The standard deviation of the sample mean (also known as the standard error of the mean) in a normal distribution is the population standard deviation divided by the square root of the sample size.
Understanding the Standard Deviation of the Mean
The standard deviation of the mean, often denoted as σx̄ or SEM (Standard Error of the Mean), quantifies the variability of sample means around the true population mean. It indicates how accurately the sample mean estimates the population mean.
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Formula: σx̄ = σ / √n
- σ: Population standard deviation
- n: Sample size
Explanation
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Population Standard Deviation (σ): This measures the spread of individual data points in the entire population.
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Sample Size (n): This is the number of observations in the sample used to calculate the sample mean.
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The Square Root of n (√n): As the sample size increases, the standard deviation of the mean decreases. This is because larger samples provide more information about the population, leading to a more precise estimate of the population mean.
Example
Suppose a population has a normal distribution with a standard deviation (σ) of 10. If we take a random sample of size 25 (n = 25), the standard deviation of the mean (σx̄) would be:
σx̄ = 10 / √25 = 10 / 5 = 2
This means the sample means will cluster more closely around the true population mean than the individual data points.
Key Takeaways
- The standard deviation of the mean is always smaller than the population standard deviation.
- A larger sample size leads to a smaller standard deviation of the mean, indicating a more precise estimate of the population mean.
- The standard deviation of the mean is a crucial measure in statistical inference for constructing confidence intervals and performing hypothesis tests.
