# 4 Pemusatan & Penyebaran Data (TM)

Chapter 3

Describing Data Using

Numerical Measures`

Business Statistics: A Decision-Making Approach

8th Edition

Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 3-1

Chapter Goals

After completing this chapter, you should be able to:

Compute and interpret the mean, median, and mode for a set

of data

Compute the range, variance, and standard deviation and

know what these values mean

Construct and interpret a box and whisker graph

Compute and explain the coefficient of variation and z scores

Compute and explain the covariance and coefficient of

correlation


Descriptive Procedures


Center and Location

Mean

Median

Mode

Other Measures of Location

Weighted Mean

Describing Data Numerically

Variation

Variance

Standard Deviation

Coefficient of Variation

RangePercentiles

Interquartile RangeQuartiles

Measures of Center and Location


Center and Location

Mean Median Mode Weighted Mean

N

x

n

x

x

N

i

i

n

i

i

1

1

i

ii

W

i

iiW

w

xw

w

xwX

Balance point

Midpoint of ranked values

Most frequently observed value

Mean (Arithmetic Average)

The most common measure of central tendency

Mean = sum of values divided by the number of values

Affected by extreme values (outliers)


0 1 2 3 4 5 6 7 8 9 10

Mean = 3

0 1 2 3 4 5 6 7 8 9 10

Mean = 4

35

15

5

54321

4

5

20

5

104321

Mean (Arithmetic Average)

The Mean is the arithmetic average of data values

Population mean

Sample mean


n = Sample Size

N = Population Size

n

xxx

n

x

x n

n

i

i

211

N

xxx

N

xN

N

i

i

211

(continued)

Median

In an ordered array (lowest to highest), the median is the

middle number, i.e., the number that splits the distribution

in half numerically

50% of the data is above the median, 50% is below

Represented as Md

The median is not affected by extreme values


0 1 2 3 4 5 6 7 8 9 10

Median = 3

0 1 2 3 4 5 6 7 8 9 10

Median = 3

Median

To find the median, sort the n data values from low to high

(sorted data is called a data array)

Find the value in the i = (1/2)n position

The ith position is the Median Index Point

If i is not an integer, round up to next highest integer

If i is an integer, the median is the average of the values in

position i and i + 1


(continued)

Median Example

Note that n = 13

Find the i = (1/2)n position:

i = (1/2)(13) = 6.5

Since 6.5 is not an integer, round up to 7

The median is the value in the 7th position:

Md = 12


Data array:4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23, 24

Finding the Median

The location of the median:

If the number of values is odd, the median is the middle

number

If the number of values is even, the median is the average of

the two middle numbers

Note that is not the value of the median, only the

position of the median in the ranked data


dataorderedtheinpositionn

positionMedian2

1

2

1n

Median Example

n = 15 (odd)

Median is the midle number = 85

n = 10 (even)

Median is (n + 1)/2 = 11/2 = 5,5

Median value is (35 + 45)/ 2 = 40


Data array:60 68 75 77 80 80 80 85 88 90 95 95 95 95 99

Data array:23 25 25 34 35 45 46 47 52 54

Shape of a Distribution

Describes how data is distributed

Symmetric or skewed

The greater the difference between the mean and the median,

the more skewed the distribution


Mean = MedianMean < Median Median < Mean

Right-SkewedLeft-Skewed Symmetric

(Longer tail extends to left) (Longer tail extends to right)

Mode

A measure of location

The value that occurs most often

Not affected by extreme values

Used for either numerical or categorical data

There may be no mode

There may be several modes (2 modes = bimodal)


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 5

0 1 2 3 4 5 6

No Mode

Weighted Mean

Used when values are grouped by frequency or relative

importance


Days to Complete

Frequency

5 4

6 12

7 8

8 2

Example: Sample of 26 Repair Projects

Weighted Mean Days to Complete:

days 6.31 26

164

28124

8)(27)(86)(125)(4

w

xwX

i

iiW

Review Example

Five houses on a hill by the beach


$2,000 K

$500 K

$300 K

$100 K

$100 K

House Prices:

$2,000,000500,000300,000100,000100,000

Summary Statistics

Mean: ($3,000,000/5)

m = $600,000

Median: middle value of ranked data

Md = $300,000

Mode: most frequent value

Mode = $100,000


House Prices:

$2,000,000500,000300,000100,000100,000

Sum 3,000,000

Which measure of location is the best?

Mean is generally used, unless extreme values (outliers) exist

Then Median is often used, since the median is not sensitive to

extreme values.

Example: Median home prices may be reported for a region

less sensitive to outliers

Mode is good for determining most likely to occur


Other Location Measures

The pth percentile in a data array:

p% are less than or equal to

this value

(100 p)% are greater than or equal to this value

(where 0 p 100)


Percentiles

1st quartile = 25th percentile

2nd quartile = 50th percentileAlso the median

3rd quartile = 75th percentile

Quartiles

Percentiles

The pth percentile in an ordered array of n values is

the value in ith position, where


Example: Find the 60th percentile in an ordered array of

19 values.

