Descriptive Statistics Prerequisite Material MGS 8110 Regression & Forecasting

Descriptive StatisticsPrerequisite Material

MGS 8110Regression & Forecasting

L00D MGS 8110 - Descriptive Statistics

2

Descriptive Analysis of Data

Given a bunch of data (simple data for only one variable), how is the best way to summarize the data?


3

Measures of Central Tendency

• Mean

• Median if n is odd

if n is even •

• Mode the xi value that occurs most frequently

• Mid Range

• Mid InterQuartile

n

xx i

2

1

2

x

nM

2/)xx(1

22

nnM

2/)xx( maxmin RM

2/)xx( 75.25. QM

2

1x nM


4

Measures of Variability• Av Deviate

• Av Absolute Deviate

• Variance

• Standard Deviation

• Range

• Inter Quartile Range (IQR)

n

xxi

)(

n

xxi

1

)(s

2

n

xxi

)xx( minmax R

)xx( 25.75. QR

1

)(s

22

n

xxi

5

Lets Review - Statistical Precepts

Use the Mean for quantitative, symmetric data.

Use the Medium for quantitative non-symmetric data.

Use the mode for categorical data.

Use the Variance when doing calculations.

Use the Standard Deviation when presenting the results of the calculations.

Major Teaching Points are frequently shown in green boxes



6

Appropriate Statisticsdiscussed in

chapter 2

Quantitative - Bell Shaped Mean St Dev HistogramQuantitative - NOT Bell Shaped Median Quartile Range Histogram

Ordinal Median Quartile Range Tables or Bar chartsNominal Mode NA Tables or Bar charts

7

More Review – Need to memorize formules


1

)(s

2

n

xxi

1

)(s

22

n

xxi Used for calculation, but not for presentations. Units are squared (e.g., inches squared).

Used for presentations. Common units (e.g., inches).

Divide by n-1 instead of n in order to get an unbiased estimate.


8

Interpretation of Standard Deviation (1 of 2)

populationtheof73.99contains)3,3(populationtheof%45.95contains)2,2(populationtheof%27.68contains),(

sxsxsxsx

sxsx

If the data is normally distributed

Statistical Precepts

Two-thirds of the data is contained in one sigma.

95% of the data is contained in two sigma.

Almost all of the data is contained in three sigma.


9

Interpretation of Standard Deviation (2 of 2)

If ever asked to explain what the Standard Deviation means, say “two-thirds of the data will be within plus or minus one Standard Deviation from the mean”.

If ever asked for the “worst case” or “best case” outcome calculate “mean – (2)sigma” and/or “mean + (2)sigma”.

Statistical Precepts

Definition of Standard Deviation - two-thirds of the data is contained in a range of values that are two sigma wide.

Worst case outcome is – .

Best case outcome is + 2.


10

Other Measures

• Percentiles xp is the xi such that(measure of tails) p% of the xi < xp

• Quartiles are percentile where p = 25, 50 or 75(measure of tails) the lower, middle or upper quartile.

• 4

1

4

3

4

1

4

1125. xorxorx

nnn

CC

2/)xx(orx1

222

1250.

nnnCC

4

1n3

4

1

4

3

4

3375. xxx

nn

CCmaximum4

minimum0

x

x

C

C


11

Other Measures

• Coefficient of Variation(percentage measure of variability)

• Correlation Coefficient(measure of linear association)

x

sCv

yxii ss

n

yyxx/

1

))((

1,1


12

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

X variable

Y v

aria

ble

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

X variable

Y v

aria

ble

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

X variable

Y v

aria

ble

Interpretation of Correlation Coefficientn=25

=0

1,1

=.8=

1


13

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

X variable

Y v

aria

ble

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

X variable

Y v

aria

ble

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

X variable

Y v

aria

ble

Interpretation of Correlation Coefficientn=25

=0

1,1

=-.8

=-1


14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Correlation Coefficient

Per

cent

age

of V

aria

bilit

y E

xpla

ined

Interpretation of Correlation Coefficient

yx

ii

ss

n

yyxx

1

))((


15

Interpretation of Correlation Coefficient (1

of 2)

Statistical tests of correlation coefficients are relatively meaningless. These tests are based on the hypothesis that “ = ”. Based on the previous graph, knowing that a correlation coefficient is greater than zero is not necessarily a valuable piece of information.

