Answer to Question #160363 in Statistics and Probability for Andrew

Question

The formula for the standard error (standard deviation of the distribution of sample means) implies that as the sample size (​n​) increases, the size of the standard error decreases. Explain the role of the standard error in comparing the sample mean and the population mean. Use the definitions and concepts on sample means distribution and standard error and show examples comparing the sample mean and the population mean of a distribution when the standard error changes in value to express your answer

Answer 1

Standard error gives the accuracy of a sample mean by measuring the sample-to-sample variability of the sample means. As the size of the sample data grows larger, the SEM decreases versus the SD, hence, as the sample size increases, the sample mean estimates the true mean of the population with greater precision.

• Standard deviation (SD) measures the dispersion of a dataset relative to its mean.
• Standard error of the mean (SEM) measured how much discrepancy there is likely to be in a sample's mean compared to the population mean.
• The SEM takes the SD and divides it by the square root of the sample size.

Example :

The formula for the SD requires a few steps:

1. First, take the square of the difference between each data point and the sample mean, finding the sum of those values.
2. Then, divide that sum by the sample size minus one, which is the variance.
3. Finally, take the square root of the variance to get the SD.

For example , data is 2,3,6

mean= (2+3+6)/3= 3.67 and

variance= $\frac{(2-3.67)^2+(3-3.67)^2+(6-3.67)^2}{3-1}$

=4.33

hence standard deviation = $\sqrt{variance}=\sqrt{4.33}=$ 2.08

Standard error = $\frac{sd}{\sqrt{n}}=\frac{2.08}{\sqrt{3}}=1.200$