Sampling Error Calculator Formula
Understand the math behind the sampling error calculator. Each variable explained with a worked example.
Formulas Used
Sampling Error
sampling_error = z * sqrt(p * (1 - p) / n)Sampling Error (%)
error_pct = z * sqrt(p * (1 - p) / n) * 100Standard Error of Proportion
se_prop = sqrt(p * (1 - p) / n)Lower CI
lower_ci = p - z * sqrt(p * (1 - p) / n)Upper CI
upper_ci = p + z * sqrt(p * (1 - p) / n)Variables
| Variable | Description | Default |
|---|---|---|
p | Proportion (p) | 0.5 |
n | Sample Size | 400 |
z | Z-Value (e.g., 1.96 for 95%) | 1.96 |
How It Works
How to Calculate Sampling Error for Proportions
Formula
Sampling Error = z * sqrt(p * (1-p) / n)
Sampling error is the margin of error in estimating a population proportion from a sample. It depends on the confidence level (z), the proportion (p), and the sample size (n). Maximum error occurs at p = 0.5. The confidence interval is p +/- sampling error.
Worked Example
A poll of 400 people finds 50% support (p = 0.5) at 95% confidence.
- 01SE = sqrt(0.5 * 0.5 / 400) = sqrt(0.000625) = 0.025
- 02Sampling error = 1.96 * 0.025 = 0.049
- 03As a percentage: +/- 4.9%
- 04CI: (0.451, 0.549) or 45.1% to 54.9%
Frequently Asked Questions
Why is sampling error maximized at p = 0.5?
The expression p*(1-p) is largest when p = 0.5 (it equals 0.25). As p moves toward 0 or 1, uncertainty decreases because the outcome is more predictable. Using p = 0.5 gives the most conservative (largest) margin of error.
How do I reduce sampling error?
Increase sample size (most effective), accept a lower confidence level (reduce z), or use stratified sampling. Quadrupling n halves the sampling error.
Does sampling error account for all sources of error?
No. Sampling error only covers random sampling variation. Non-sampling errors (bias, non-response, measurement error, question wording) are separate and often more problematic. A large sample does not fix non-sampling errors.
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