numerical descriptive measures...title bbs13e_chapter03 created date 12/5/2016 5:52:29 pm

Copyright © 2015, 2012, 2009 Pearson Education, Inc. Chapter 3, Slide 1

Numerical Descriptive Measures

Chapter 3


In this chapter, you learn:n To describe the properties of central tendency,

variation, and shape in numerical dataTn To construct and interpret a boxplotn To compute descriptive summary measures for a

populationn To calculate the covariance and the coefficient of

correlation

Learning Objectives


Summary Definitions

§ The central tendency is the extent to which all the data values group around a typical or central value.

§ The variation is the amount of dispersion or scattering of values

§ The shape is the pattern of the distribution of values from the lowest value to the highest value.

DCOVA


Measures of Central Tendency:The Mean

n The arithmetic mean (often just called the “mean”) is the most common measure of central tendency

n For a sample of size n:

Sample size

nXXX

n

XX n21

n

1ii +++==

∑= !

Observed values

The ith valuePronounced x-bar

DCOVA


Measures of Central Tendency:The Mean (con’t)

n The most common measure of central tendencyn Mean = sum of values divided by the number of valuesn Affected by extreme values (outliers)

11 12 13 14 15 16 17 18 19 20

Mean = 13

11 12 13 14 15 16 17 18 19 20

Mean = 14

31565

55141312111

==++++ 41

570

52041312111

==++++

DCOVA


Measures of Central Tendency:The Median

n In an ordered array, the median is the “middle” number (50% above, 50% below)

n Less sensitive than the mean to extreme values

Median = 13 Median = 13

11 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20

DCOVA


Measures of Central Tendency:Locating the Median

n The location of the median when the values are in numerical order (smallest to largest):

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

n 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

dataorderedtheinposition21npositionMedian +

=

21n +

DCOVA


Measures of Central Tendency:The Mode

n Value that occurs most oftenn Not affected by extreme valuesn Used for either numerical or categorical (nominal)

datan There may may be no moden There may be several modes

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

Mode = 9

0 1 2 3 4 5 6

No Mode

DCOVA


Measures of Central Tendency:Review Example

House Prices:

$2,000,000$ 500,000$ 300,000$ 100,000$ 100,000

Sum $ 3,000,000

§ Mean: ($3,000,000/5) = $600,000

§ Median: middle value of ranked data

= $300,000§ Mode: most frequent value

= $100,000

DCOVA


Measures of Central Tendency:Which Measure to Choose?

§ The mean is generally used, unless extreme values (outliers) exist.

§ The median is often used, since the median is not sensitive to extreme values. For example, median home prices may be reported for a region; it is less sensitive to outliers.

§ In some situations it makes sense to report both the mean and the median.

DCOVA


Measure of Central Tendency For The Rate Of Change Of A Variable Over Time:The Geometric Mean & The Geometric Rate of Return

§ Geometric mean§ Used to measure the rate of change of a variable over time

§ Geometric mean rate of return§ Measures the status of an investment over time

§ Where Ri is the rate of return in time period i

n/1n21G )XXX(X ×××= !

1)]R1()R1()R1[(R n/1n21G −+××+×+= !

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The Geometric Mean & The Mean Rate of Return: Example

An investment of $100,000 declined to $50,000 at the end of year one and rebounded to $100,000 at end of year two:

The overall two-year return is zero, since it started and ended at the same level.

000,100$X000,50$X000,100$X 321 ===

50% decrease 100% increase

DCOVA


The Geometric Mean & The Mean Rate of Return: Example (con’t)

Use the 1-year returns to compute the arithmetic mean and the geometric mean:

%2525.2

)1()5.(==

+−=X

Arithmetic mean rate of return:

Geometric mean rate of return: %0111)]2()50[(.

1))]1(1())5.(1[(

1)]1()1()1[(

2/12/1

2/1

/121

=−=−×=

−+×−+=

−+××+×+= nnG RRRR !

Misleading result

More

representative

result

DCOVA


Measures of Central Tendency:Summary

Central Tendency

Arithmetic Mean

Median Mode Geometric Mean

n

XX

n

ii∑

== 1

n/1n21G )XXX(X ×××= !

Middle value in the ordered array

Most frequently observed value

Rate of change ofa variable over time

DCOVA


Same center, different variation

Measures of Variation

n Measures of variation give information on the spread or variability or dispersion of the data values.

