MBA7020_04.ppt/June 120, 2005/Page 1 Georgia State University - Confidential MBA 7020 Business...

MBA7020_04.ppt/June 120, 2005/Page 1Georgia State University - Confidential

MBA 7020

Business Analysis Foundations

Descriptive Statistics

June 20, 2005


Agenda

Confidence Interval

Descriptive Summary Measures

1. Measures of Central LocationMean, Median, Mode

2. Measures of VariationThe Range, Variance and

Standard Deviation

3. Measures of AssociationCovariance and Correlation


1. Measures of Central Location Mean, Median, Mode

2. Measures of Variation The Range, Variance and Standard

Deviation

3. Measures of Association Covariance and Correlation

Describing Data: Summary Measures


1. It is the Arithmetic Average of data values:

2. The Most Common Measure of Central Tendency

3. Affected by Extreme Values (Outliers)

n

xn

ii

1 n

xxx ni 2

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

Mean


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

1. Important Measure of Central Tendency

2. 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.

3. Not Affected by Extreme Values

Median


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

Mode = 9

1. A Measure of Central Tendency

2. Value that Occurs Most Often

3. Not Affected by Extreme Values

4. There May Not be a Mode

5. There May be Several Modes

6. Used for Either Numerical or Categorical Data

0 1 2 3 4 5 6

No Mode

Mode


Agenda

Confidence Interval




Standard Deviation



Variation

Variance / Standard Deviation

Coefficient of Variation

Population

Sample

Range / Percentiles

100%

X

SCV

Measures of Variability


• Measure of Variation

• Difference Between Largest & Smallest

Observations: Range =

• Ignores How Data Are Distributed:

The Range

SmallestLa xx rgest

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5


Percentile Scores

1. Arrange data in ascending order.

2. The middle number is the median.

3. The number halfway to the median is the first quartile.

4. The number halfway past the median is the 3rd quartile.

5. A number with (no more than) 66% of the values less than it is the 66th percentile, and so forth.


Box Plot

Median

Q1 Q3Smallest Largest


• Important Measure of Variation

• Shows Variation About the Mean:

• For the Population:

• For the Sample:

Variance

N

X i

2

2

1

2

2

n

XXs i

For the Population: use N in the denominator.

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


• 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.


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

)1624()1618()1617()1615()1614()1612()1610( 2222222

= 4.2426

s

:iX


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


Agenda

Confidence Interval




Standard Deviation



• Measure of Relative Variation

• Always a %

• Shows Variation Relative to Mean

• Used to Compare 2 or More Groups

• Formula (for Sample):

100%

X

SCV

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%

Comparing Coefficient of Variation


• Describes How Data Are Distributed

• Measures of Shape:

Symmetric or skewed

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

Shape


Agenda

Confidence Interval



Confidence Interval

• Sample Mean + Margin of Error (MOE)

• Called a Confidence Interval

• To Compute Margin of Error, One of Two Conditions Must Be True:

• The Distribution of the Population of Incomes Must Be Normal, or

• The Distribution of Sample Means Must Be Normal.


A Side-Trip Before Constructing Confidence Intervals

1. What is a Population Distribution?

2. What is a Distribution of the Sample Mean?

3. How Does Distribution of Sample Mean Differ From a Population Distribution?

4. What is the Central Limit Theorem?


190 105 254 345 340 363 79 197 334 117149 89 85 124 141 80 161 178 241 182187 182 348 215 203 166 368 317 372 152361 91 287 320 367 215 165 300 180 30897 135 94 183 221 228 187 371 87 14476 353 105 152 308 279 318 292 101 115302 263 127 196 241 288 242 129 366 281234 314 317 154 128 335 109 93 303 297371 353 346 238 225 277 222 119 86 314276 295 250 121 343 188 135 137 175 173165 316 284 156 346 87 288 211 230 152162 316 312 278 302 360 261 292 365 186330 242 337 207 140 333 159 286 287 188174 101 368 161 235 197 374 343 318 348247 287 195 108 344 191 104 308 310 275272 153 305 285 333 76 279 354 88 230349 361 253 242 365 220 152 320 224 330275 353 211 125 94 77 237 260 223 249256 354 235 115 100 248 324 95 156 285199 185 206 174 138 297 232 344 256 232

Assume Small Population of Lexus Owners’ Incomes (N = 200)

Estimating the Population Mean Income of Lexus Owners


Distribution of N = 200 Incomes (Population Mean )

75 125 175 225 275 325 75 125 175 225 275 325

3030

MeanMean


Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Mean

362 79 197 333 116 217.480 160 177 241 182 168.0

166 367 316 372 151 274.4214 165 300 180 307 233.2228 187 370 87 144 203.2278 317 292 100 114 220.2288 241 129 366 281 261.0335 109 92 303 296 227.0277 221 118 86 313 203.0188 135 136 175 172 161.286 287 211 229 151 192.8

359 260 291 365 185 292.0332 159 285 287 187 250.0

Constructing a Distribution of Samples of Size 5 from N = 200 Owners

x


Distribution of Sample Mean Incomes (Column #7)

125 175 225 275 325

2

6

21

8

3

0

5

10

15

20

25

125 175 225 275 325

Distribution of Sample Means Near Normal! ( )X

Estimated Std. ErrorEstimated Std. Error


Central Limit Theorem

• Even if Distribution of Population is Not Normal, Distribution of Sample Mean Will Be Near Normal Provided You Select Sample of Five or Ten or Greater From the Population.

