Transcript 12.4 - Standard Deviation

12.4 – Standard Deviation
Measures of Variation


The range of a set of data is the
difference between the greatest and
least values.
The interquartile range is the
difference between the third and
first quartiles
Example
There are 9 members of the Community Youth Leadership Board.
Find the range and interquartile range of their ages: 22, 16, 24, 17,
16, 25, 20, 19, 26.
greatest value – least value = 26 – 16
Find the range.
= 10
median
16
16
17
19
Q1 =
(16 + 17)
= 16.5
2
20
Find the median.
22
24
25
26
Q3 =
(24 + 25)
= 24.5
2
Find Q1 and Q3.
Q3 – Q1 = 24.5 – 16.5
Find the interquartile range.
=8
The range is 10 years. The interquartile range is 8 years.
More Measures of Variation

Standard deviation is a measure
of how each value in a data set
varies or deviates from the mean
Steps to Finding Standard Deviation
1. Find the mean of the set of data: x
2. Find the difference between each
value and the mean: x  x
3. Square the difference: ( x  x) 2
4. Find the average (mean) of these
squares:  ( x  x ) 2
n
5. Take the square root to find the
standard deviation
 ( x  x) 2
n
Standard Deviation
Find the mean and the standard deviation for the values 78.2, 90.5,
98.1, 93.7, 94.5.
x = (78.2 + 90.5 + 98.1 +93.7 +94.5) = 91
Find the mean.
5
x
78.2
90.5
98.1
93.7
94.5
x
91
91
91
91
91
x–x
–12.8
–0.5
7.1
2.7
3.5
(x – x)2
163.84
.25
50.41
7.29
12.25
Organize the next
steps in a table.
=
=
(x – x)2
Find the standard
n
deviation.
234.04
5
The mean is 91, and the standard deviation is about 6.8.
6.8
Let’s Try One – No Calculator!
Find the mean and the standard deviation for the values 9, 4, 5, 6
x
x
Find the mean.
x
x–x
Organize the next
steps in a table.
(x – x)2
=
(x – x)2
Find the standard
n
deviation.
Let’s Try One – No Calculator
Find the mean and the standard deviation for the values 9, 4, 5, 6
(9+4+5+6)
4
x=
x
9
4
5
6
x
6
6
6
6
x–x
3
-2
-1
0
sum
= 6 Find the mean.
(x – x)2
9
4
1
0
14
Organize the next
steps in a table.
=
(x – x)2
Find the standard
n
deviation.
14  4
56
14


 1.87
4 4
4
2
The mean is 6, and the standard deviation is about 1.87.
More Measures of Variation

Z-Score: The Z-Score is the number
of standard deviations that a value
is from the mean.
Z-Score
A set of values has a mean of 22 and a standard deviation of 3. Find the
z-score for a value of 24.
value – mean
z-score = standard deviation
=
24 – 22
3
Substitute.
=
2
3
Simplify.
= 0.6
Z-Score
A set of values has a mean of 34 and a standard deviation of 4. Find the
z-score for a value of 26.
value – mean
z-score = standard deviation
=
26 – 34
4
Substitute.
=
-8
4
Simplify.
= -2
Standard Deviation
Use the data to find the mean and standard deviation for daily
energy demands on the weekends only.
S
33
39
33
33
M
53
40
47
45
40
T
52
41
49
45
40
W
47
44
54
42
41
Th
47
47
53
43
42
F
50
49
46
39
S
39
43
36
33
The mean is about 36.1 MWh;
the standard deviation
is about 3.6 MWh.
Step 1: Use the STAT feature
to enter the data as L1.
Step 2: Use the CALC menu of
STAT to access the
1-Var Stats option.
The mean is x.
The standard
deviation is x.