This is “The Mean and Standard Deviation of the Sample Mean”, section 6.1 from the book Beginning Statistics (v. 1.0). For details on it (including licensing), click here.
For more information on the source of this book, or why it is available for free, please see the project's home page. You can browse or download additional books there. To download a .zip file containing this book to use offline, simply click here.
Suppose we wish to estimate the mean μ of a population. In actual practice we would typically take just one sample. Imagine however that we take sample after sample, all of the same size n, and compute the sample mean of each one. We will likely get a different value of each time. The sample mean is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty. We will write when the sample mean is thought of as a random variable, and write for the values that it takes. The random variable has a meanThe number about which means computed from samples of the same size center., denoted , and a standard deviationA measure of the variability of means computed from samples of the same size., denoted Here is an example with such a small population and small sample size that we can actually write down every single sample.
A rowing team consists of four rowers who weigh 152, 156, 160, and 164 pounds. Find all possible random samples with replacement of size two and compute the sample mean for each one. Use them to find the probability distribution, the mean, and the standard deviation of the sample mean
Solution
The following table shows all possible samples with replacement of size two, along with the mean of each:
Sample | Mean | Sample | Mean | Sample | Mean | Sample | Mean | |||
---|---|---|---|---|---|---|---|---|---|---|
152, 152 | 152 | 156, 152 | 154 | 160, 152 | 156 | 164, 152 | 158 | |||
152, 156 | 154 | 156, 156 | 156 | 160, 156 | 158 | 164, 156 | 160 | |||
152, 160 | 156 | 156, 160 | 158 | 160, 160 | 160 | 164, 160 | 162 | |||
152, 164 | 158 | 156, 164 | 160 | 160, 164 | 162 | 164, 164 | 164 |
The table shows that there are seven possible values of the sample mean The value happens only one way (the rower weighing 152 pounds must be selected both times), as does the value , but the other values happen more than one way, hence are more likely to be observed than 152 and 164 are. Since the 16 samples are equally likely, we obtain the probability distribution of the sample mean just by counting:
Now we apply the formulas from Section 4.2.2 "The Mean and Standard Deviation of a Discrete Random Variable" in Chapter 4 "Discrete Random Variables" for the mean and standard deviation of a discrete random variable to For we obtain.
For we first compute :
which is 24,974, so that
The mean and standard deviation of the population {152,156,160,164} in the example are μ = 158 and The mean of the sample mean that we have just computed is exactly the mean of the population. The standard deviation of the sample mean that we have just computed is the standard deviation of the population divided by the square root of the sample size: These relationships are not coincidences, but are illustrations of the following formulas.
Suppose random samples of size n are drawn from a population with mean μ and standard deviation σ. The mean and standard deviation of the sample mean satisfy
The first formula says that if we could take every possible sample from the population and compute the corresponding sample mean, then those numbers would center at the number we wish to estimate, the population mean μ.
The second formula says that averages computed from samples vary less than individual measurements on the population do, and quantifies the relationship.
The mean and standard deviation of the tax value of all vehicles registered in a certain state are and Suppose random samples of size 100 are drawn from the population of vehicles. What are the mean and standard deviation of the sample mean ?
Solution
Since n = 100, the formulas yield
Random samples of size 225 are drawn from a population with mean 100 and standard deviation 20. Find the mean and standard deviation of the sample mean.
Random samples of size 64 are drawn from a population with mean 32 and standard deviation 5. Find the mean and standard deviation of the sample mean.
A population has mean 75 and standard deviation 12.
A population has mean 5.75 and standard deviation 1.02.
,