Calculate sample proportion with confidence intervals. Estimate population proportions from sample data with statistical precision.
Last updated: March 2026
p̂ (p-hat) is the sample proportion—the estimated proportion of successes in a sample. It's calculated as p̂ = x/n, where x is the number of successes and n is the sample size. The "hat" notation (^) indicates it's a sample estimate of the true population proportion p.
p̂ serves as our best point estimate for the unknown population proportion. However, because samples vary randomly, we compute a confidence interval around p̂ to capture the likely range where the true population proportion lies. The interval accounts for sampling variability and provides a measure of precision.
The confidence interval is constructed as p̂ ± (z* × SE), where SE = √(p̂q̂/n) is the standard error and z* is the critical value for the desired confidence level. For example, a 95% confidence interval means if we repeated the sampling process many times, approximately 95% of the intervals would contain the true population proportion.
Political Poll Analysis
The sample proportion is 24%, and we're 95% confident that the true proportion of all voters who support the candidate is between 20.25% and 27.75%. This range accounts for sampling variability.
p̂ = sample proportion = x/n. Best estimate of true population proportion p using sample data. Example: 120 successes in 500 trials → p̂ = 0.24 (24% estimated success rate). Hat (^) indicates it's a sample estimate.
Range of plausible values for true population proportion: [p̂ - ME, p̂ + ME]. 95% CI means: if we repeated sampling 100 times, ~95 intervals would contain true p. Wider CI = more confidence but less precision.
ME = z* × SE. Uncertainty buffer around p̂. z* = critical value (1.96 for 95%, 2.576 for 99%). SE = √(p̂q̂/n) where q̂ = 1-p̂. Larger sample → smaller SE → smaller ME (more precise estimate).
Tradeoff: 90% = narrower interval, less confident; 99% = wider interval, more confident. Standard: 95% balances precision & confidence. Choose 99% for critical decisions (medicine, law), 90% for quick estimates.
Rule of thumb: np̂ ≥ 10 AND n(1-p̂) ≥ 10. Ensures normal approximation is accurate. Small samples or extreme p̂ (near 0 or 1) need alternative methods like Wilson score or Clopper-Pearson exact CI.
p = true population proportion (unknown, fixed). p̂ = sample proportion (calculated, varies by sample). Goal: use p̂ ± ME to estimate range containing p. Distinction critical: p never changes, p̂ varies across samples.
Poll: surveyed 1000 voters, 550 support candidate (p̂=0.55). 95% CI: [0.52, 0.58]. Meaning: 'Likely 52-58% of all voters support candidate' vs claiming exactly 55% (false precision). Accounts for sampling uncertainty.
Larger n → smaller SE → smaller ME → narrower CI (more precise). Example: n=100 vs n=1,000 with same p̂, CI width decreases by ~√10 ≈ 3.16×. Doubling sample cuts ME by ~29%. Returns diminish with larger n.
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