Download - 1 M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 2-4 Measures of Center
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MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH
EDITIONEDITION
ELEMENTARY STATISTICSSection 2-4 Measures of Center
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Objectives Day 1
•Given a data set, determine the mean, median, and mode.
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a value at the
center or middle of a data set
Measures of Center
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Mean
(Arithmetic Mean)
AVERAGE
the number obtained by adding the values and dividing the total by the number of values
Definitions
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Notation
denotes the addition of a set of values
x is the variable usually used to represent the individual data values
n represents the number of data values in a sample
N represents the number of data values in a population
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Notationis pronounced ‘x-bar’ and denotes the mean of a set of sample values
x =n
xx
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Notation
µ is pronounced ‘mu’ and denotes the mean of all values in a population
is pronounced ‘x-bar’ and denotes the mean of a set of sample values
Calculators can calculate the mean of data
x =n
xx
Nµ =
x
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Definitions Median
the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude
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Definitions Median
the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude
often denoted by x (pronounced ‘x-tilde’)~
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Definitions Median
the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude
often denoted by x (pronounced ‘x-tilde’)
is not affected by an extreme value
~
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6.72 3.46 3.60 6.44
3.46 3.60 6.44 6.72 no exact middle -- shared by two numbers
3.60 + 6.44
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(even number of values)
MEDIAN is 5.02
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6.72 3.46 3.60 6.44 26.70
3.46 3.60 6.44 6.72 26.70
(in order - odd number of values)
exact middle MEDIAN is 6.44
6.72 3.46 3.60 6.44
3.46 3.60 6.44 6.72 no exact middle -- shared by two numbers
3.60 + 6.44
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(even number of values)
MEDIAN is 5.02
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Definitions Mode
the score that occurs most frequently
Bimodal
Multimodal
No Mode
denoted by M
the only measure of central tendency that can be used with nominal data
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a. 5 5 5 3 1 5 1 4 3 5
b. 1 2 2 2 3 4 5 6 6 6 7 9
c. 1 2 3 6 7 8 9 10
Examples
Mode is 5
Bimodal - 2 and 6
No Mode
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Midrange
the value midway between the highest and lowest values in the original data set
Definitions
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Midrange
the value midway between the highest and lowest values in the original data set
Definitions
Midrange =highest score + lowest score
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Carry one more decimal place than is present in the original set of values
Round-off Rule for Measures of Center
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use class midpoint of classes for variable x
Mean from a Frequency Table
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use class midpoint of classes for variable x
Mean from a Frequency Table
x = Formula 2-2f
(f • x)
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use class midpoint of classes for variable x
Mean from a Frequency Table
x = class midpoint
f = frequency
f = n
x = Formula 2-2f
(f • x)
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214)( fx
Example Qwerty Keyboard Word Ratings
Word
Ratings
Interval
Midpoints
Frequency
0-2 1 20 20
3-5 4 14 56
6-8 7 15 105
9-11 10 2 20
12-14 13 1 13
x f x f
52f
2144 11 4 1 points
52. .x
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• Pages 65-66 3,5,9,11
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Homework Solutionspg 65 #3
• Mean
• Median
35 46 55 65 74 83 88 93 99 107 108 119
Occurs between the 6th and 7th data values
Mode- none
972
1281 0 seconds.
xx
n
x
x
83 8885 5 seconds
2.x
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Homework Solutionspg 65 #5
• Mean
• Median
• Mode
7 15 minutes.x
7 20.x Jefferson Valley
Providence7.70
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Homework Solutionspg 65 #9
40-49
50-59
60-69
70-79
80-89
90-99
100-109
freq x*fmidpts
200f
1487074 35 74 4 minutes
200. .x
14870 fx
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Homework Solutionspg 65 #11
• Mean
233946 78 46 8 mph
50. .x
Midptsx
Freqf x*f
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Objectives Day 2
• Given a data set where the scores vary in importance, compute a weighted mean.
• Determine how extreme values affect measures of center.
• Understand the relationship between the shape of a distribution and the relative location of the mean and median.
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Weighted Mean – used whenscores vary in importance
where represents the scores and the corresponding weights
w xx x w
w
Formula
1 1 2 2
1 2
...
...n n
n
w x w x w xx
w w w
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Example Weighted Mean
• The final grade computation for a freshman statistics course is based on weighted components.
Tests 20% each
Final 40%
If the Test scores are 83%, 73%, 82% , and a final exam score of 91% … What is the final grade?
20 83 20 73 20 82 40 91
20 20 20 40wx
840084
100wx
A straight percentage grade based on all tests being 100 pNote oints
3298225 82
400. %
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Weighted Mean common error
The weights do not need to sum to 100
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Example Weighted Mean
• Two algebra classes had the following average test scores. Period 2 had a mean score of 40 with 24 students and period 8 had a mean score of 34 with 16 students. What is the mean of the two classes combined?
24 140 346
24 16wx
150437 6 points
40.wx
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Example Weighted Mean
• Your first semester grades at college are as follows:
Subject Grade Credits
Biology A 3
Calculus B 4
College Writing B 3
Archery C 1
Chemistry A 3
What is your GPA?
3 4 3 1 3
3 4 3
4 3 3 2 4
1 3GPA
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3 35714
.GPA
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Advantages - Disadvantages
Table 2-13
Best Measure of Center
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Extreme Values and Measures of Center
Consider the following data set of salaries at a small shipping company.
30,000 30,000 30,000 30,000 30,000 40,000 45,000 125,000
What is the mean? The median? The mode?
What is the mean? The median? The mode?
36000045 000 30 000 mode 30 000
8$ , $ , $ ,x x
Now change the $125,000 to $250,000
485 00060 625 30 000 mode 30 000
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,$ , $ , $ ,x x
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SymmetricData is symmetric if the left
half of its histogram is roughly a mirror of its right half.
SkewedData is skewed if it is not
symmetric and if it extends more to one side than the other.
Definitions
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Skewness
Mode = Mean = Median
SYMMETRIC
Figure 2-13 (b)
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Skewness
Mode = Mean = Median
SKEWED LEFT(negatively)
SYMMETRIC
Mean Mode Median
Figure 2-13 (b)
Figure 2-13 (a)
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Skewness
Mode = Mean = Median
SKEWED LEFT(negatively)
SYMMETRIC
Mean Mode Median
SKEWED RIGHT(positively)
Mean Mode Median
Figure 2-13 (b)
Figure 2-13 (a)
Figure 2-13 (c)
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Review of Concepts
• The mean of a data set is the balance point of the values. Think of the mean as a give and take
• The median is the middle value of an ordered data set.
• The mode is the data value that occurs most frequently. There may be no mode, one mode, or more than one mode.
• If a distribution has few values or it is skewed, then the measures of center may not actually be near the center of the distribution. You must make appropriate decisions as to use which measure of center is most appropriate.
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Page 68 #20, 21, 24
Page 107 # 2, 3
Handout data analysis of mean