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Mth115
Measure of Dispersion (4.3)
Ten students from the Northern Washington University were asked how many minutes they
studied each night. Also, ten students from Southern Washington University were asked the
same question. Here’s the results:
[A]
Let’s find the Standard Deviation for the NWU data! You’ll need the mean,
which is ______
Minutes Deviation
(x)
(x – x )
50
60
75
120
80
95
120
110
80
80
(Deviation)2
(x – x )2
i) After finishing the table,
we have (x - x )2 =
___________
ii) The Variance is
s2 = (x - x )2
n–1
=
___________
iii) The Standard Deviation
is s =
s2
=
___________
Note:
 The Standard Deviation is the square root of the Variance.
 The larger the Standard Deviation, the more spread out the data is from the mean.
 There’s another way of finding Variance, which we’ll try on the next page.
[B]
What percentage of the NWU data is …
…within 1 Standard Deviation of the Mean: [ x – 1s, x + 1s] ?
___________
…within 2 Standard Deviation of the Mean: [ x – 2s, x + 2s] ?
___________
1
[C]
Let’s find the Standard Deviation for the SWU data!
Minutes
(x)
75
80
80
85
85
85
90
90
95
105
(Minutes 2)
(x 2)
(i) After finishing the table,
you’ll need (x)
__________
(ii) Now find [ (x) ]2
=
__________
(iii) You want [ (x) ]2
n
=
__________
(iv) Also, you’ll need (x2 )
=
__________
(v) And now the Variance
s2 = (iv) – (iii)
n–1
=
2
[(x )]
(x2 )
n
__________________
n–1
n
=
__________
=
__________
(vi) The Standard Deviation
is s =
s2
[D]
What percentage of the SWU data is …
…within 1 Standard Deviation of the Mean: [ x – 1s, x + 1s] ?
__________
…within 2 Standard Deviation of the Mean: [ x – 2s, x + 2s] ?
__________
…within 3 Standard Deviation of the Mean: [ x – 3s, x + 3s] ?
__________
2
[C]
We can use a similar technique as we did in [B] to find the Standard Deviation
for Grouped Data. Suppose the following table shows data for the ages of people who
earned the minimum wage. Let’s find the Standard Deviation for this data.
y = Age
f = Frequency
(In Thousands)
15  y < 19
514
19  y < 25
570
25  y < 30
411
35  y < 40
309
40  y < 50
225
50  y < 70
108
x = Midpoint
(Of Age Interval)
fx
DON’T FORGET n =  f
=
__________
(i) After finishing the table,
you’ll need ( f  x)
=
__________
(ii) Now find [ ( f  x) ]2
=
__________
(iii) You want [ ( f  x) ]2
n
=
__________
(iv) Also, you’ll need ( f  x2 )
=
__________
f  x2
(v) And now the Variance
s2 = (iv) – (iii)
n–1
=
( f x2 ) [(f x )]
n
__________________
n–1
n
2
=
__________
=
__________
(vi) The Standard Deviation
is s =
s2
3