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Transcript
Standard Deviation, ZScores, Variance
ALGEBRA 1B LESSON 42
INSTRUCTIONAL MATERIAL 2
Standard Deviation
 Standard
Deviation is a number that
tells us how a value in the data set
differs from the mean. It also tells us
how spread out the numbers are.
The bigger the standard deviation,
the more widespread the data is.
 The Greek letter sigma, , will
represent Standard Deviation.
Standard Deviation
 To
find Standard Deviation, there are 2 ways:
 One
way is the “long” way:
Calculate
Subtract
Square
the mean
the mean from each data value
the differences (results from 2nd step)
Find
the average of the squared differences
(This is called “Variance”)
Take
the square root of the variance
Example
 Given
the following Algebra 1 test scores,
find the standard deviation:
75, 77, 86, 89, 93, 82
 First,
find the mean:
75+77+86+89+93+82

= 83.67
6
Cont.
 Subtract
value:
the mean from each data
 75
– 83.67= -8.67
 77
– 83.67 = -6.67
 86
– 83.67 = 2.33
 89
– 83.67 = 5.33
 93
– 83.67 = 9.33
 82
– 83.67 = 1.67
Cont.
 Square
the difference and find the
average:
75 + 44 + 5 + 28 + 87 + 2
(-8.67)2 = 75
(-6.67)2 = 44
6
(2.33)2 = 5
(5.33)2 = 28
(9.33)2 = 87
(1.67)2 = 2
75+44+5+28+87+2

=
6
40
40
is the Variance. To find the
Standard Deviation, take the
square root. 40 = 6.3
So, the standard deviation is 6. 3
 The
2nd way to find Standard Deviation (the shortcut):
 Type
the data values in your calculator
Stat,
Edit, L1
Stat,
Calc, 1, Calculate – look for the sigma, .
Standard
Deviation = 6.4 (Between the calculator
and paper method, the standard deviation may be
slightly different. That is due to rounding. The more
places you round to, the more accurate your answer
Example 2
 Find
the Standard Deviation for the
following average temperatures:
63, 61, 66, 67, 63, 65, 68
 Standard
Deviation (  = 2.3)
Z-Score
 The
Z-Score tells how many standard deviations a
specific number in the data set is away from the
mean.
 If
the z-score is positive, the data value will be above
the mean. If the z-score is negative, the data value will
be below the mean.
 To
find the Z-Score use this formula:
𝑥−µ
=
, where 𝑥 = the data value, µ = mean, and  =

standard deviation.
𝑧
Example 3
A
student scored an 84 on the Algebra 1
benchmark. The average benchmark score for
that class was a 90, while the standard
deviation was 2. Find the student’s z-score.
𝑥
= 84
µ
= 90
 = 2
𝑧
=
𝑥−µ

=
84−90
2
=
−6
2
= −3
Cont.
 With
a z-score of -3, that means the
student scored 3 standard deviations
below the mean.
Example 4
A
student had a z-score of 3 after her math test. The
class average was a 82 and the standard deviation
was 2. What was the student’s score on the math test?
𝑧
=3
𝑥
=𝑥
µ
= 82

=2
𝑧
=
𝑥−µ

3
6
=
𝑥−82
2
= 𝑥 − 82
 88 = 𝑥
 The student’s scored an 88 on the
math test.
Example 5
 The
class average on the box-and-whisker quiz
was a 92. The variance was 4. Find the interval
for the score of a student that scored within 1.5
standard deviations from the mean.
𝑧
= −1.5 𝑎𝑛𝑑 1.5
𝑥
=𝑥
µ
= 92
 =
4=2
Cont.
 −1.5
 −3
=
𝑥−92
2
1.5 =
𝑥−92
𝑥
= 𝑥 − 92
3 = 𝑥 − 92
 𝑥 = 89
𝑥 = 95
 The student must have scored
between an 89 and 95.
Summary
 Standard
Deviation, Variance, and Z-Score are 3
different ways to analyze data.
 The
Standard Deviation and Variance describe how
spread out data values are from each other, as well as
from the mean. Given the variance, take the square
root to find the standard deviation.
Summary Cont.
 The
Z-Score measures how far a data
value is away from the mean in terms of
Standard Deviation.
 The
Standard Deviation is what you
add/subtract to the mean while the zscore is how many times you
add/subtract to the mean.