Chapter 3 - Part B Descriptive Statistics: Numerical Methods

Measures of Relative Location and Detecting OutliersExploratory Data AnalysisMeasures of Association Between Two VariablesThe Weighted Mean and Working with Grouped Data

%%xx

Measures of Relative Locationand Detecting Outliers

z-ScoresChebyshev’s TheoremEmpirical RuleDetecting Outliers

z-Scores

The z-score is often called the standardized value.It denotes the number of standard deviations a data value xi is from the mean.

A data value less than the sample mean will have a z-score less than zero.A data value greater than the sample mean will have a z-score greater than zero.A data value equal to the sample mean will have a z-score of zero.

zx xsii

zx xsii

z-Score of Smallest Value (425)

Standardized Values for Apartment Rents

74.54

80.490425

s

xxz i

74.54

80.490425

s

xxz i

-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27

-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27

Example: Apartment Rents

Chebyshev’s Theorem

At least (1 - 1/z2) of the items in any data set will bewithin z standard deviations of the mean, where z isany value greater than 1.

– At least 75% of the items must be withinz = 2 standard deviations of the mean.

– At least 89% of the items must be withinz = 3 standard deviations of the mean.

– At least 94% of the items must be withinz = 4 standard deviations of the mean.

Example: Apartment Rents

Chebyshev’s Theorem

Let z = 1.5 with = 490.80 and s = 54.74

At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56%

of the rent values must be between - z(s) = 490.80 - 1.5(54.74) =

_______ and

+ z(s) = 490.80 + 1.5(54.74) =_______

xx

xx

xx

Chebyshev’s Theorem (continued) Actually, 86% of the rent values

are between ____ and _____.

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

Example: Apartment Rents

Empirical Rule

For data having a bell-shaped distribution:

– Approximately 68% of the data values will be within one standard deviation of the mean.

Empirical Rule

For data having a bell-shaped distribution:

– Approximately 95% of the data values will be within two standard deviations of the mean.

Empirical Rule

For data having a bell-shaped distribution:

– Almost all (99.7%) of the items will be within

three standard deviations of the mean.

Example: Apartment Rents

Empirical Rule Interval % in

IntervalWithin +/- 1s 436.06 to 545.54 48/70 = 69%Within +/- 2s 381.32 to 600.28 68/70 = 97%Within +/- 3s 326.58 to 655.02 70/70 = 100%

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

Detecting Outliers

An outlier is an unusually small or unusually large value in a data set.A data value with a z-score less than -3 or greater than +3 might be considered an outlier. It might be:– an incorrectly recorded data value– a data value that was incorrectly included in

the data set– a correctly recorded data value that belongs

in the data set

Example: Apartment Rents

Detecting OutliersThe most extreme z-scores are -1.20 and

2.27.Using |z| > 3 as the criterion for an

outlier, there are no outliers in this data set.

Standardized Values for Apartment Rents-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27

-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27

Exploratory Data Analysis

Five-Number SummaryBox Plot

Five-Number Summary

Smallest ValueFirst QuartileMedianThird QuartileLargest Value

Example: Apartment Rents

Five-Number SummaryLowest Value = 425 First Quartile

= 445 Median = 475

Third Quartile = 525 Largest Value = 615

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

Box Plot

A box is drawn with its ends located at the first and third quartiles.A vertical line is drawn in the box at the location of the median.Limits are located (not drawn) using the interquartile range (IQR).– The lower limit is located 1.5(IQR) below Q1.– The upper limit is located 1.5(IQR) above

Q3.– Data outside these limits are considered

outliers.… continued

Box Plot (Continued)

Whiskers (dashed lines ) are drawn from the ends of the box to the smallest and largest data values inside the limits.The locations of each outlier is shown with the symbol * .

Example: Apartment Rents

Box Plot

Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(75) = 332.5

Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5

There are no outliers.

