2 Ukuran Numerik Dan Deskriptif

8/14/2019 2 Ukuran Numerik Dan Deskriptif

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Pertemuan 02

Ukuran Numerik Deskriptif

Matakuliah : I0262-Statistik Probabilitas

Tahun : 2007


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Outline Materi: Ukuran Pemusatan

Ukuran Variasi

Ukuran Posisi (Letak)


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Basic Business Statistics

Numerical DescriptiveMeasures


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Chapter Topics

Measures of Central Tendency Mean, Median, Mode, Geometric Mean

Quartile

Measure of Variation Range, Interquartile Range, Variance and

Standard Deviation, Coefficient of Variation

Shape Symmetric, Skewed, Using Box-and-Whisker

Plots


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Chapter Topics

The Empirical Rule and the Bienayme-Chebyshev Rule

Coefficient of Correlation

Pitfalls in Numerical Descriptive Measures

and Ethical Issues

(continued

)


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Summary Measures

Central Tendency

MeanMedian

Mode

Quartile

Geometric Mean

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range


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Measures of Central Tendency

Central Tendency

Mean Median Mode

Geometric Mean1

1

n

i

i

N

i

i

X

X n

X

N

=

=

=

=

(1/

1

2

n

GXL


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Mean (Arithmetic Mean)

Mean (Arithmetic Mean) of Data Values Sample mean

Population mean

1 1 2

n

i

i n

X X X X X

n n

= + + += = L

1 1 2

N

i

i N

X X X X

N N

=+ + +

= =

L

Sample Size

Population Size


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The Most Common Measure of CentralTendency

Affected by Extreme Values (Outliers)

(continued

)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6


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Approximating the Arithmetic Mean

Used when raw data are not available


(continued

)

1

sample size

number of classes in the frequency distribution

midpoint of the th class

frequencies of the th class

c

j j

j

j

j

m fX

n

n

c

m j

f j

==

=

=

=

=


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Median

Robust Measure of Central Tendency

Not Affected by Extreme Values

In an Ordered Array, the Median is the Middle

Number If n or N is odd, the median is the middle number

If n or N is even, the median is the average of the 2middle numbers

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


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Mode

A Measure of Central Tendency

Value that Occurs Most Often


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 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode


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Geometric Mean

Useful in the Measure of Rate of Changeof a Variable Over Time

Geometric Mean Rate of Return

Measures the status of an investment over

time

( )1/

1 2

n

G n X X X X = L

( ) ( ) ( )1/

1 21 1 1 1n

G n R R R R= + + + L


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Example

An investment of $100,000 declined to $50,000 at theend of year one and rebounded back to $100,000 atend of year two:

1 20.5 (or 50%) 1 (or 100% )R R= =

( ) ( )

( ) ( )

1/ 2

1/ 2 1/ 2

Average rate of return:

( 0.5) (1)0.25 (or 25%)

2

Geometric rate of return:

1 0.5 1 1 1

0.5 2 1 1 1 0 (or 0%)

G

R

R

+= =

= +

= = =


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Quartiles

Split Ordered Data into 4 Quarters

Position of i-th Quartile

and are Measures of Noncentral

Location = Median, a Measure of Central Tendency

25% 25% 25% 25%

( )1Q ( )2Q ( )3Q

Data in Ordered Array: 11 12 13 16 16 17 18 21 22

( ) ( )1 1

1 9 1 12 13Position of 2.5 12.5

4 2Q Q

+ += = = =

1Q 3Q

2Q

( ) ( )1

4i

i nQ +=


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Measures of Variation

Variation

Variance Standard Deviation Coefficientof Variation

Population

Variance

Sample

Variance

Population

Standard

Deviation

Sample

Standard

Deviation

Range

Interquartile Range


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Range

Measure of Variation Difference between the Largest and the

Smallest Observations:

Ignores How Data are Distributed

Largest SmallestRange X X=

7 8 9 10 11

12

Range = 12 - 7 = 5

7 8 9 10 11

12

Range = 12 - 7 = 5


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Measure of Variation Also Known as Midspread

Spread in the middle 50%

Difference between the First and ThirdQuartiles


3 1Interquartile Range 17.5 12.5 5Q Q= = =

Interquartile Range

Data in Ordered Array: 11 12 13 16 16 17 17 18 21


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( )2

2 1

N

i

i

X

N

=

=

Important Measure of Variation Shows Variation about the Mean

Sample Variance:

Population Variance:

( )2

2 1

1

n

i

i

X X

Sn

=

=

Variance


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Standard Deviation

Most Important Measure of Variation

Shows Variation about the Mean

Has the Same Units as the Original Data

Sample Standard Deviation:

Population Standard Deviation:

( )2

1

1

n

i

i

X X

Sn

=

=

( )2

1

N

i

i

X

N

=

=

Standard Deviation


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Approximating the Standard Deviation

Used when the raw data are not availableand the only source of data is a frequency

distribution

Standard Deviation

( )2

1

1

sample size

number of classes in the frequency distr ibution

midpoint of the th class

frequencies of the th class

c

j j

j

j

j

m X f

Sn

n

c

m j

f j

=

=

=

=

=

=

ompar ng an ar


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ompar ng an arDeviations

Mean = 15.5s = 3.338

11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5

s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5

s = 4.57

Data C


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Coefficient of Variation

Measure of Relative Variation

Always in Percentage (%)

Shows Variation Relative to the Mean

Used to Compare Two or More Sets of Data

Measured in Different Units

Sensitive to Outliers100%

S

CV X

=


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Shape of a Distribution

Describe How Data are Distributed Measures of Shape

Symmetric or skewed

Mean = Median =ModeMean < Median < Mode Mode


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Exploratory Data Analysis

Box-and-Whisker Graphical display of data using 5-number

summary

Median( )

4 6 8 10 12

XlargestXsmallest1Q 3Q

2Q


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Distribution Shape &Box-and-Whisker

Right-SkewedLeft-Skewed Symmetric

1Q 1Q 1Q2Q 2Q 2Q3Q 3Q3Q


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The Empirical Rule

For Most Data Sets, Roughly 68% of theObservations Fall Within 1 StandardDeviation Around the Mean

Roughly 95% of the Observations Fall Within

2 Standard Deviations Around the Mean

Roughly 99.7% of the Observations FallWithin 3 Standard Deviations Around the

Mean

The Biena me Cheb she


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The Bienayme-Chebyshev

Rule

The Percentage of Observations Contained

Within Distances ofkStandard DeviationsAround the Mean Must Be at Least Applies regardless of the shape of the data set

At least 75% of the observations must becontained within distances of 2 standarddeviations around the mean

At least 88.89% of the observations must be

contained within distances of 3 standarddeviations around the mean

At least 93.75% of the observations must becontained within distances of 4 standard

deviations around the mean

( )21 1/ 100%k

C ffi i t f C l ti


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Coefficient of Correlation

Measures the Strength of the LinearRelationship between 2 QuantitativeVariables

( ) ( )

( ) ( )

1

2 2

1 1

n

i i

i

n n

i i

i i

X X Y Y

r

X X Y Y

=

= =

=

F t f C l ti


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Features of CorrelationCoefficient

Unit Free Ranges between 1 and 1

The Closer to 1, the Stronger the Negative

Linear Relationship

The Closer to 1, the Stronger the Positive

Linear Relationship

The Closer to 0, the Weaker Any Linear

Relationship


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Scatter Plots of Data with Various

Correlation Coefficients

Y

X

Y

X

Y

X

Y

X

Y

X

r = -1 r = -.6 r = 0

r = 6 r = 1

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