In terms of “Practical Significance” (compared to “Statistical Significance”), the correlation coefficient has to be at least greater than .5. From the previous graph it can be seen that = only explains 25% of the variability in the data.

Statistical Precept

must be greater than .5 to be of Practical Significance


16

Other Measures(page 3 of 3)

• Skew(measure of symmetry)

• Kurtosis(measure of peakedness)

2

32

3

3

)(

)(

n

xx

n

xx

i

i

3)(

)(

22

4

4

n

xx

n

xx

i

i

3,3y typicallmore ,3

3,3y typicallmore ,3


17

Verifying Bell Shape (Normal Distribution) • Negative Skew if the distribution has a ‘long’ tail to the left,

measured as skewness -1

• Positive Skew if the distribution has a ‘long’ tail to the right (more common situation), measured as skewness +1

• Symmetric if -1 skewness +1

• Peaked Distribution if Kurtosis is a large positive number ( +1).

• Flat Distribution if Kurtosis is a large negative number ( -1).

• Normal shape (proportionally S-shaped sides) if Kurtosis near zero.


18

Verifying Bell Shape (Normal Distribution)

Statistical Precept

Bell shaped (Normally distributed)

if -1 skewness +1 and

if -1 kurtosis +1


19

How is the best way to summarize data?(our original question)

• Central Tendency (Mean, Median & Mode)

• Variability (Variance & Standard Deviation)

• Shape (Percentiles, Skewness & Kurtosis)

• Association (correlation)


20

Notations

pnCorrelatio

sVariance

Deviation

sStandard

average

XMean

ValueTrue

ValueSamplePopulationParameter

22

)(


21

Standard Deviation of Sample Mean

n

ss XX

Called the “Standard Error” of the Mean


22

Insert / Function examples (1 of 3)

• Mean Average(A1:A10)

• Median Median(A1:A10)

• Mode Mode(A1:A10)

• Mid Range ( MAX(A1:A10) + MIN(A1:A10) ) / 2

• InterQuartile ( Quartile(A1:A10,1) + Quartile(A1:A10,3) ) / 2


23


• Av Deviate NA

• Av Absolute Deviate AveDev(A1:A10)

• Variance Var(A1:A10)

• Standard Deviation StDev(A1:A10)

• Range MAX(A1:A10) - MIN(A1:A10)

• Inter Quartile Range (IQR) Quartile(A1:A10,3) - Quartile(A1:A10,1)


24


• Percentiles Percentile(A1:A10,.05)

• Quartiles Quartile(A1:A10,q) where q=0,1,2,3,4

• Coef. of Variation StDev(A1:A10)/Average(A1:A10)

• Correlation Correl(A1:A10,B1:B10)

• Skew Skew(A1:A10)

• Kurtosis Kurt(A1:A10)


25

Example Calculations

Q. Should I use the Mean or the Median to state the Central value of this data?

-0.31 =SKEW(Height)-1.27 =KURT(Height)

0.17 =SKEW(Weight)-1.03 =KURT(Weight)

Answer – Both variables have a somewhat peaked distributions (Kurtosis greater than 1), but both variables have very symmetric distributions (non-skewed distribution); hence, use Mean.


26


Q. The Standard Deviation for Height is almost 2 inches, what is the practical interpretation of this value?

Answer – The height of 2/3 of the population will vary by less than 4 inches (3.87”).

67.1 =B14-B1571.0 =B14+B153.87 =F10-F9


27


Q. What is the height of the shortest person and the tallest person that I may meet today (worst case and best case)?

Answer – The shortest person will be 5’-5” (65.2”) and the tallest person will be 6’-1” (72.9”).

65.2 =AvHt-2*StDevHt72.9 =AvHt+2*StDevHt


28


Q. What is the height of the shortest person that I may meet over the next year?

Answer – The shortest person that I am likely to meet in the foreseeable future will be 5’-6” (66.0”).

66.0 =PERCENTILE(Height,0.01)


29


Q. The answers to the two previous questions are not consistent. The 5% values calculated as Mean – 2(Sigma) was 5’-5” where as the 1% value calculated as a Percentile was 5’-6”.

Answer – These types of inconsistencies (i.e., errors) will occur with small samples. The procedure used by the PERCENTILE function is based on an interpolated calculation with the two smallest values in the sample.


