) = p.
Deriving the Standard Error
The variance of a binomial random variable is Var(X) = np(1-p). The variance of the sample proportion is Var(p̂) = Var(X/n) = (1/n²)Var(X) = (1/n²)(np(1-p)) = p(1-p)/n. The standard deviation (standard error) is the square root of the variance, so σp̂ = √[p(1-p)/n].