Standard Deviation and Standard Error

Statistics Tutorial

Standard Deviation and Standard Error

Intro

As a math and statistic tutor I have often been asked “What is the difference between standard deviation and standard error?” My objective in this tutorial is to demonstrate with an example what the difference is. To fully comprehend the difference we need to know what a sampling distribution is and the definitions of standard deviation and stand error.

A sampling distribution of a statistic is the distribution of the sample statistic of all possible samples of the same size n taken from the same population.

Standard deviation is a measure of how spread out our data is from the mean or a measure of variation about the mean. It is a type of average deviation (Triola).

Standard error is the standard deviation of a sampling distribution.

Sample Problem

Suppose our population of interest is the number facing up when a six sided die is rolled. Since the numbers 1, 2, 3, 4, 5 and 6 are equally likely to come up consider the population A = {1, 2, 3, 4, 5, 6}.

a) Find the population mean and population standard deviation.

b) Create a table and graph of the sampling distribution of the means with replacement using a sample size of n=2.

c) Find the mean and standard error of the sampling distribution of the means and compare them to the population mean and population standard deviation.

d) Draw a dotplot of the population and sampling distribution of the means and compare them.

Solution

a) The population mean is:

\bar{x} = \frac{\sum x}{N}\qquad\bar{x}=\frac{1+2+3+4+5+6}{6} = 3.5

The population standard deviation is:

\sigma = \sqrt{\frac{\sum(x - \mu)^2}{N}} = \sqrt{\frac{35}{12}}

\sigma \thickapprox 1.7078

b) The sampling distribution of the means with replacement is:

\begin{matrix} Sample & \bar{x} & Sample & \bar{x} & Sample & \bar{x} & Sample & \bar{x} & Sample & \bar{x}\\ (1,1) & 1 & (2,1) & 1.5 & (3,1) & 2 & (4,1) & 2.5 & (5,1) & 3\\ (1,2) & 1.5 & (2,2) & 2 & (3,2) & 2.5 & (4,2) & 3 & (5,2) & 3.5\\ (1,3) & 2 & (2,3) & 2.5 & (3,3) & 3 & (4,3) & 3.5 & (5,3) & 4\\ (1,4) & 2.5 & (2,4) & 3 & (3,4) & 3.5 & (4,4) & 4 & (5,4) & 4.5\\ (1,5) & 3 & (2,5) & 3.5 & (3,5) & 4 & (4,5) & 4.5 & (5,5) & 5\\ (1,6) & 3.5 & (2,6) & 4 & (3,6) & 4.5 & (4,6) & 5 & (5,6) & 5.5 \end{matrix}

\begin{matrix} Sample & \bar{x} \\ (6,1) & 3.5\\ (6,2) & 4\\ (6,3) & 4.5\\ (6,4) & 5\\ (6,5) & 5.5\\ (6,6) & 6 \end{matrix}

c) The mean of the sampling distribution of the means \mu_{\bar{x}} is:

\mu_{\bar{x}} = \frac{1 + 1.5 + 2 + \ldots + 5 + 5.5 + 6}{36} = 3.5

The standard error or standard deviation of the sampling distribution of the means \sigma_{\bar{x}} is:

\sigma_{\bar{x}} = \frac{\sqrt{N\sum{x^2} - \left (\sum{x} \right )^2}}{N} = \frac{\sqrt{210}}{12}} = 1.2076

We see that the means of both distributions is the same 3.5 and the standard error is the population standard deviation divided by \sqrt{2}. That is

    \[ 1.2076 = \frac{1.7076}{\sqrt{2}} \]

d) The dotplot of the population distribution is

Rendered by QuickLaTeX.com

The dotplot of the sampling distribution of the means is

Rendered by QuickLaTeX.com

Each dot of the population distribution represents one individual of the population. The mean and standard deviation uses all of the individual values.

Each dot of the sampling distribution of the means represents the mean of two individuals. The mean is the mean of the mean of all samples of size n = 2. The standard error is the standard deviation of the sampling distribution but uses the mean of all samples of size n = 2 instead of individual values.

The mean of each distribution is the same.

When we study the Central Limit Theorem (CLT) we find that for any population distribution that the distribution of the sample means approaches a normal distribution as the sample size increases and the standard error equals the population standard deviation divided by the square root of the sample size n. We don’t need to sample the entire population to infer the population mean provided we sample randomly and know that the underlying population is normally distributed or n is greater than or equal to thirty then

    \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\qquad \mu_{\bar{x}} = \mu} \]



About The Author

College Math, Statistics And Trigonometry
I am a graduate of Cleveland State University. After earning my Bachelors of Electrical Engineering I went on to earn a Master’s of Science in Electrical Engineering. Also I am a graduate of Lorain County Community College (LCCC) earning an Associates of Science Pre-Professional Engineering and ...
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