Sampling Error Calculator
Calculate margin of error, sample size, and sampling error with step-by-step solutions
What is Sampling Error?
Sampling error is the difference between a sample statistic and the corresponding population parameter. It occurs because we can only study a subset of the population, not the entire population. The margin of error quantifies this uncertainty and helps us understand how reliable our sample estimates are.
Types of Calculations
Margin of Error:
Calculate the maximum expected difference between sample and population
ME = z × √(p(1-p)/n)
Sample Size:
Determine required sample size for desired margin of error
n = (z² × p(1-p)) / ME²
Confidence Level:
Calculate confidence level from margin of error
z = ME / √(p(1-p)/n)
Sampling Error Formulas
For Proportions:
ME = z × √(p(1-p)/n)
Where:
- ME = Margin of Error
- z = Critical value (z-score)
- p = Sample proportion
- n = Sample size
For Means:
ME = z × (σ/√n)
Where:
- ME = Margin of Error
- z = Critical value (z-score)
- σ = Population standard deviation
- n = Sample size
Sample Size Formula
To determine the required sample size for a desired margin of error:
For maximum variability (p = 0.5), this becomes:
Critical Values (z-scores)
| Confidence Level | z-score | α/2 |
|---|
| 80% | 1.282 | 0.10 |
| 85% | 1.440 | 0.075 |
| 90% | 1.645 | 0.05 |
| 95% | 1.960 | 0.025 |
| 99% | 2.576 | 0.005 |
How to Use the Calculator
- Choose the calculation type (margin of error, sample size, or confidence level)
- Enter the required values based on your calculation type
- Select the confidence level
- Click "Calculate" to get the result
- View the step-by-step solution and interpretation
Example Calculations
Margin of Error Example:
Sample proportion = 0.6
Sample size = 400
Confidence level = 95%
ME = 1.96 × √(0.6 × 0.4 / 400)
ME = 1.96 × 0.0245 = 0.048
Margin of Error = ±4.8%
Sample Size Example:
Desired margin of error = 3%
Confidence level = 95%
Expected proportion = 0.5
n = (1.96² × 0.5 × 0.5) / 0.03²
n = 1,067.11 ≈ 1,068
Factors Affecting Sampling Error
- Sample Size: Larger samples reduce sampling error
- Population Variability: More variable populations require larger samples
- Confidence Level: Higher confidence levels increase margin of error
- Sampling Method: Random sampling reduces bias
- Population Size: For large populations, sample size matters more than population size
Applications
- Survey Design: Determining appropriate sample sizes for polls
- Market Research: Estimating customer preferences and behaviors
- Quality Control: Monitoring product quality from samples
- Political Polling: Estimating voting preferences
- Scientific Research: Designing experiments and studies
- Business Analytics: Making decisions based on sample data
Best Practices
- Use random sampling methods when possible
- Consider the population size when determining sample size
- Account for non-response bias in your calculations
- Use appropriate confidence levels for your application
- Consider the cost-benefit trade-off of larger samples
- Report both the point estimate and margin of error
Common Mistakes
- Confusing margin of error with standard error
- Using the wrong confidence level for the application
- Ignoring the finite population correction for small populations
- Not accounting for non-response in sample size calculations
- Using convenience samples and assuming random sampling
- Interpreting margin of error as absolute certainty
Finite Population Correction
For small populations relative to sample size, use the finite population correction:
ME = z × √(p(1-p)/n) × √((N-n)/(N-1))
Where N is the population size. This correction is important when n/N > 0.05.