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Summarizing and Describing
Numerical Data
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Lectures 3+4+5 Topics
Measures ofCentral Tendency
Mean, Median, Mode
Measures ofVariationThe Range, Variance and
Standard Deviation
ShapeSymmetric, Skewed, Skewness, Kurtosis
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Summary Measures
Central Tendency
MeanMedian
Mode
Summary Measures
Variation
Variance
Standard Deviation
Coefficient of
Variation
Range
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Measures of Central Tendency
Central Tendency
Mean Median Mode
n
xn
ii
!1
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Sum of the observations
Number of observationsMean =
This is the most popular and usefulThis is the most popular and usefulmeasure of central locationmeasure of central location
The ArithmeticThe Arithmetic
MeanMean
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n
xx i
n1i!!
Sample mean Population mean
N
x iN
1i!!Q
Sample size Population size
n
xx i
n1i!!
The ArithmeticThe Arithmetic
MeanMean
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!
!
! !
10
...
10
102110
1 xxxxx ii
Example 1The reported time spent on the Internet of10 adults are 0, 7,12, 5,
33,14, 8,0, 9,22hours. Find the mean time spent on the Internet.
00 77 222211.0hours11.0hours
Example 2
Suppose the telephone bills representthe population of measurements ( 200). The population meanis
!
!
!Q !
200
x...xx
200
x 20021i200
1i 42.1942.19 38.4538.45 45.7745.7743.5943.59
The ArithmeticThe Arithmetic
MeanMean
The arithmetic
mean
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Weighted mean for data groupedWeighted mean for data grouped
by categories or variantsby categories or variants
i
ii
k
i
ffxx
! !1
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When many of the measurements have the same value, the
measurement can be summarized in a frequency table. Suppose
the number ofchildrenin a sample of16 families were recorded
as follows:
NUMBER OF CHILDREN 0 1 2 3
NUMBER OF FAMILIES 3 4 7 2
16 families
5.116
)3(2)2(7)1(4)0(3
16
....
16
16162211
16
1 !
!
!
! !fxfxfxfx
x iii
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The Median
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5
Important Measure of Central Tendency
In an ordered array, the median is the
middle number.If n is odd, the median is the middle number.If n is even, the median is the average of the 2
middle numbers.
Not Affected by Extreme Values
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Odd number of observations
0, 0, 5, 7, 8 9, 12, 14, 220, 0, 5, 7, 8, 9, 12, 14, 22, 330, 0, 5, 7, 8, 9, 12, 14, 22, 33
Even number of observations
Example 4.3
Find the median of the time spent on the internet
for the adults of example 1
TheThe MedianMedian of a set of observations is theof a set of observations is thevalue that falls in the middle when thevalue that falls in the middle when theobservations are arranged in order ofobservations are arranged in order ofmagnitude or ranked increasinglymagnitude or ranked increasingly
The MedianThe Median
Suppose only 9 adults were sampled
(exclude, say, the longest time (33))
Comment
8
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The Mode
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
A Measure of Central Tendency
Value that Occurs Most Often
Not Affected by Extreme Values
There May Not be a Mode
There May be Several Modes
Used for Either Numerical or Categorical Data
0 1 2 3 4 5 6
No Mode
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TheThe ModeMode of a set of observations is theof a set of observations is thevariable value that occurs most frequently.variable value that occurs most frequently.
Set of data may have one mode (or modalSet of data may have one mode (or modalclass), or two or more modes.class), or two or more modes.
The modal classFor large data sets
the modal class is
much more relevant
than a single-value
mode.
The ModeThe Mode
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Approximating DescriptiveApproximating Descriptive
Measures for groupedMeasures for groupedData by CLASSESData by CLASSESApproximating descriptive measures forApproximating descriptive measures for
grouped data may be needed in twogrouped data may be needed in twocases:cases:
when approximated values.suffices the needs,when approximated values.suffices the needs,
when only secondary grouped data arewhen only secondary grouped data areavailable.available.
i
k
i
ii
k
i
f
fxx
1
1
!
!
!
x midpoint
f frequency
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Class Class Frequency Midpointi limits fi xi xi fi1 2-5 3 3.5 10.5
2 5-8 6 6.5 39.0
3 8-11 8 9.5 76.0
. . . . .
6 17-20 2 18.5 37.0
n =sample size= 30=f1++fn 312.0
Example 3Example 3Approximate the mean (calculate the mean) ofApproximate the mean (calculate the mean) ofthe telephone call durations problem asthe telephone call durations problem asrepresented by the frequency distributionrepresented by the frequency distribution
5 8 11 14 17 20 More6.5
26.10
:valueReal
!x
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Median and ModeMedian and Mode
MedianMedian
Me
1-Me
1i
i
0n
n-1)(21
Kx
!
