Sampling Error Calculator

Calculate margin of error and confidence intervals for sample means and proportions

Calculate Sampling Error

Number of observations in your sample

Confidence level for the interval

Proportion as decimal (e.g., 0.8 for 80%)

Sampling Error Formulas

Sample Proportion

e = zα/2 × √[p̂(1-p̂)/n]

e: Sampling error

zα/2: Critical z-value

p̂: Sample proportion

n: Sample size

Sample Mean

e = tα/2 × (s/√n)

e: Sampling error

tα/2: Critical t-value or z-value

s: Sample standard deviation

n: Sample size

Common Examples

Election Poll

Sample: 500 voters

Support: 80% (0.8)

Confidence: 95%

Result: ±3.51%

Product Testing

Sample: 30 products

Mean: 600 units

Std Dev: 70 units

Result: ±26.14 units

Key Concepts

Margin of Error: The range of values below and above the sample statistic

Standard Error: Standard deviation of the sampling distribution

Confidence Level: The probability that the interval contains the true parameter

Sample Size Effect: Larger samples reduce sampling error

Understanding Sampling Error

What is Sampling Error?

Sampling error is the difference between a sample statistic and the corresponding population parameter. It occurs because we're using a sample to estimate population characteristics, and samples naturally vary from the population they represent.

Why Does It Matter?

  • Quantifies uncertainty in statistical estimates
  • Essential for constructing confidence intervals
  • Helps determine required sample sizes
  • Critical for making statistical inferences

Key Relationships

Sample Size Impact

Sampling error decreases as sample size increases. It's inversely proportional to √n.

Confidence Level Trade-off

Higher confidence levels produce wider margins of error but greater certainty.

Population Variability

More variable populations lead to larger sampling errors for the same sample size.

Types of Sampling Error

For Proportions

Used when estimating percentages, success rates, or categorical outcomes.

SE = √[p̂(1-p̂)/n]

For Means

Used when estimating averages of continuous variables like height, weight, or income.

SE = s/√n or σ/√n

Reducing Sampling Error

Increase Sample Size

The most effective way to reduce sampling error is to collect more data points.

Improve Sampling Method

Use stratified or systematic sampling to ensure better representation.

Control Variability

When possible, reduce measurement error and control for confounding variables.