375375

400400

425425

450450

475475

500500

525525

550550 575575 600600 625625

Measures of Association Between Two Variables

CovarianceCorrelation Coefficient

Covariance

The covariance is a measure of the linear association between two variables.Positive values indicate a positive relationship.Negative values indicate a negative relationship.

If the data sets are samples, the covariance is denoted by sxy.

If the data sets are populations, the covariance is denoted by .

Covariance

sx x y ynxy

i i

( )( )

1s

x x y ynxy

i i

( )( )

1

xyi x i yx y

N

( )( )

xy

i x i yx y

N

( )( )

xyxy

x = Number of y = Number ofInterceptions Points Scored

1 14 3 24 2 18 1 17 3 27

----------------- --------------------

Example: Panthers Football Team

Correlation Coefficient

The coefficient can take on values between -1 and +1.Values near -1 indicate a strong negative linear relationship.Values near +1 indicate a strong positive linear relationship.If the data sets are samples, the coefficient is rxy.

If the data sets are populations, the coefficient is xy.

rs

s sxyxy

x yrs

s sxyxy

x y

xyxy

x y

xyxy

x y

The Weighted Mean andWorking with Grouped Data

Weighted MeanMean for Grouped DataVariance for Grouped DataStandard Deviation for Grouped Data

Weighted Mean

When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean.In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade.When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.

Weighted Mean

wi xi

x = ___________ wi

where: xi = value of observation i

wi = weight for observation i

Grouped Data

The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data.To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class.We compute a weighted mean of the class midpoints using the class frequencies as weights.Similarly, in computing the variance and standard deviation, the class frequencies are used as weights.

Sample Data

Population Data

where: fi = frequency of class i

Mi = midpoint of class i

Mean for Grouped Data

i

ii

f

Mfx

i

ii

f

Mfx

N

Mf iiN

Mf ii

Example: Apartment Rents

Given below is the previous sample of monthly rents

for one-bedroom apartments presented here as grouped

data in the form of a frequency distribution.

Rent ($) Frequency420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Rent ($) Frequency420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Example: Apartment Rents

Mean for Grouped Data

This approximation differs by $2.41 from

the actual sample mean of $_______.

Rent ($) f i M i f iM i

420-439 8 429.5 3436.0440-459 17 449.5 7641.5460-479 12 469.5 5634.0480-499 8 489.5 3916.0500-519 7 509.5 3566.5520-539 4 529.5 2118.0540-559 2 549.5 1099.0560-579 4 569.5 2278.0580-599 2 589.5 1179.0600-619 6 609.5 3657.0

Total 70 34525.0

Rent ($) f i M i f iM i

420-439 8 429.5 3436.0440-459 17 449.5 7641.5460-479 12 469.5 5634.0480-499 8 489.5 3916.0500-519 7 509.5 3566.5520-539 4 529.5 2118.0540-559 2 549.5 1099.0560-579 4 569.5 2278.0580-599 2 589.5 1179.0600-619 6 609.5 3657.0

Total 70 34525.0

21.49370

525,34_

x 21.49370

525,34_

x

Sample Data

Population Data

Variance for Grouped Data

sf M xn

i i22

1

( )s

f M xn

i i22

1

( )

22

f M

Ni i( ) 2

2

f M

Ni i( )

Example: Apartment Rents

Variance for Grouped Data

Standard Deviation for Grouped Data

This approximation differs by only $_____ from the actual standard deviation of $______.

s2 3 017 89 , .s2 3 017 89 , .

s 3 017 89 54 94, . .s 3 017 89 54 94, . .

A 5-Minute In-Class Exercise

With = 490.80 and s = 54.74:1. What is the z-score for an observation value Xi=

600?

Z = 2. According to Chebyshev’s Theorem, if z =

3.16228, then What Percentage of the data set values must be between what Lower Limit and what Upper Limit?

Percentage = Lower Limit =Upper Limit =

xx

End of Chapter 3, Part B