• For a Sample Sizes of 30 or More, Distribution of the Sample Mean Will Be Normal, with

– mean of sample means = population mean, and

– standard error = [population deviation] / [sqrt(n)]

• Thus Can Use Expression:

MOEX


Why Does Central Limit Theorem Work?

As Sample Size Increases:

1. Most Sample Means will be Close to Population Mean,

2. Some Sample Means will be Either Relatively Far Above or Below Population Mean.

3. A Few Sample Means will be Either Very Far Above or Below Population Mean.


Impact of Side-Trip on MOE

1. Determine Confidence, or Reliability, Factor.

2. Distribution of Sample Mean Normal from Central Limit Theorem.

3. Use a “Normal-Like Table” to Obtain Confidence Factor.

4. Determine Spread in Sample Means (Without Taking Repeated Without Taking Repeated Samples)Samples)

sXin Spreadfactor) e(Confidenc= MOE


Drawing Conclusions about a Population Mean Using a Sample Mean

Select Simple Random SampleSelect Simple Random Sample

Compute Sample Mean andCompute Sample Mean andStd. Dev. For n < 10, Sample Bell-Shaped?Std. Dev. For n < 10, Sample Bell-Shaped?For n >10 For n >10 CLT Ensures Dist of NormalCLT Ensures Dist of Normal

x MOEx MOE

Draw Conclusion about Draw Conclusion about Population MeanPopulation Mean

x


Federal Aid Problem

• Suppose a census tract with 5000 families is eligible for aid under program HR-247 if average income of families of 4 is between $7500 and $8500 (those lower than 7500 are eligible in a different program). A random sample of 12 families yields data on the next page.


Federal Aid Study Calculations

7,300 7,700 8,100 8,4007,800 8,300 8,500 7,600 7,400 7,800 8,300 8,600

x

s

$7,

$441( ) .. ( )

983

7300 7983 8600 7983

12 1

2 2

Representative Sample

x MOE 7 983, MOE


Estimated Standard Error

• Measures Variation Among the Sample Means If We Took Repeated If We Took Repeated SamplesSamples.

• But We Only Have One Sample! How Can We Compute Estimated Standard Error?

• Based on Constructing Distribution of Sample Mean Slide, Will Estimated Standard Error Be Smaller or Larger Than Sample Standard Deviation (s)?

• Estimated Std. Error ______ than s.

• Estimated Standard Error Expression:

sizesample

deviationdardssample

tan

s

nFor Federal

Aid Study


Confidence Factor for MOE:

Df = n-1 2-Sided90%

2-Sided95%

2-Sided99%

8 1.86 2.306 3.355

11 1.796 2.201 3.106

14 1.761 2.145 2.977

17 1.740 2.110 2.898

Can Use t-Table Provided Distribution of Sample Mean is Normal


95% Confidence Interval

ErrordardSEstConfidenceX tan .

7 983 2 201441

127 703 8 263

, ( . )

$ , $ ,

Interpretation of Confidence Interval

• 95% Confident that Interval $7,983 + $280 Contains Unknown PopulationPopulation (Not SampleNot Sample) ) Mean Income.

• If We Selected 1,000 Samples of Size 12 and Constructed 1,000 Confidence Intervals, about 950 Would Contain Unknown Population Mean and 50 Would Not.

• So Is Tract Eligible for Aid???


Sample Means versus Sample Proportion

• Income/Loss

• Time to Complete Loan Papers

• Number of Fat Calories in Burger

• Breaking Strength of Cellular Phone Housing

• Americans Who Believe that Japan is #1 Economic Power

• Circuit Boards with One or More Failed Solder Connections

• African-Americans Who Pass CPA

Mean Proportion of

Means and Proportions Not the Same!!!!


Similarities and Differences Between Sample Means and Proportions

• Sample Means Computed from Data that Are MeasuredMeasured. Estimate Population Means.

• Sample Proportions Computed from Data that Are CountedCounted. Estimate Population Proportions.


Drawing Conclusions about a Population Proportion From a Sample Proportion

Select Simple Random SampleSelect Simple Random Sample

Compute Sample ProportionCompute Sample ProportionCheck for Normality - Table 7.8Check for Normality - Table 7.8

p MOE ( ) /p z p p n 1

Draw Conclusion AboutDraw Conclusion About

Population Proportion, pPopulation Proportion, p


Japan Business Survey

• N =200 Californians

• Yes = 116

• No = 84

Is Japan the ForemostEconomic Power Today?

.p 116

2000 58


90% Confidence Interval on P

.

.(. )(. )

.

. .

58

164558 42

200057

58 057

MOE

MOE

(.523, .637)

90% Confident that Between 52.3%and 63.7% of Californians Believe Japan is Leading Economic Power.

Common Practice to Report Proportions as Percentages. After After Calculations Done.Calculations Done.

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