Week 4 - The Mean of Means Flashcards
Suppose you have taken several samples of 10 units each from a population of 500 students, and calculated the mean
of each sample. How might you use the data you now have to estimate a mean for the entire population?
-read
In statistics, you often need to take data from a small number of samples and use it to extrapolate an estimate of the
parameters of the population the samples were pulled from. Since one of the more common parameters of interest
is the mean, it is common to see a distribution of the means of a number of samples (I realize this may be confusing,
“sample” here actually refers to the results of several individual samples) from the same population. This distribution
is called, appropriately, the “sampling distribution of the sample mean”. We will be investigating the sampling
distribution of the sample mean in more detail in the next lesson “The Central Limit Theorem”, but in essence it is
simply a representation of the spread of the means of several samples.
-read
The mean of means, notated here as µx, is actually a pretty straightforward calculation. Simply sum the means of all
your samples and divide by the number of means.
As a formula, this looks like:
-formula
The second common parameter used to define sampling distribution of the sample means is the “standard deviation
of the distribution of the sample means”. The only significant difference between the standard deviation of a
population and the standard deviation of sample means is that you need to divide the population standard deviation
by the square root of the sample size.
As a formula, this looks like:
-formula
I recognize that the terminology in this lesson may be getting a bit scary, but the actual concept and the required
calculations are actually not particularly difficult. Work your way through the examples below, and I think you
will find that the hardest part of this lesson is getting past the wording!
-read
Given the following sample means, what is the mean of means?
x1 = 4.35, x2 = 4.62, x3 = 4.29, x4 = 4.39, x5 = 4.55
To calculate the mean of means, sum the sample means and divide by the number of samples
4.44
Brian works at a pizza restaurant, and has been carefully monitoring the weight of cheese he puts on each pizza for
the past week. Each day, Brian tracks the weight of the cheese on each pizza he makes, and calculates the mean
weight of cheese on each pizza for that day. If the weights below represent the mean weights for each day, what is
the mean of means weight of cheese over the past week? If Brian makes 25 pizzas per day and knows the standard
deviation of cheese weight per pizza is 0.5 oz, what is the standard deviation of the sample distribution of the sample
means?
First calculate the mean of means by summing the mean from each day and dividing by the number of days:
Then use the formula to find the standard deviation of the sampling distribution of the sample means:
Where s is the standard deviation of the population, and n is the number of data points in each sampling
Brian’s research indicates that the cheese he uses per pizza has a mean weight of 7.96 oz, with a standard
deviation of .01 oz.
Calculate µx, given the following:
x1 = 352.7
x2 = 351.9
x3 = 349.97
x4 = 352.33
x5 = 353.1
x6 = 349.63
The µx (mean of means) of the given data is:
The µx (mean of means) of the given data is:
µx = 352.7+351.9+349.97+352.33+353.1+349.63
6
= 2109.63
6
µx = 351.61
Suppose you have taken several samples of 10 units each from a population of 500 students, and calculated the mean
of each sample. How might you use the data you now have to estimate a mean for the entire population?
You could sum the means of each 10-unit sampling, and divide by the number of samples to get the mean of the
means. You could further divide the standard deviation of the entire 500 students (if known) by p
10 (since each
sampling contained 10 data points), to find the standard deviation of the distribution of the sample mean.
Calculate µx and sx, given the following data:
x1 = 251.6
x2 = 242.8
x3 = 248.79
x4 = 245.33
x5 = 253.21
x6 = 256.31
Sample size = 9, s = 5.8
First calculate µx, using µx = x1+x2+x3…+xn
n :
251.6+242.8+248.79+245.33+253.21+256.31
6
1498.04
6
µx = 249.67
Next calculate sx, using sx = ps
n
:
sx = 5.8
p9
= 5.8
3
sx = 1.933
Example 2
If the s of a population is 2.94 and 25 samples of 12 samples each are taken, what is sx?
To calculate sx, use sx = ps
n
:
sx = 2.94
p
12
= 2.94
4
sx = .735
Given the population [1, 2, 3, 4, 5], create a sampling distribution by finding the mean of all possible samples that
include four units. How does µx compare to µ?
This one requires a few steps, first we need to find the mean of each possible sample of four units:
x1(1234) has a mean of 2.5
x2(1235) has a mean of 2.75
x3(1245) has a mean of 3
x4(1345) has a mean of 3.25
x5(2345) has a mean of 3.5
Next we calculate the mean of means, µx, using µx = x1+x2+x3…+xn
n :
µx = 2.5+2.75+3+3.25+3.5
5
= 15
5
µx = 3
Now we need to calculate µ, using all the population data:
µ = 1+2+3+4+5
5
= 15
5
µ = 3
With the given data, µx = µ
- Find µx, given x1 = 21.0, x2 = 24.3, x3 = 25.0, x4 = 20.6, x5 = 22.3, and x6 = 22.3
- Find µx, given x1 = 15.1, x2 = 15.77, x3 = 15.55, x4 = 15.99, x5 = 15.42, and x6 = 15.37
- Find µx, given x1 = 341.52, x2 = 345.16, x3 = 343.66, x4 = 345.86, and x5 = 336.10
- Find µx, given x1 = 1.41, x2 = 0.59, x3 = 1.44, x4 = 0.93, x5 = 1.44, x6 = 1.01, and x7 = 0.74
- Find µx, given x1 = 218.19, x2 = 279.70, x3 = 262.86, and x4 = 243.88
- If the s of a population is 292.66, and samples of 17 units each are taken, what is sx?
- If the s of a population is 41.39, and 23 samples of 30 samples each are taken, what is sx?
- If the s of a population is 193.61, and samples of 19 units each are taken, what is sx?
- If the s of a population is 91.85, and 129 samples of 11 samples each are taken, what is sx?
- If the s of a population is 255.19, and 43 samples of 31 samples each are taken, what is sx?
- Given the population: {1, 2, 3, 4, 5, 6}, create a sampling distribution by finding the mean of all possible
samples that include two units. How does µx compare to µ? - Given the population: {1, 2, 3, 4}, create a sampling distribution by finding the mean of all possible samples
that include two units. What is µx? - Given the population: {1, 2, 3, 4, 5}, create a sampling distribution by finding the mean of all possible samples
that include two units. How does µx compare to µ? - Given the population: {1, 2, 3, 4, 5}, create a sampling distribution by finding the mean of all possible samples
that include three units. What is µx? - Given the population: {1, 2, 3, 4}, create a sampling distribution by finding the mean of all possible samples
that include three units. What is µx?
Review
