Task 1 student:
Your teacher has a problem and needs your input. She has to give one math award this year to a
deserving student, but she can’t make a decision. Here are the test grades for her two best
students:
Bryce: 90, 90, 80, 100, 99, 81 98, 82
Brianna: 90, 90, 91, 89, 91, 89, 90, 90
Write down which of the two students should get the math award and discuss why they should be
the one to receive it.
Calculate the mean of Bryce’s distribution.
Calculate the mean deviation, variance, and standard deviation of Bryce’s distribution.
The formulas for mean deviation, variance, and standard deviation are below. Fill out the table
to help you calculate them by hand.
n

mean deviation :
i 1
n
n
 ( X i  X )2
Xi  X
variance :
i 1
n
which is the square root of the variance.
X i for Bryce
Xi  X
90
90
80
100
99
81
98
82
Mean deviation for Bryce:
Variance for Bryce:
Standard deviation for Bryce:
What do these measures of spread tell you?
Calculate the mean of Brianna’s distribution.
n
Xi  X
(X
i 1
standard deviation :
X
i
X
i
 X )2
n
2
Calculate the mean deviation, variance, and standard deviation of Brianna’s distribution.
2
X i for Brianna
Xi  X
Xi  X
X  X 
i
90
90
91
89
91
89
90
90
Mean deviation for Brianna:
Variance for Brianna:
Standard deviation for Brianna:
What do these measures of spread tell you?
Solution:
Note to teacher: In task #1, students will learn how to calculate the population variance and
population standard deviation by hand. They will compare it to the mean deviation as a measure
of spread. They should have learned how to calculate the mean deviation last year in Math 1.
Your teacher has a problem and needs your input. She has to give one math award this year to a
deserving student, but she can’t make a decision. Here are the test grades for her two best
students:
Bryce: 90, 90, 80, 100, 99, 81 98, 82
Brianna: 90, 90, 91, 89, 91, 89, 90, 90
Write down which of the two students should get the math award and discuss why they should be
the one to receive it. Your students will probably first calculate the average of each student.
They will soon discover that both students have an average of 90. They will then either say that
both deserve the award or go on to say that Bryce should get it because he had higher A’s or
Brianna should get it because she was more consistent. This should open up a discussion that it
is very important to use a measure of spread (or variability) to describe a distribution. Many
times we only look at a measure of center to describe a distribution.
Calculate the mean of Bryce’s distribution. 90
Calculate the mean deviation, variance, and standard deviation of Bryce’s distribution.
The formulas for mean deviation, variance, and standard deviation are below. Fill out the table
to help you calculate them by hand. Last year, students calculated the mean deviation. This is
probably the first time for them to calculate the variance and the standard deviation. Remind
them that the sum of the column X i  X will always be zero and that is why they have to take the
absolute value or square those values before they average them to get the mean deviation or the
variance. They will probably discover that there is a huge discrepancy between the mean
deviation and the variance. That should lead into the discussion of why they take the square root
of the variance to get the standard deviation.
n

mean deviation :
i 1
n
variance :
n
which is the square root of the variance.
X i for Bryce
Xi  X
90
90
80
100
99
81
0
0
-10
10
9
-9
n
 ( X i  X )2
Xi  X
i 1
n
Xi  X
0
0
10
10
9
9
(X
i 1
standard deviation :
X
i
0
0
100
100
81
81
X
i
 X )2
n
2
98
82
8
-8
8
8
64
64
Mean deviation for Bryce: 54/8 = 6.75
Variance for Bryce: 490/8 = 61.25
Standard deviation for Bryce:
61.25 = 7.826
What do these measures of spread tell you? All of these values tell you how your data deviates
from the mean. The variance is much larger than the mean deviation or the standard deviation
because the deviations from the mean were squared. That’s why you take the square root of the
variance to get the standard deviation. The standard deviation and mean deviation are pretty
close in value.
Calculate the mean of Brianna’s distribution.
Calculate the mean deviation, variance, and standard deviation of Brianna’s distribution.
2
X i for Brianna
Xi  X
Xi  X
X  X 
i
90
90
91
89
91
89
90
90
0
0
1
-1
1
-1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
Mean deviation for Brianna: 4/8 = .5
Variance for Brianna: 4/8 = .5
Standard deviation for Brianna:
.5 = .7071
What do these measures of spread tell you? All of these values tell you how your data deviates
from the mean. In this case, the mean deviation and the variance are the same. Unlike Bryce,
and probably surprising to students, the standard deviation is larger than the variance.