30


Q. Which variable, Height or Weight, has the greatest relative variability?

Answer – In agreement with our intuition, Weight is 3 to 4 times more variable than height (11/3 = 3.67).

Coef of Variation3% =StDevHt/AvHt

11% =StDevWt/AvWt


31


Q. Is there a relationship between Height and Weight and if so how large is the relationship?

Answer – The correlation between Height and Weight is .78 which means that about 60% (.61) of the variability in weight is due to differences in Height.

0.783 =CORREL(Height,Weight)0.61 =G23^2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Correlation Coefficient

Pe

rce

nta

ge

of

Va

ria

bili

ty E

xpla

ine

d


32

Example CalculationsQ. Given that there is a

relationship between Height and Weight, is the relationship linear or non-linear?

Answer – Simple statistics cannot be used to determine linear versus non-linear, would need to plot the data. The correlation indicates that there is a relatively strong linear relationship, but a plot of the data (Weight vs. Height) may indicate that there is an even stronger non-linear relationship

0.783 =CORREL(Height,Weight)0.61 =G23^2


33


Q. Are Height and Weight Normally distributed?

Answer – Based on out Rule-of-Thumb test (-1 < Skew < +1 and -1 < Kurt < +1), neither of these variables are normally distributed.

-0.31 =SKEW(Height)-1.27 =KURT(Height)

0.17 =SKEW(Weight)-1.03 =KURT(Weight)


34


Q. Given that the variables are NOT Normally distributed, why do I care?

Answer – You previous interpretation of the Standard Deviation maybe somewhat inaccurate (“The height of 2/3 of the population will vary by less than 4 inches “). Also, you previous interpretation of Worst Case and Best Case maybe somewhat inaccurate (“The shortest person will be 5’-5” and the tallest person will be 6’-1”).


35


Q. The average Height is estimated to be 69.1”, how good is that estimate?

Answer – The true average height could be anywhere between 67.8 inches to 70.3 inches. A better estimate could be obtained if a large sample was available.

67.8 =AvHt-2*StErrorHt70.3 =AvHt+2*StErrorHt


36

More about Variability

1

)(s

2

n

xxi Use StDev (or Var) in Excel

N

xxi

2)(s

Use StDevP (or VarP) in Excel

Alternative formulation

1) if every item in the Universe is included in the Sample or

2) The Mean is know with certainty.

n

xi

2)(s


37

Cumulative Distribution Function for Male Height

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Height

Cu

mu

lati

ve D

istr

ibu

tio

n

.95

Probability Density Function for Male Height

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Height

Pro

bab

ility

Den

sity

.95

Normal Calculations

NORMINV(0.95,68.8,2.6)=73.08


38


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Height

Pro

bab

ility

Den

sity

.244


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Height

Cu

mu

lati

ve D

istr

ibu

tio

n

.244

Normal Calculations

NORMDIST(67,68.8,2.6,TRUE)=.244


39

Standardized Normal Calculations

NORMSINV(0.95)=1.645

NORMSINV(.05)=-1.645

s

xxz i

X variable has mean and StDev of and which are estimated by x bar and s.

Z variable has mean=0 and StDev=1. Z is a “standardized normal”.


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Pro

bab

ility

Den

sity

.95


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Cu

mu

lati

ve D

istr

ibu

tio

n

.95


40

Standardized Normal Calculations

NORMSDIST(-1)=.159

NORMSDIST(+1)=.841


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Pro

bab

ility

Den

sity

.159


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Cu

mu

lati

ve D

istr

ibu

tio

n

.159


41

t-distribution Probability Density Function for Male Height

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Pro

bab

ility

Den

sity

t-Dist df=4 t-Dist df=16 z Normal

.95

t-distribution is needed if is not know and estimated by s and n<30.


42

t-distribution Calculations

one-tail

TDIST(2,4,1)=.058

TDIST(X, d.f., # tails)


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Pro

bab

ility

Den

sity

.058


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Cu

mu

lati

ve D

istr

ibu

tio

n

.058

“t” with tails=1 sums from + infinity.

“Z” and “Normal” sums from –infinity.


43

t-distribution Calculations

two-tail

TDIST(2,4,2)=.116

TDIST(X, d.f., # tails)


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Pro

bab

ility

Den

sity

.058.058


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-5 -4 -3 -2 -1 0 1 2 3 4 5

Height

Cu

mu

lati

ve D

istr

ibu

tio

n

.116

“t (2 tail)” sums simultaneously from both – infinity and + infinity. Undefined for negative values of t.


