Standard Deviation of Sample Mean Calculator
Calculate the standard deviation of the sampling distribution of the sample mean (σX̄)
Calculate Standard Deviation of Sample Mean
The standard deviation of the entire population
Number of observations in each sample
Standard Deviation Results
Formula: σX̄ = σ / √n
Interpretation
Enter values to see interpretation of results.
Example: American Women Height Study
Population Parameters
Population: Adult American women height
Population mean (μ): 161.3 cm
Population standard deviation (σ): 7.1 cm
Sample size (n): 100 women per sample
Calculation
σX̄ = σ / √n
σX̄ = 7.1 / √100
σX̄ = 7.1 / 10
σX̄ = 0.71 cm
Interpretation
If you take many samples of 100 women each, the mean heights of these samples will typically vary by about ±0.71 cm from the population mean of 161.3 cm. This is much smaller than the individual variation (±7.1 cm), showing how sample means are more precise estimates of the population mean.
Key Concepts
Standard Deviation of Sample Mean
Measures how sample means vary around the population mean
Standard Error
Another name for the standard deviation of sample mean
Central Limit Theorem
Sample means are normally distributed for n ≥ 30
Effect of Sample Size
Small Sample (n=10)
Higher variability in sample means
Medium Sample (n=30)
CLT begins to apply reliably
Large Sample (n=100+)
Low variability, precise estimates
Remember: σX̄ decreases as sample size increases (∝ 1/√n)
Quick Tips
Larger samples = smaller standard error
Used to construct confidence intervals
Essential for hypothesis testing
Always smaller than population σ
Understanding Standard Deviation of Sample Mean
What is Standard Deviation of Sample Mean?
The standard deviation of the sample mean (σX̄) is a measure of how much sample means vary around the population mean. It quantifies the precision of sample means as estimates of the population mean.
Alternative Names
- •Standard error of the mean (SEM)
- •Standard deviation of the sampling distribution
- •Standard deviation of sample means
Formula and Components
σX̄ = σ / √n
- σX̄: Standard deviation of sample mean
- σ: Population standard deviation
- n: Sample size
- √n: Square root of sample size
Key Insight: The standard error decreases as sample size increases, making larger samples more reliable for estimation.
Applications
- •Constructing confidence intervals
- •Hypothesis testing (z-tests, t-tests)
- •Quality control and process monitoring
- •Sample size determination
- •Margin of error calculations
Important Properties
- •Always smaller than population standard deviation
- •Decreases by factor of 1/√n
- •Central to the Central Limit Theorem
- •Independent of population distribution shape (for large n)
- •Used to quantify estimation precision