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Chapter 2
Descriptive Statistics
§ 2.4
Measures of
Variation
Range
The range of a data set is the difference between the maximum and
minimum date entries in the set.
Range = (Maximum data entry) – (Minimum data entry)
Example:
The following data are the closing prices for a certain stock
on ten successive Fridays. Find the range.
Stock
56 56 57 58 61 63
63 67 67 67
The range is 67 – 56 = 11.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
3
Deviation
The deviation of an entry x in a population data set is the difference
between the entry and the mean μ of the data set.
Deviation of x = x – μ
Example:
The following data are the closing
prices for a certain stock on five
successive Fridays. Find the
deviation of each price.
The mean stock price is
μ = 305/5 = 61.
Stock
x
56
58
61
63
67
Σx = 305
Deviation
x–μ
56 – 61 = – 5
58 – 61 = – 3
61 – 61 = 0
63 – 61 = 2
67 – 61 = 6
Σ(x – μ) = 0
Larson & Farber, Elementary Statistics: Picturing the World, 3e
4
Variance and Standard Deviation
The population variance of a population data set of N entries is
(x  μ )2
2
Population variance =  
.
N
“sigma
squared”
The population standard deviation of a population data set of N
entries is the square root of the population variance.
2
Population standard deviation =    
(x  μ )2
“sigma”
Larson & Farber, Elementary Statistics: Picturing the World, 3e
N
.
5
Finding the Population Standard Deviation
Guidelines
In Words
In Symbols
1. Find the mean of the population
data set.
μ  x
N
2. Find the deviation of each entry.
x μ
3. Square each deviation.
x  μ2
4. Add to get the sum of squares.
SS x   x  μ
5. Divide by N to get the population
variance.
6. Find the square root of the
variance to get the population
standard deviation.
2
 

Larson & Farber, Elementary Statistics: Picturing the World, 3e
 x  μ
2
2
N
 x  μ
2
N
6
Finding the Sample Standard Deviation
Guidelines
In Words
In Symbols
1. Find the mean of the sample data
set.
x  x
n
2. Find the deviation of each entry.
x x
3. Square each deviation.
x  x 2
4. Add to get the sum of squares.
SS x   x  x 
5. Divide by n – 1 to get the sample
variance.
6. Find the square root of the
variance to get the sample
standard deviation.
 x  x 
s 
n 1
2
s
Larson & Farber, Elementary Statistics: Picturing the World, 3e
2
2
 x  x 
n 1
2
7
Finding the Population Standard Deviation
Example:
The following data are the closing prices for a certain stock on five
successive Fridays. The population mean is 61. Find the population
standard deviation.
Always positive!
Stock
x
56
58
61
63
67
Σx = 305
Deviation
x–μ
–5
–3
0
2
6
Σ(x – μ) = 0
Squared
(x – μ)2
25
9
0
4
36
Σ(x – μ)2 = 74
SS2 = Σ(x – μ)2 = 74
2 

 x  μ
2
N
 x  μ
N

74
 14.8
5
2
 14.8  3.8
3.85
σ  $3.85
Larson & Farber, Elementary Statistics: Picturing the World, 3e
8
Interpreting Standard Deviation
When interpreting standard deviation, remember that is a
measure of the typical amount an entry deviates from the
mean. The more the entries are spread out, the greater
the standard deviation.
14
12
x=4
s = 1.18
10
8
6
4
Frequency
Frequency
14
12
10
8
6
4
2
2
0
0
2
4
Data value
6
x =4
s=0
2
4
Data value
Larson & Farber, Elementary Statistics: Picturing the World, 3e
6
9
Standard Deviation for Grouped Data
(x  x )2f
Sample standard deviation = s 
n 1
where n = Σf is the number of entries in the data set, and x is the
data value or the midpoint of an interval.
Example:
The following frequency distribution represents the ages
of 30 students in a statistics class. The mean age of the
students is 30.3 years. Find the standard deviation of the
frequency distribution.
Continued.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
10
Standard Deviation for Grouped Data
The mean age of the students is 30.3 years.
(x – x )2
(x – x )2f
– 8.8
77.44
1006.72
8
– 0.8
0.64
5.12
37.5
4
7.2
51.84
207.36
42 – 49
45.5
3
15.2
231.04
693.12
50 – 57
53.5
2
23.2
538.24
1076.48
Class
x
f
18 – 25
21.5
13
26 – 33
29.5
34 – 41
n = 30
x–x
  2988.80
(x  x )2f
2988.8
s

 103.06  10.2
n 1
29
The standard deviation of the ages is 10.2 years.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
11
Homework
Page 72, 51 & 52
Read and take notes on Chapter 4 section 4 (74-83)
Pg 84-91 # 1-12
Larson & Farber, Elementary Statistics: Picturing the World, 3e
12
Homework Page 72, 51 & 52
Pg 84-91 # 1-12
Page 72,
51 a Mean 6.005 Median 6.01
b Mean 5.945 Median 6.01
c the mean is affected by the data entry error
52 a Mean 29.63 Median 18.3
b Mean 22.34 Median 17.25
c the mean is affected by the Canadian exports
Read and take notes on Chapter 4 section 4 (74-83)
Pg 84-91 # 1-12
Larson & Farber, Elementary Statistics: Picturing the World, 3e
13
Homework Page 72, 51 & 52
Pg 84-91 # 1-12
Pg 84-91 # 1-12
2 range 10
mean 16.6 variance 10.2 standard
deviation 3.2
4 range 19
mean 17.9 variance 59.6 standard
deviation 7.7
6 Range 34-24 = 10
8 The deviation is the difference between the value and
the mean. The sum of all of the deviations is zero
10 Neither measure can be zero the standard deviation is
the square root of the variance
77777
12 3 3 3 7 7 7
Larson & Farber, Elementary Statistics: Picturing the World, 3e
14
Homework
Pg 84-91 # 13-14, 17-19
Larson & Farber, Elementary Statistics: Picturing the World, 3e
15
Homework Pg 84-91 # 13-14, 17-19
13. For graph b, the more of the data is in the
center and it is less spread out, therefore, it will
have the smaller standard deviation (16). Graph a
will have the larger standard deviation (24)
14 Graph b has more variability, therefore it
would have a standard deviation of 5, graph a has
a standard deviation of 2.4
17 Company B, larger standard deviation, more
offers at the ends, but you would also be more
likely to get an offer less than 29,000
Larson & Farber, Elementary Statistics: Picturing the World, 3e
16
Homework Pg 84-91 # 13-14, 17-19
18. The lower standard deviation suggests less
variability…more consistency, so player B is more
consistent.
19
Range
Variance
Standard Deviation
Los Angeles
17.6
37.35
6.11
Long Beach
8.7
8.71
2.95
Based on the data it would suggest that the annual
salaries in LA are more variable than the salaries in
Long Beach
Larson & Farber, Elementary Statistics: Picturing the World, 3e
17
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