Sample Size Calculator

Calculate the required sample size for your statistical study based on confidence level, margin of error, and population parameters.

Study Parameters

Sample Size Formulas

For Proportion Estimation

The formula for calculating sample size for a proportion is:

n=z2×p×(1−p)e2n = \frac{z^2 \times p \times (1-p)}{e^2}

Where:

  • nn = Required sample size
  • zz = Z-score for the desired confidence level
  • pp = Expected proportion
  • ee = Margin of error

For Mean Estimation

The formula for calculating sample size for a mean is:

n=(z×Īƒ)2e2n = \frac{(z \times \sigma)^2}{e^2}

Where:

  • nn = Required sample size
  • zz = Z-score for the desired confidence level
  • Īƒ\sigma = Population standard deviation
  • ee = Margin of error

Finite Population Correction

When the population size is known and finite, the sample size can be adjusted using:

n′=n×Nn+N−1n' = \frac{n \times N}{n + N - 1}

Where:

  • n′n' = Adjusted sample size
  • nn = Sample size calculated for infinite population
  • NN = Population size

Common Z-scores

Confidence LevelZ-score
90%1.645
95%1.96
99%2.576
99.9%3.291

Understanding Sample Size Calculation

Sample size calculation is a critical step in research design that determines how many subjects or observations are needed to achieve reliable and valid results.

Why Sample Size Matters:

  • Statistical Power: Larger sample sizes increase the ability to detect true effects.
  • Precision: Larger samples provide more precise estimates with narrower confidence intervals.
  • Resource Efficiency: Calculating the optimal sample size helps avoid wasting resources on unnecessarily large samples or collecting too few data points to draw meaningful conclusions.

Key Factors in Sample Size Calculation:

  • Confidence Level: The probability that the true population parameter falls within the confidence interval. Common values are 90%, 95%, and 99%.
  • Margin of Error: The amount of error you're willing to tolerate in your results, expressed as a percentage.
  • Expected Proportion/Standard Deviation: Your best estimate of the parameter you're measuring. For proportions, if you have no prior information, using 50% gives the most conservative (largest) sample size.
  • Population Size: For finite populations, the total size of the population affects the required sample size, especially when the sample is a significant fraction of the population.

Practical Considerations:

  • Non-response and Attrition: In practice, you may need to recruit more participants than the calculated sample size to account for non-response or dropout.
  • Subgroup Analysis: If you plan to analyze subgroups separately, you'll need a sufficient sample size for each subgroup.
  • Feasibility: Balance statistical requirements with practical constraints like budget, time, and available participants.

Example:

Suppose you want to estimate the proportion of voters who support a particular candidate with a 95% confidence level and a 3% margin of error. You have no prior information about the proportion, so you use 50% as your expected proportion.

Using the formula for proportion estimation:

n=1.962×0.5×(1−0.5)0.032=0.9604×0.250.0009=0.24010.0009≈1068n = \frac{1.96^2 \times 0.5 \times (1-0.5)}{0.03^2} = \frac{0.9604 \times 0.25}{0.0009} = \frac{0.2401}{0.0009} \approx 1068

You would need a sample size of at least 1,068 voters to achieve your desired level of precision.

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