Lesson03_static11

Lesson. 3 - 1

Statistics

for Management

Summarizing and Describing Numerical Data

Lesson. 3 - 2

Lesson Topics•Measures of Central Tendency Mean, Median, Mode, Midrange, Midhinge•Quartile

•Measures of Variation The Range, Interquartile Range, Variance and Standard Deviation, Coefficient of variation•Shape Symmetric, Skewed, using Box-and-Whisker Plots

Lesson. 3 - 3

Summary Measures

Central Tendency

MeanMedian

Mode

Midrange

Quartile

Midhinge

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of Variation

Range

Lesson. 3 - 4

Measures of Central Tendency

Central Tendency

Mean Median Mode

Midrange

Midhinge

n

xn

ii

1

Lesson. 3 - 5

The Mean (Arithmetic Average)

•It is the Arithmetic Average of data values:

•The Most Common Measure of Central Tendency

•Affected by Extreme Values (Outliers)

n

xn

1ii

n

xxx n2i

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

xSample Mean

Lesson. 3 - 6

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

Lesson. 3 - 7

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

Lesson. 3 - 8

Midrange

•A Measure of Central Tendency

•Average of Smallest and Largest

Observation:

•Affected by Extreme Value

2

xx smallestestl arg

Midrange

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Midrange = 5 Midrange = 5

Lesson. 3 - 9

Quartiles

• Not a Measure of Central Tendency• Split Ordered Data into 4 Quarters

• Position of i-th Quartile: position of point

25% 25% 25% 25%

Q1 Q2 Q3

Q i(n+1)i 4

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

Position of Q1 = 2.50 Q1 =12.5= 1•(9 + 1)4

Lesson. 3 - 10

Midhinge

• A Measure of Central Tendency

• The Middle point of 1st and 3rd Quarters

• Not Affected by Extreme Values

Midhinge = 2

31 QQ

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

Midhinge = 162

519512

231

..QQ

Lesson. 3 - 11

Summary Measures

Central Tendency

MeanMedian

Mode

Midrange

Quartile

Midhinge

n

xn

ii

1

Summary Measures

Variation

Variance

Standard Deviation


Range

1n

xxs

2i2

Lesson. 3 - 12

Measures of Variation

Variation

Variance Standard Deviation Coefficient of Variation

PopulationVariance

Sample

Variance

PopulationStandardDeviationSample

Standard

Deviation

Range

Interquartile Range

100%

X

SCV

Lesson. 3 - 13

• Measure of Variation

• Difference Between Largest & Smallest Observations:

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

Lesson. 3 - 14

• Measure of Variation

• Also Known as Midspread: Spread in the Middle 50%

• Difference Between Third & First

Quartiles: Interquartile Range =

• Not Affected by Extreme Values

Interquartile Range

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

13 QQ = 17.5 - 12.5 = 5

Lesson. 3 - 15

•Important Measure of Variation

•Shows Variation About the Mean:

•For the Population:

•For the Sample:

Variance

N

X i

22

1

22

n

XXs i

For the Population: use N in the denominator.

For the Sample : use n - 1 in the denominator.

Lesson. 3 - 16

•Most Important Measure of Variation

•Shows Variation About the Mean:

•For the Population:

•For the Sample:

Standard Deviation

N

X i

2

1

2

n

XXs i

For the Population: use N in the denominator.

For the Sample : use n - 1 in the denominator.

Lesson. 3 - 17

Sample Standard Deviation

1

2

n

XX i For the Sample : use n - 1 in the denominator.

Data: 10 12 14 15 17 18 18 24

s =

n = 8 Mean =16

18

1624161816171615161416121610 2222222

)()()()()()()(

= 4.2426

s

:X i

Lesson. 3 - 18

Comparing Standard Deviations

1

2

n

XX is =

= 4.2426

N

X i

2 = 3.9686

Value for the Standard Deviation is larger for data considered as a Sample.

Data : 10 12 14 15 17 18 18 24:X i

N= 8 Mean =16

Lesson. 3 - 19

Comparing Standard Deviations

Mean = 15.5 s = 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

Lesson. 3 - 20


•Measure of Relative Variation

•Always a %

•Shows Variation Relative to Mean

•Used to Compare 2 or More Groups

•Formula ( for Sample):

100%

X

SCV

Lesson. 3 - 21

Comparing Coefficient of Variation

Stock A: Average Price last year = $50

Standard Deviation = $5

Stock B: Average Price last year = $100

Standard Deviation = $5

100%

X

SCV

Coefficient of Variation:

Stock A: CV = 10%

Stock B: CV = 5%

Lesson. 3 - 22

Shape

• Describes How Data Are Distributed

• Measures of Shape: Symmetric or skewed

Right-SkewedLeft-Skewed SymmetricMean = Median = ModeMean Median Mode Median MeanMode

Lesson. 3 - 23

Box-and-Whisker Plot

Graphical Display of Data Using50%-Number Summary

Median

4 6 8 10 12

Q3Q1 XlargestXsmallest

Lesson. 3 - 24

Distribution Shape & Box-and-Whisker Plots

Right-SkewedLeft-Skewed Symmetric

Q1 Median Q3Q1 Median Q3 Q1

Median Q3

Lesson. 3 - 25

Lesson Summary• Discussed Measures of Central Tendency Mean, Median, Mode, Midrange, Midhinge

• Quartiles• Addressed Measures of Variation The Range, Interquartile Range, Variance, Standard Deviation, Coefficient of Variation• Determined Shape of Distributions

Symmetric, Skewed, Box-and-Whisker Plot

Mean = Median = ModeMean Median Mode Mode Median Mean

Lesson03_static11

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