Variation

Standard Deviation

Coefficient of Variation

Range Variance

DCOVA


Measures of Variation:The Range

§ Simplest measure of variation§ Difference between the largest and the smallest values:

Range = Xlargest – Xsmallest

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

Range = 13 - 1 = 12

Example:

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Measures of Variation:Why The Range Can Be Misleading

§ Does not account for how the data are distributed

§ Sensitive to outliers

7 8 9 10 11 12Range = 12 - 7 = 5

7 8 9 10 11 12Range = 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

DCOVA


n Average (approximately) of squared deviations of values from the mean

n Sample variance:

Measures of Variation:The Sample Variance

1-n

)X(XS

n

1i

2i

2∑=

−=

Where = arithmetic mean

n = sample size

Xi = ith value of the variable X

X

DCOVA


Measures of Variation:The Sample Standard Deviation

n Most commonly used measure of variationn Shows variation about the meann Is the square root of the variancen Has the same units as the original data

n Sample standard deviation:

1-n

)X(XS

n

1i

2i∑

=

−=

DCOVA


Measures of Variation:The Standard Deviation

Steps for Computing Standard Deviation

1. Compute the difference between each value and the mean.

2. Square each difference.3. Add the squared differences.4. Divide this total by n-1 to get the sample variance.5. Take the square root of the sample variance to get

the sample standard deviation.

DCOVA


Measures of Variation:Sample Standard Deviation:Calculation Example

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

n = 8 Mean = X = 16

4.30957130

1816)(2416)(1416)(1216)(10

1n)X(24)X(14)X(12)X(10S

2222

2222

==

−

−++−+−+−=

−

−++−+−+−=

!

!

A measure of the “average” scatter around the mean

DCOVA


Measures of Variation: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 = 15.5S = 0.926

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5S = 4.567

Data C

DCOVA


Measures of Variation:Comparing Standard Deviations

Smaller standard deviation

Larger standard deviation

DCOVA


Measures of Variation:Summary Characteristics

§ The more the data are spread out, the greater the range, variance, and standard deviation.

§ The more the data are concentrated, the smaller the range, variance, and standard deviation.

§ If the values are all the same (no variation), all these measures will be zero.

§ None of these measures are ever negative.

DCOVA


Measures of Variation:The Coefficient of Variation

n Measures relative variationn Always in percentage (%)n Shows variation relative to meann Can be used to compare the variability of two or

more sets of data measured in different units

100%XSCV ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

DCOVA


Measures of Variation:Comparing Coefficients of Variation

n Stock A:n Average price last year = $50n Standard deviation = $5

n Stock B:n Average price last year = $100n Standard deviation = $5

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

10%100%$50$5100%

XSCVA =⋅=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

5%100%$100$5100%

XSCVB =⋅=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

DCOVA


Measures of Variation:Comparing Coefficients of Variation (con’t)

n Stock A:n Average price last year = $50n Standard deviation = $5

n Stock C:n Average price last year = $8n Standard deviation = $2

Stock C has a much smaller standard deviation but a much higher coefficient of variation

10%100%$50$5100%

XSCVA =⋅=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

25%100%$8$2100%

XSCVC =⋅=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

DCOVA


Locating Extreme Outliers:Z-Score

§ To compute the Z-score of a data value, subtract the mean and divide by the standard deviation.

§ The Z-score is the number of standard deviations a data value is from the mean.

§ A data value is considered an extreme outlier if its Z-score is less than -3.0 or greater than +3.0.

§ The larger the absolute value of the Z-score, the farther the data value is from the mean.

DCOVA



where X represents the data valueX is the sample meanS is the sample standard deviation

SXXZ −

=

DCOVA



§ Suppose the mean math SAT score is 490, with a standard deviation of 100.

§ Compute the Z-score for a test score of 620.

3.1100130

100490620

==−

=−

=SXXZ

A score of 620 is 1.3 standard deviations above the mean and would not be considered an outlier.