44

Loading “Data Analysis” in Office 2003

/ Tools / Add-Ins /

Will need to have original MS Office CD.

Both

NO


45

Example ofTools / Data Analysis / Descriptive Statistics


46

Example Output ofTools / Data Analysis / Descriptive Statistics

Mean 47.84Standard Error 2.63Median 48.00Mode 41.00Standard Deviation 11.46Sample Variance 131.25Kurtosis -0.73Skewness 0.29Range 41Minimum 30Maximum 71Sum 909Count 19


47


1) Click the Office button in the upper left hand corner of the Excel.

2) Click the “Excel Options” tab in the bottom right-hand cornet of the drop-down menu gotten from step #1.

Both

NO

NO

1) click


48


3)Click Add-Ins in the left banner of the Excel Options menu.

4)Click “Analysis ToolPak” in the Add-ins menu. Then, select BOTH “Analysis ToolPak” and “Analysisi ToolPak – VBA”

5)Click the Go button at the bottom right hand corner of the Excel Options menu. Don’t click the “OK” button”.

Both

NO

NO

Both

5) click

4) click3) click

6) click


49


1) click


50

Example ofTools / Data Analysis / Descriptive Statistics


51

Precision of numerical results – state “3 Significant Digits”

Wt Wt Wt

Mean 76.506 Mean 76.5 Mean 76.5Standard Error 4.780314 Standard Error 4.8 Standard Error 4.78

Median 73.58 Median 73.6 Median 73.6Mode #N/A Mode #N/A Mode #N/A

Standard Deviation 15.11668 Standard Deviation 15.1 Standard Deviation 15.1Sample Variance 228.514 Sample Variance 228.5 Sample Variance 228.5

Kurtosis -0.67472 Kurtosis -0.7 Kurtosis -0.675Skewness 0.362102 Skewness 0.4 Skewness 0.362

Range 47.98 Range 48.0 Range 48.0Minimum 54.28 Minimum 54.3 Minimum 54.3Maximum 102.26 Maximum 102.3 Maximum 102.3

Sum 765.06 Sum 765.1 Sum 765.1Count 10 Count 10.0 Count 10

Too Many Digits Too Few Digits Correct 3+ significant digits

meaningless

extra digits

too f ew digits

(.651 to .744)

OK


52

Precision of numerical results – state “3 Significant Digits” (continued)

Wt Wt

Mean 76.5 Mean 76.506Standard Error 4.78 Standard Error 4.780

Median 73.6 Median 73.580Mode #N/A Mode #N/A

Standard Deviation 15.1 Standard Deviation 15.117Sample Variance 229 Sample Variance 228.514

Kurtosis -0.675 Kurtosis -0.675Skewness 0.362 Skewness 0.362

Range 48.0 Range 47.980Minimum 54.3 Minimum 54.280Maximum 102 Maximum 102.260

Sum 765 Sum 765.060Count 10 Count 10.000

3 significant digits 3 decimal places

CORRECT I NCORRECT


53

Data is a potential outlier

Symmetric distributionxi < mean – 3s or xi > mean + 3 s

Skewed distribution xi < Q1 – (1.5)RQ or xi > Q3 + (1.5)RQ

Some Great Rules of Thumb

Data is Normally distributed (Bell shaped)

if -1 skewness +1 andif -1 kurtosis +1

Let’s Review


54

Prerequisite Spreadsheet Skills• Cut, Copy, Paste & Paste Special

• Cell corner Copy• Add or delete Rows or Columns• Change width/height of row/column• Font, alignment, boarder & number of cell• Referencing and calculations with cells.• Data / Sort• Naming cell or range of cells• Insert / Function / Average• Sum, Max, Min, Count, Small and Large• (Tools / Add-ins / Data analysis)• Tools / Data Analysis / Descriptive statistics• Single quote for equation statement. • REPLACE command.• DATA / Group & FORMAT: Column, Hide. • Grab an entire column of data (CTRL+SHIFT, down arrow).

See also “L99A MBA7025.ppt” in folder “L00A MGS8110”

Documents

Descriptive Statistics Prerequisite Material MGS 8110 Regression & Forecasting