1 a value at the center or middle of a data set measures of center
TRANSCRIPT
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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
2
(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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Measures of Variation
Range
valuehighest lowest
value
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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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5% trimmed mean
the mean of the middle 90% of the scores
Definitions
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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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a measure of variation of the scores about the mean
(average deviation from the mean)
Measures of Variation
Standard Deviation
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Sample Standard Deviation Formula
calculators can compute the sample standard deviation of data
(x - x)2
n - 1S =
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Sample Standard Deviation Shortcut Formula
n (n - 1)
s =n (x2) - (x)2
calculators can compute the sample standard deviation of data
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Population Standard Deviation
calculators can compute the population standard deviation
of data
2 (x - µ)
N =
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Symbolsfor Standard DeviationSample Population
x
xn
s
Sx
xn-1
Book
Some graphicscalculators
Somenon-graphicscalculators
Textbook
Some graphicscalculators
Somenon-graphics
calculators
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Measures of Variation
Variance
standard deviation squared
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Measures of Variation
Variance
standard deviation squared
s
2
2
}
use square keyon calculatorNotation
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SampleVariance
PopulationVariance
Variance
(x - x )2
n - 1s2 =
(x - µ)2
N 2 =
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Find the range, midrange, mean, median, and 5% trimmed
mean
Data Set A
Range = 98Mean = 50
Midrange = 50Median = 50
5% trimmed = 50
Data Set B
Range = 98Mean = 50
Midrange = 50Median = 50
5% trimmed= 50
A B
1 1
2 2
40 3
40 4
40 5
60 95
60 96
60 97
98 98
99 99
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A x- µ (x - µ)2
1 -49 2401
2 -48 2304
40 -10 100
40 -10 100
40 -10 100
60 10 100
60 10 100
60 10 100
98 48 2304
99 49 2401
(x - µ)2
N =
10
=
= 31.63610010
Find the Standard Deviation
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A x- µ (x - µ)2
1 -49 2401
2 -48 2304
3 -47 2209
4 -46 2116
5 -45 2025
95 45 2025
96 46 2116
97 47 2209
98 48 2304
99 49 2401
(x - µ)2
N =
10
=
= 47.02122110
Find the Standard Deviation
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Coefficient of Variance
Used to compare the variability of two data setsthat do not measure the same thing.
vmeanpopulation
deviationstandardpopulation
varianceoft coefficien
v
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Coefficient of Variance
v
Age Shoe Size
14 7
18 9
16 9 1/2
15 8
20 10
Use the coefficient of variance todetermine which varies more.
Age µ = 16.6 = 2.154
Shoe Size µ = 8.7
= 1.077
Age varies more than shoe size
.13016.6
2.154v
.1248.7
1.077v
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Histograma bar graph in which the horizontal scale represents classes and the vertical scale
represents frequencies
Sturgess’ RuleThe number of intervals in a histogram be
approximately 1 +3.3 log(N)•Less-Than cummulative frequency curve ( ] •Greater-Than cummulative frequency curve [ )
•Frequency Polygon A straight line graph that is formed by connecting the midpoints of the top of the bars of the
histogram•Frequency Curve A smoothed out frequency polygon•Ogive A frequency curve formed from a cummulative frequency histogram by connecting the corners with a
smooth curve
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Qwertry Keyboard Word RatingsTable 2-1
2 2 5 1 2 6 3 3 4 2
4 0 5 7 7 5 6 6 8 10
7 2 2 10 5 8 2 5 4 2
6 2 6 1 7 2 7 2 3 8
1 5 2 5 2 14 2 2 6 3
1 7
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Frequency Table of Qwerty Word RatingsTable 2-3
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency
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Histogram of Qwerty Word Ratings
0 - 2 20
3 - 5 14
6 - 8 15
9 - 11 2
12 - 14 1
Rating Frequency
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Relative Frequency Histogram of Qwerty Word Ratings
Figure 2-3
0 - 2 38.5%
3 - 5 26.9%
6 - 8 28.8%
9 - 11 3.8%
12 - 14 1.9%
RatingRelative
Frequency
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Histogram and
Relative Frequency Histogram
Figure 2-2 Figure 2-3
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Figure 2-4
Frequency Polygon
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Figure 2-5
Ogive
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Bachelor High SchoolDegree Diploma
Figure 1-1 Salaries of People with Bachelor’s Degrees and with High School Diplomas
$40,000
30,000
25,000
20,000
$40,500
$24,400
35,000
$40,000
20,000
10,000
0
$40,500
$24,40030,000
Bachelor High SchoolDegree Diploma
(a) (b)
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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