Module 20: Distribution of Sample Means

Importance of Population Size

• The size of the population is not critical in discussing the center, spread, and shape of the sampling distribution for sample means.
• Key points to remember:
  • Mean of Distribution of Sample Means: Equal to the population mean (μ).
  • Standard Deviation of Distribution of Sample Means: Equal to the population standard deviation (σ) divided by the square root of sample size (n).
  • Effect of Sample Size (n):
    • Larger sample sizes result in smaller standard deviations.
    • Means from larger samples show less variability.
    • Large samples provide more accurate estimates of the population mean.

Shape of the Distribution

• The distribution of sample means is approximately normal when sample size (n) > 30.
• True even if the population distribution is skewed.
• If the variable is normally distributed, the sample means' distribution will be normal regardless of n.
• These properties hold as long as the population is large.

Illustration with Different Population Sizes

• Two populations compared:
  • Population A: 10,000 newborns.
  • Population B: 20,000 newborns.
• Both have the same mean (3500) and standard deviation (500) for individual birth weights.
• Conducted with 525 random samples of 100 babies from each population.
• Created histograms of the sample means.
• Results:
  • Histograms showed some variation due to random sampling.
  • Histogram from larger Population B had a similar shape, center, and spread as Population A.
  • The size of Population B did not affect the sampling distribution.

Main Points

• Population size does not affect the variability of the sample means as long as the population is large.
• Sample size is crucial for reducing variability in sample means, not population size.
• Large population ensures the properties of sample distributions hold true.