The Mean, Standard Deviation, and Sampling Distribution of the Sample Mean
Site: | Saylor Academy |
Course: | MA121: Introduction to Statistics |
Book: | The Mean, Standard Deviation, and Sampling Distribution of the Sample Mean |
Printed by: | Guest user |
Date: | Friday, December 1, 2023, 6:39 PM |
Description
This section gives several concrete examples of calculating the exact distributions of the sample mean. The corresponding means and standard deviations are computed for demonstration based on these distributions. Next, it discusses sampling distributions of sample means when the sample size is large. It also considers the case when the population is normal. Finally, it uses the central limit theorem for large sample approximations.
The Mean and Standard Deviation of the Sample Mean
Learning Objectives
- To become familiar with the concept of the probability distribution of the sample mean.
- To understand the meaning of the formulas for the mean and standard deviation of the sample mean.
Suppose we wish to estimate the mean 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
, 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 mean, denoted
, and a standard deviation, denoted
. Here is an example with such a small population and small sample size that we can actually write down every single sample.
Example 1
A rowing team consists of four rowers who weighSolution
The following table shows all possible samples with replacement of size two, along with the mean of each:
The table shows that there are seven possible values of the sample mean . The value
happens only one way (the rower weighing
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 observec than
and
are. Since the
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.
The mean and standard deviation of the population in the example are
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 are drawn from a population with mean
and standard deviation
. 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 .
The second formula says that averages computed from samples vary less than individual measurements on the population do, and quantifies the relationship.
Example 2
The mean and standard deviation of the tax value of all vehicles registered in a certain state are and
. Suppose random samples of size
are drawn from the population of vehicles. What are the mean
and standard deviation
of the sample mean
?
Solution
This text was adapted by Saylor Academy under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor.
Key Takeaways
- The sample mean is a random variable; as such it is written
, and
stands for individual values it takes.
- As a random variable the
sample mean has a probability distribution, a mean
, and a standard deviation
.
- There are formulas that relate the mean and standard deviation of the sample mean to the mean and standard deviation of the population from which the sample is drawn.
Exercises
1. Random samples of size
are drawn from a population with mean
and standard deviation
. Find the mean and standard deviation of the sample mean.
3. A population has mean and standard deviation
.
b. How would the answers to part (a) change if the size of the samples were instead of
?
The Sampling Distribution of the Sample Mean
Learning Objectives
The Central Limit Theorem
In Note 6.5 "Example 1" in Section 6.1 "The Mean and Standard Deviation of the Sample Mean" we constructed the probability distribution of the sample mean for samples of size two drawn from the population of four rowers. The probability distribution is:
Figure 6.1 "Distribution of a Population and a Sample Mean" shows a side-by-side comparison of a histogram for the original population and a histogram for this distribution. Whereas the distribution of the population is uniform, the sampling distribution of the mean has a shape approaching the shape of the familiar bell curve. This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general. Here is a somewhat more realistic example.
Figure 6.1 Distribution of a Population and a Sample Mean
Suppose we take samples of size ,
,
, or
from a population that consists entirely of the numbers
and
, half the population
, half
, so that the population mean is
. The sampling distributions are:
Histograms illustrating these distributions are shown in Figure 6.2 "Distributions of the Sample Mean".
Figure 6.2 Distributions of the Sample Mean
As increases the sampling distribution of
evolves in an interesting way: the probabilities on the lower and the upper ends shrink and the probabilities in the middle become larger in relation to them. If we were to continue to increase
then the shape of the sampling distribution would become smoother and more bell-shaped.
What we are seeing in these examples does not depend on the particular population distributions involved. In general, one may start with any distribution and the sampling distribution of the sample mean will increasingly resemble the bell-shaped normal curve as the sample size increases. This is the content of the Central Limit Theorem.
The Central Limit Theorem
For samples of size or more, the sample mean is approximately normally distributed, with mean
and standard deviation
, where
is the sample size. The larger the sample size, the better the approximation.
The Central Limit Theorem is illustrated for several common population distributions in Figure 6.3 "Distribution of Populations and Sample Means".
Figure 6.3 Distribution of Populations and Sample Means
The dashed vertical lines in the figures locate the population mean. Regardless of the distribution of the population, as the sample size is increased the shape of the sampling distribution of the sample mean becomes increasingly bell-shaped, centered on the population mean. Typically by the time the sample size is the distribution of the sample mean is practically the same as a normal distribution.
The importance of the Central Limit Theorem is that it allows us to make probability statements about the sample mean, specifically in relation to its value in comparison to the population mean, as we will see in the examples. But to use the result properly we must first realize that there are two separate random variables (and therefore two probability distributions) at play:
, the measurement of a single element selected at random from the population; the distribution of
is the distribution of the population, with mean the population mean
and standard deviation the population standard deviation
;
, the mean of the measurements in a sample of size
; the distribution of
is its sampling distribution, with mean
and standard deviation
.
