Explain How Much More Or Less The Standard Deviation Of A Sa

explain How Much More Or Less The Standard Deviation Of Sample Means

1. Explain how much more or less the standard deviation of sample means was than the population standard deviation. According to the formula for standard deviation of sample means, it should be far less. (That formula is σₓ̄ = σ / √n = σ / √10 = σ / 3.16) Does your computed σₓ̄ agree with the formula?

2. According to the Empirical Rule, what percentage of your sample means should be within 1 standard deviation of the population mean? Using your computed σₓ̄, do your sample means seem to conform to the rule?

Paper For Above instruction

The standard deviation of sample means, often referred to as the standard error (SE), quantifies the variability of the sample means around the true population mean. Theoretical understanding suggests that the standard error of the mean (SEM) should be less than the population standard deviation, σ, because averaging across multiple observations tends to cancel out random fluctuations. According to the central limit theorem, the formula for SEM is σₓ̄ = σ / √n, where n is the sample size. In this case, with n = 10, the SEM should be σ / 3.16.

Empirically, if one computes the standard deviation of a set of sample means, it should approximate this theoretical value, verifying the adequacy of the sampling process and confirming the law of large numbers. Any significant discrepancy might suggest randomness in sampling was not representative or that other factors influenced the results.

In a practical scenario, suppose the population standard deviation (σ) is known or estimated from prior data. When calculating the sample means, the standard deviation of these means (the sample standard deviation of the means, sₓ̄) should be close to σ / √n. For example, if σ = 10, then SEM ≈ 10 / 3.16 ≈ 3.16. If the computed SEM from the actual sample data significantly deviates from this value, it warrants further investigation into sampling design, randomness, or data integrity.

The Empirical Rule states that, for a normally distributed data set, approximately 68% of the data falls within one standard deviation of the mean. Extending this concept to the distribution of sample means, it implies that about 68% of the sample means should lie within one SEM of the population mean. Therefore, if the computed SEM is accurate and the distribution of sample means is approximately normal, roughly 68% of the sample means should fall within this range.

Evaluating actual sample data involves calculating the proportion of sample means within one SEM of the population mean. If the observed percentage aligns closely with 68%, it indicates that the sample data conform well to the assumptions underlying the Empirical Rule and the central limit theorem. Deviations from this percentage could suggest issues with sample size, data distribution, or sampling procedures.

Overall, understanding the relationship between the population standard deviation and the standard deviation of sample means is crucial for inferential statistics. Confirming the theoretical SEM with empirical calculations reinforces confidence in conclusions drawn from sample data, supports accurate estimation of population parameters, and enhances the robustness of statistical inference.

References

  • Freund, J. E., & Williams, F. J. (2010). Modern business statistics. Pearson Education.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W.H. Freeman.
  • Gravetter, F. J., & Wallnau, L. B. (2016). Statistics for the Behavioral Sciences. Cengage Learning.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Brooks Cole.
  • Schunk, D. H. (2012). Learning Theories: An Educational Perspective. Pearson.
  • Triola, M. F. (2018). Elementary Statistics. Pearson.
  • Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data. Pearson.
  • Gerald, M., & Barr, R. (2014). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis. Pearson.

Related Assignments