!
in
Me
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Median and ModeMedian and Mode
ModeMode
21
1
0 Kx
((
(!Mo
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If a distribution is symmetrical, theIf a distribution is symmetrical, themean, median and mode coincidemean, median and mode coincide
If a distribution is non symmetrical, andIf a distribution is non symmetrical, andskewed to the left or to the right, theskewed to the left or to the right, thethree measures differ.three measures differ.
A positively skeweddistribution(skewed to the right)
Mean
Median
Mode MeanMedian
Mode
A negatively skeweddistribution(skewed to the left)
Relationship among Mean, Median,Relationship among Mean, Median,and Modeand Mode
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Summary Measures
Central Tendency
MeanMedian
Mode
n
xn
ii!1
Summary Measures
Variation
Variance
Standard Deviation
Coefficient of
Variation
Range
1n
xxs
2
i2
!
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Measures of Variation
Variation
Variance Standard Deviation Coefficient of
VariationPopulation
Variance
Sample
Variance
Population
Standard
Deviation
Sample
Standard
Deviation
Range
100%
! X
S
CV
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Measure of Variation
Difference Between Largest & Smallest
Observations:
Absolute Range =
Relative Range =
Ignores How Data Are Distributed:
The Range
SmallestrgestLa xx
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
meanxxSmallestLa
/)( rgest
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DeviationDeviation
Individual deviation from the mean =Individual deviation from the mean =
Overall deviation = 0, becauseOverall deviation = 0, because
Summing squared deviationsSumming squared deviations
ororabsolute values of the deviationsabsolute values of the deviations
meanxi
! 0XXi
2
XXi
|| xxi
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Important Measure of Variation
Shows Variation About the Mean
Computed as an arithmetic mean of
squared deviations or as a square mean ofindividual deviations
For the Population:
For the Sample:
Variance
N
Xi !
2
2 QW
1
2
2
!
n
XXs
i
For the Population: use N in the
denominator.
For the Sample : use n - 1
in the denominator.
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Most Important Measure of Variation
Shows Variation About the Mean:
For the Population:
For the Sample:
Standard Deviation
N
Xi !
2Q
W
1
2
!
nXXs i
For the Population: use N in the
denominator.
For the Sample : use n - 1
in the denominator.
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Sample Standard Deviation
1
2
!
n
XXi
Data: 10 12 14 15 17 18 18 24
s =
n = 8 Mean =16
18
16241618161716151614161216102222222
)()()()()()()(
= 4.2426
s
:Xi
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Comparing Standard Deviations
1
2
n
XXis = = 4.2426
N
Xi
!
2Q
W = 3.9686
Value for the Standard Deviation is larger for data considered as a Sample.
Data : 10 12 14 15 17 18 18 24:Xi
N= 8 Mean =16
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Comparing Standard Deviations
Mean = 15.5
s = 3.33811 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B - AGE
Data A - AGE
Mean = 15.5
s = .9258
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 4.57
Data C - AGE
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Coefficient of VariationCoefficient of Variation
Measure ofMeasure of Relative VariationRelative Variation
Always aAlways a % or coefficient% or coefficient
Shows Variation Relative to MeanShows Variation Relative to Mean
Used toUsed to Compare 2 or More GroupsCompare 2 or More Groups
Formula ( for Sample):Formula ( for Sample):
100%
!
X
SCV
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Comparing Coefficient of VariationComparing Coefficient of Variation
Stock A:Stock A: Average Price last year =Average Price last year = $50$50 Standard Deviation (sd)Standard Deviation (sd) == $5$5 Stock B:Stock B: Average Price last yearAverage Price last year == $100$100 (sd) =(sd) = $5$5
100%
!
X
SCV
Coefficient of Variation:
Stock A: CV = 10%Stock B: CV = 5%
Both average prices are
representatives
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ShapeShape
Describes How Data Are DistributedDescribes How Data Are Distributedbetween smallest and largest valuesbetween smallest and largest values
Measures of Shape:Measures of Shape: Symmetric or skewedSymmetric or skewed
Right-Skewed or
Positively SkewedLeft-Skewed or
Positive Skew-ness Symmetric
Mean = Median = ModeMean Median Mode Median MeanMod
e
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Box plotBox plot graphical presentation ofgraphical presentation of
CTMCTM
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Central tendencyCentral tendency
measures summarymeasures summary Discussed Measures ofDiscussed Measures of Central TendencyCentral Tendency Mean, Median, ModeMean, Median, Mode Addressed Measures ofAddressed Measures ofVariationVariation The RangeThe Range,, Variance,Variance, Standard Deviation, Coefficient ofStandard Deviation, Coefficient ofVariationVariation DeterminedDetermined ShapeShape of Distributionsof Distributions Symmetric or SkeweSymmetric or SkeweddCoefficient of skewnessCoefficient of skewness
Mean= Median =ModeMean Median Mode Mode Median Mean