Example 3
Let be the mean of a random sample of size
drawn from a population with mean
and standard deviation
.
a. Find the mean and standard deviation of .
b. Find the probability that assumes a value between
and
.
c. Find the probability that assumes a value greater than
.
Solution:
a. By the formulas in the previous section
b. Since the sample size is at least , the Central Limit Theorem applies:
is approximately normally distributed. We compute probabilities using Figure 12.2 "Cumulative Normal Probability" in the usual way, just being careful to use
and not
when we standardize:
c. Similarly
Note that if in Note 6.11 "Example 3" we had been asked to compute the probability that the value of a single randomly selected element of the population exceeds , that is, to compute the number
, we would not have been able to do so, since we do not know the distribution of
, but only that its mean is
and its standard deviation is
. By contrast we could compute
even without complete knowledge of the distribution of
because the Central Limit Theorem guarantees that
is approximately normal.
Example 4
The numerical population of grade point averages at a college has mean and standard deviation
. If a random sample of size
is taken from the population, what is the probability that the sample mean will be between
and
?
Solution
The sample mean has mean
and standard deviation
, so
Normally Distributed Populations
The Central Limit Theorem says that no matter what the distribution of the population is, as long as the sample is "large," meaning of size or more, the sample mean is approximately normally distributed. If the population is normal to begin with then the sample mean also has a normal distribution, regardless of the sample size.
For samples of any size drawn from a normally distributed population, the sample mean is normally distributed, with mean and standard deviation
, where
is the sample size.
The effect of increasing the sample size is shown in Figure 6.4 "Distribution of Sample Means for a Normal Population".
Figure 6.4 Distribution of Sample Means for a Normal Population
Example 5
A prototype automotive tire has a design life of miles with a standard deviation of
miles. Five such tires are manufactured and tested. On the assumption that the actual population mean is
miles and the actual population standard deviation is
miles, find the probability that the sample mean will be less than
miles. Assume that the distribution of lifetimes of such tires is normal.
Solution:
For simplicity we use units of thousands of miles. Then the sample mean has mean
and standard deviation
. Since the population is normally distributed, so is
, hence
That is, if the tires perform as designed, there is only about a chance that the average of a sample of this size would be so low.
Example 6
An automobile battery manufacturer claims that its midgrade battery has a mean life of months with a standard deviation of
months. Suppose the distribution of battery lives of this particular brand is approximately normal.
a. On the assumption that the manufacturer's claims are true, find the probability that a randomly selected battery of this type will last less than months.
b. On the same assumption, find the probability that the mean of a random sample of such batteries will be less than
months.
Solution
a. Since the population is known to have a normal distribution
Exercises
Basic
1. A population has mean and standard deviation
.
b. Find the probability that the mean of a sample of size
3. A population has mean and standard deviation
.
a. Find the mean and standard deviation of for samples of size
.
b. Find the probability that the mean of a sample of size will be less than
.
5. A normally distributed population has mean and standard deviation
.
a. Find the probability that a single randomly selected element of the population exceeds
.
b. Find the mean and standard deviation of for samples of size
.
c. Find the probability that the mean of a sample of size drawn from this population exceeds
.
7. A population has mean and standard deviation
.
a. Find the mean and standard deviation of for samples of size
.
b. Find the probability that the mean of a sample of size will be more than
.
9. A normally distributed population has mean and standard deviation
.
a. Find the probability that a single randomly selected element of the population is between
and
.
b. Find the mean and standard deviation of for samples of size
.
c. Find the probability that the mean of a sample of size drawn from this population is between
and
.
11. A population has mean and standard deviation
.
a. Find the mean and standard deviation of for samples of size
.
b.
Find the probability that the mean of a sample of size will
differ from the population mean
by at least
units, that is,
is either less than
or more than
. (Hint: One way to solve
the problem is to first find the probability of the complementary
event.)
Applications
13. Suppose the mean number of days to germination of a variety of seed is15. Suppose the mean amount of
cholesterol in eggs labeled "large" is milligrams, with standard
deviation
milligrams. Find the probability that the mean amount
of cholesterol in a sample of
eggs will be within
milligrams of the population mean.
17. Suppose speeds of vehicles on
a particular stretch of roadway are normally distributed with mean
mph and standard deviation
mph.
a. Find the probability that the speed of a randomly selected vehicle is between
and
mph.
b. Find the probability that the mean speed of
randomly selected vehicles is between
and
mph.
19. Suppose the mean cost across
the country of a -day supply of a generic drug is
,
with standard deviation
. Find the probability that the mean
of a sample of
prices of
-day supplies of this drug will
be between
and
.
21. Scores on a common final exam
in a large enrollment, multiple-section freshman course are normally
distributed with mean and standard deviation
.
a. Find the probability that the score on a randomly selected exam paper is between
and
.
b. Find the probability that the mean score of
randomly selected exam papers is between
and
.
23. Suppose that in a certain
region of the country the mean duration of first marriages that end in
divorce is years, standard deviation
years. Find the
probability that in a sample of
divorces, the mean age of the
marriages is at most
years.