(n)100

pi

11.4(19)100

60(n)

100

pi

If i is not an integer, round up to the next higher integer value

So use value in the i = 12th position

Percentile Location

Index

Quartiles

Quartiles split the ranked data into 4 equal groups:

Note that

Q1 is the value for which 25% of the observations are

smaller & 75% are larger.

Q2 quartile (the 50th percentile) is the median

Only 25% of the observations are greater than the Q3 IQR (interquartile range) = Q3 Q1


25%

Q1 Q2 Q3

25% 25% 25%

Quartile Formulas


Find a quartile by determining the value in the appropriate position in the ranked data, where

First quartile position: Q1 = (n+1)/4

Second quartile position: Q2 = (n+1)/2 (the median position)

Third quartile position: Q3 = 3(n+1)/4

where n is the number of observed values

Quartiles


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

Example: Find the first quartile

(n = 9)

Q1 = 25th percentile, so find i : i = (9) = 2.25

so round up and use the value in the 3rd position: Q1 = 13

25100

Round up to 3 since not an

integer

Interpretation: 25% of the data is below 13

Quartiles


(n = 9)

Q1 is in the (9+1)/4 = 2.5 position of the ranked data,

so Q1 = 12.5

Q2 is in the (9+1)/2 = 5th position of the ranked data,

so Q2 = median = 16

Q3 is in the 3(9+1)/4 = 7.5 position of the ranked data,

so Q3 = 19.5

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

Example:

(continued)

Box and Whisker Plot

A graphical display of data using a central box and extended

whiskers


Example:

25% 25% 25% 25%

Outliers Lower 1st Median 3rd UpperLimit Quartile Quartile Limit

**

Constructing the Box and Whisker Plot


Outliers Lower 1st Median 3rd UpperLimit Quartile Quartile Limit

**

The lower limit is Q1 1.5 (Q3 Q1)

The upper limit is Q3+ 1.5 (Q3 Q1)

The center box extends from Q1 to Q3 The line within the box is the median

The whiskers extend to the smallest and largest values within the calculated limits

Outliers are plotted outside the calculated limits

Shape of Box and Whisker Plots

The Box and central line are centered between the endpoints if data is symmetric around the median

A Box and Whisker plot can be shown in either vertical or horizontal format


Distribution Shape and Box and Whisker Plot


Right-SkewedLeft-Skewed Symmetric

Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3

Constructing a Box and Whisker Plot

1. Sort values from lowest to highest

2. Find Q1, Q2, Q33. Draw the box so that the ends are at Q1 and Q34. Draw a vertical line through the median

5. Calculate the interquartile range (Q3 Q1)

6. Extend dashed lines from each end to the highest and

lowest values within the limits

7. Identify outliers with an asterisk (*)


Box-and-Whisker Plot Example

Below is a Box-and-Whisker plot for the following data:

0 2 2 2 3 3 4 5 6 11 27

This data is right skewed, as the plot depicts


0 2 3 6 12 27

Min Q1 Q2 Q3 Max

*

Upper limit = Q3 + 1.5 (Q3 Q1)

= 6 + 1.5 (6 2) = 12

27 is above the upper limit so is shown as an outlier

Measures of Variation


Variation

Variance Standard Deviation Coefficient of Variation

PopulationVariance

Sample Variance

PopulationStandardDeviation

Sample Standard Deviation

Range

Interquartile Range

Variation

Measures of variation give information on the spread or

variability of the data values.

Smaller value Less variation

Larger value More variation


Same center,

different variation

Range

Simplest measure of variation

Difference between the largest and the smallest observations:


Range = xmaximum xminimum

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Range = 14 - 1 = 13

Example:

Disadvantages of the Range

Ignores the way in which data are distributed

Sensitive to outliers


7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120

Range = 5 - 1 = 4

Range = 120 - 1 = 119

Interquartile Range

Can eliminate some outlier problems by using the interquartile range

Eliminate some high-and low-valued observations and calculate the range from the remaining values.

Interquartile range = Q3 Q1


Interquartile Range Example


Median(Q2)

XmaximumX minimum Q1 Q3

Example:

25% 25% 25% 25%

12 30 45 57 70

Interquartile range = 57 30 = 27

Variance

Average of squared deviations of values from the

mean (in squared units) Population variance:

Sample variance:


N

)(x

N

1i

2

i2

1- n

)x(x

s

n

1i

2

i2

Standard Deviation

Most commonly used measure of variation

Shows variation about the mean

Has the same units as the original data

Population standard deviation:

Sample standard deviation:


N

)(x

N

1i

2

i

1-n

)x(x

s

n

1i

2

i

Calculation Example : Sample Standard Deviation


Sample Data (Xi) : 10 12 14 15 17 18 18 24

n = 8 Mean = x = 16

4.30957

130

18

16)(2416)(1416)(1216)(10

1n

)x(24)x(14)x(12)x(10s

2222

2222

A measure of the average scatter around the mean

Measuring variation


Small standard deviation

Large standard deviation

Comparing Standard Deviations


Mean = 15.5s = 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

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

Same mean, but different standard deviations:

Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean

Is used to compare two or more sets of data measured in

different units


100%x

sCV

100%

CV

Population Sample

Comparing Coefficients of Variation

Stock A:

Average price last year = $50

Standard deviation = $5

Stock B:

Average price last year = $100

Standard deviation = $5


5% 100%*$100

$5100%*

x

sCVB

10% 100%*$50

$5100%*

x

sCVA

Both stocks have the same standard deviation, but stock B is less variable relative to its price

The Empirical Rule

If the data distribution is bell-shaped, then the interval (mean

= median thus not skewed):

contains about 68% of the values in the

population or the sample


1

68%

1

The Empirical Rule

contains about 95% of the values in the population

or the sample

contains about 99.7% of the values in the

population or the sample


2

3

3

99.7%

95%

2

Tchebysheffs Theorem

Regardless of how the data are distributed, at least (1 - 1/k2)

of the values will fall within k standard deviations of the mean

Examples:

(1 - 1/12) = 0% ..... k=1 ( 1)

(1 - 1/22) = 75% ........ k=2 ( 2)

(1 - 1/32) = 89% . k=3 ( 3)


withinAt least

Standardized Data Values

A standardized data value refers to the number of standard deviations a value is from the mean

Standardized data values are sometimes referred to as z-scores

Can be used to compare datasets

Will be addressed in more detail later in the text


Standardized Population Values

where:

x = original data value

= population mean

= population standard deviation

z = standard score

(number of standard deviations x is from )


x z

Standardized Sample Values

where:

x = original data value

x = sample mean

s = sample standard deviation

z = standard score

(number of standard deviations x is from )


s

xx z

Standardized Value Example

IQ scores in a large population have a bell-shaped

distribution with mean = 100 and standard deviation

= 15

Find the standardized score (z-score) for

a person with an IQ of 121.

Someone with an IQ of 121 is 1.4 standard deviations above

the mean


1.415

100121

x z

Answer:

Using Microsoft Excel

Descriptive Statistics are easy to obtain from Microsoft Excel

Use menu choice:

Data / data analysis / descriptive statistics

Enter details in dialog box


Using Excel


Select:

Data / data analysis / descriptive statistics

Using Excel

Enter dialog box details

Check box for summary statistics

Click OK


(continued)

Excel output


Microsoft Excel

descriptive statistics output

using the house price data:

House Prices:

$2,000,000500,000300,000100,000100,000

The Sample Covariance

The sample covariance measures the strength of the

linear relationship between two variables (called

bivariate data)

The sample covariance:

Only concerned with the strength of the relationship

No causal effect is implied


1n

)YY)(XX(

)Y,X(cov

n

1i

ii

Interpreting Covariance

Covariance between two random variables:

cov(X,Y) > 0 X and Y tend to move in the same direction

cov(X,Y) < 0 X and Y tend to move in opposite directions

cov(X,Y) = 0 X and Y are independent


Coefficient of Correlation

Measures the relative strength of the linear relationship

between two variables

Sample coefficient of correlation:

where


YXSS

Y),(Xcovr

1n

)X(X

S

n

1i

2

i

X

1n

)Y)(YX(X

Y),(Xcov

n

1i

ii

1n

)Y(Y

S

n

1i

2

i

Y

Features of Correlation Coefficient, r

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 the linear relationship


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 = +.3r = +1

Y

Xr = 0

Using Excel to Find the Correlation Coefficient

Select

Tools/Data Analysis

Choose Correlation

from the selection

menu

Click OK . . .


Using Excel to Find the Correlation Coefficient

Input data range and select appropriate options

Click OK to get output


(continued)

Interpreting the Result

r = .733

There is a relatively strong positive linear relationship between test score #1 and test score #2

Students who scored high on the first test tended to score high on second test, and students who scored low on the first test tended to score low on the second test


Scatter Plot of Test Scores

70

75

80

85

90

95

100

70 75 80 85 90 95 100

Test #1 ScoreT

est

#2 S

co

re

Pitfalls in Numerical Descriptive Measures

Data analysis is objective

Should report the summary measures that best meet the

assumptions about the data set

Data interpretation is subjective

Should be done in fair, neutral and clear manner


Ethical Considerations

Numerical descriptive measures:

Should document both good and bad results

Should be presented in a fair, objective and neutral manner

Should not use inappropriate summary measures to distort

facts


Chapter Summary

Described measures of center and location

Mean, median, mode, weighted mean

Discussed percentiles and quartiles

Created Box and Whisker Plots

Illustrated distribution shapes

Symmetric, skewed


Chapter Summary

Described measures of variation

Range, interquartile range, variance,

standard deviation, coefficient of variation

Discussed Tchebysheffs Theorem

Calculated standardized data values


(continued)

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# 4 Pemusatan & Penyebaran Data (TM)