Download Lesson 8 Standard Deviation as Ruler - math-b

Standard Deviation
Standard Deviation
 Variance (𝒔𝟐 ): Sum of
squared deviations from
the mean

𝑠2
=
(𝑦−𝑦)2
𝑛−1
 Standard Deviation is the
square root of the variance
 𝑠=
(𝑦−𝑦)2
𝑛−1
Standard Deviation as a “Ruler”
 How can you compare
measures – be it scores,
athletic performance, etc.,
across widely different
groups?
 Book Example: Who wins a
heptathlon?
 The trick to comparing
very different looking
values is to use the
standard deviation
 You are asking, in a sense,
how far is a given value
from the mean?
A Quick Example
Long Jump
Shot Put
Mean
6.16m
13.29m
SD
.23m
1.24m
n
26
28
Kluft
6.78m
14.77m
Skujyte
6.30m
16.4m
 Kluft’s 6.78m long jump is
0.62m longer than the
mean jump of 6.16m
 The SD for the event was
0.23m, so her jump was
(6.78-6.16)/0.23 = 2.70
standard deviations better
than the mean
A Quick Example (Cont.)
Long Jump
Shot Put
Mean
6.16m
13.29m
SD
.23m
1.24m
n
26
28
Kluft
6.78m
14.77m
Skujyte
6.30m
16.4m
 Skujyte’s winning shot put
was 16.4-13.29 = 3.11
meters longer than the
mean shot put distance.
 That is 3.11/1.24=2.51
standard deviations better
than the mean
A Quick Example (Cont.)
Long Jump
Shot Put
Mean
6.16m
13.29m
SD
.23m
1.24m
n
26
28
Kluft
6.78m
14.77m
Skujyte
6.30m
16.4m
 Kluft’s long jump was 2.70
standard deviations better
than the mean
 Skujyte’s shot put was 2.51
standard deviations better
than the mean.
 Who had the more
impressive performance?
Standardizing with z-Scores
 Expressing distance in
standard deviations
standardizes performances.
 To standardize a value we
subtract the mean
performance from the
individual performance then
divide by the standard
deviation
𝑧 =
𝑦−𝑦
𝑠
 These values are called
standardized values and are
commonly denoted with the
letter z, so are often called zscores
 No units
z-Scores
 A z-Score of 2 tells us that
a data value is 2 standard
deviations above the
mean.
 The farther a data value
from the mean the more
impressive it is, regardless
of sign.
 A z-Score of -1.6 tells us
the data value was 1.6
standard deviations below
the mean
 Kluft’s long jump with zscore 2.7 is more
impressive than Skujyte’s
shot put with z-score 2.51
Just Checking (pg 107)
The lower of your two tests will be dropped.
You got a 90 on test 1, an 80 on test 2
You are all set to drop the 80…
BUT WAIT! Your teacher announces she grades
“on a curve.” She standardizes the scores in order
to decide the lower one.
The mean of the first test was 88 with sd=4, the
mean on the second was 75 with sd=5
a) Which one will be dropped?
b) Does this seem “fair” ?
On first test, mean = 88, sd = 4
z= (90-88)/4 = .5
On second test, mean=75, sd=5
z=(80-75)/5 = 1.0
The first test has a lower z-score so it will be the
one that gets dropped
No, this doesn’t seem fair. The second test is 1 sd
above the mean, farther away than the first, so
it’s the better score relative to the class.
Shifting Data
 When we standardize data to
get z-scores we do two
things:
 We shift the data by
subtracting the mean
 We rescale the values by
dividing by their standard
deviation
 What happens to a grade
distribution if everyone
gets 5 extra points?
 If we switch feet to meters,
what happens to the
distribution of the heights
of students?
Shifting
 When we shift the data by
adding (or subtracting) a
constant to each value all
measures of position
(center, percentiles, min,
max) will increase (or
decrease) by the same
constant
 Spread is not affected.
 Shape doesn’t change,
spread doesn’t change:
 Not range, not IQR, not the
SD
Rescaling
 Converting from
something like kilograms
to pounds is an act of
rescaling the data:
 To move from kg to lbs we
multiply kg*2.2lbs/kg
 This will not change the
shape of the distribution
 Mean gets multiplied by
2.2
 In fact, all measures of
position are multiplied by
the same constant
Rescaling
 What do you think
happens to spread?
 By how much?
 2.2 times larger!
 The spread of pounds
would be larger than the
spread of kg after
rescaling.
Rescaling
 When we multiply or
divide all the data values
by a constant all measures
of position (mean, median,
percentiles) are multiplied
or divided by that same
constant.
 The same is true for
measures of spread: all
measures of spread are
multiplied or divided by
that same constant
Just Checking (pg 110)
Before re-centering some SAT scores, the mean of all test
scores was 450
- How would adding 50pts to each score affect the mean?
- The SD = 100pts, what would it be after adding 50 pts?
- Mean would increase to 500
- SD is still 100 pts
Back to z-Scores
 Standardizing into zScores:
 Shift them by the mean
 Rescale by the Standard
Deviation
 When we divide by s, the
standard deviation gets
divided by s as well
 The new SD becomes 1
Z-Scores
 Z-Scores have a mean of 0 and a standard deviation of 1
 Standardizing into z-Scores does not change the shape of the
distribution of a variable
 Standardizing into z-Scores changes the center by making it 0
 Standardizing into z-Scores changes the spread by making
the SD = 1
When is a z-score BIG?
 As a rule, z-scores are big
at around 3, definitely big
around 6 or 7…
 But that isn’t nearly
enough!
Homework
129, # 1, 2, 3, 5, 7, 9, 24
Page 130, # 26, 29, 30, 34, 43
(Previously Assigned)