DCOVA


Shape of a Distribution

n Describes how data are distributedn Two useful shape related statistics are:

n Skewnessn Measures the extent to which data values are not

symmetrical

n Kurtosisn Kurtosis affects the peakedness of the curve of

the distribution—that is, how sharply the curve rises approaching the center of the distribution

DCOVA


Shape of a Distribution (Skewness)

n Measures the extent to which data is not symmetrical

Mean = MedianMean < Median Median < MeanRight-SkewedLeft-Skewed Symmetric

DCOVA

SkewnessStatistic < 0 0 >0


Shape of a Distribution -- Kurtosis measures how sharply the curve rises approaching the center of the distribution)

Sharper PeakThan Bell-Shaped

(Kurtosis > 0)

Flatter ThanBell-Shaped

(Kurtosis < 0)

Bell-Shaped(Kurtosis = 0)

DCOVA


General Descriptive Stats Using Microsoft Excel Functions DCOVA

HousePrices2,000,000$ Mean 600,000$ =AVERAGE(A2:A6)500,000$ StandardError 357,770.88$ =D6/SQRT(D14)300,000$ Median 300,000$ =MEDIAN(A2:A6)100,000$ Mode 100,000.00$ =MODE(A2:A6)100,000$ StandardDeviation 800,000$ =STDEV(A2:A6)

SampleVariance 640,000,000,000 =VAR(A2:A6)Kurtosis 4.1301 =KURT(A2:A6)Skewness 2.0068 =SKEW(A2:A6)Range 1,900,000$ =D12-D11Minimum 100,000$ =MIN(A2:A6)Maximum 2,000,000$ =MAX(A2:A6)Sum 3,000,000$ =SUM(A2:A6)Count 5 =COUNT(A2:A6)

DescriptiveStatistics


General Descriptive Stats Using Microsoft Excel Data Analysis Tool

1. Select Data.

2. Select Data Analysis.

3. Select Descriptive Statistics and click OK.

DCOVA


General Descriptive Stats Using Microsoft Excel

4. Enter the cell range.

5. Check the Summary Statistics box.

6. Click OK

DCOVA


Excel output

Microsoft Excel descriptive statistics output, using the house price data:

House Prices:

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

DCOVAHousePrices

Mean 600000StandardError 357770.8764Median 300000Mode 100000StandardDeviation 800000SampleVariance 640,000,000,000Kurtosis 4.1301Skewness 2.0068Range 1900000Minimum 100000Maximum 2000000Sum 3000000Count 5


Minitab OutputMinitab descriptive statistics output using the house price data:House Prices:

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

DCOVA

Descriptive Statistics: House Price

TotalVariable Count Mean SE Mean StDev Variance Sum MinimumHouse Price 5 600000 357771 800000 6.40000E+11 3000000 100000

N forVariable Median Maximum Range Mode Mode Skewness KurtosisHouse Price 300000 2000000 1900000 100000 2 2.01 4.13


Quartile Measures

n Quartiles split the ranked data into 4 segments with an equal number of values per segment

25%

n The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger

n Q2 is the same as the median (50% of the observations are smaller and 50% are larger)

n Only 25% of the observations are greater than the third quartile

Q1 Q2 Q3

25% 25% 25%

DCOVA


Quartile Measures:Locating Quartiles

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

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

Second quartile position: Q2 = (n+1)/2 ranked value

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

where n is the number of observed values

DCOVA


Quartile Measures:Calculation Rules

n When calculating the ranked position use the following rulesn If the result is a whole number then it is the ranked

position to use

n If the result is a fractional half (e.g. 2.5, 7.5, 8.5, etc.) then average the two corresponding data values.

n If the result is not a whole number or a fractional half then round the result to the nearest integer to find the ranked position.

DCOVA


(n = 9)Q1 is in the (9+1)/4 = 2.5 position of the ranked dataso use the value half way between the 2nd and 3rd values,

so Q1 = 12.5

Quartile Measures:Locating Quartiles

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

Q1 and Q3 are measures of non-central locationQ2 = median, is a measure of central tendency

DCOVA


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

so Q1 = (12+13)/2 = 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 = (18+21)/2 = 19.5

Quartile MeasuresCalculating The Quartiles: Example

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

Q1 and Q3 are measures of non-central locationQ2 = median, is a measure of central tendency

DCOVA


Quartile Measures:The Interquartile Range (IQR)

n The IQR is Q3 – Q1 and measures the spread in the middle 50% of the data

n The IQR is also called the midspread because it covers the middle 50% of the data

n The IQR is a measure of variability that is not influenced by outliers or extreme values

n Measures like Q1, Q3, and IQR that are not influenced by outliers are called resistant measures

DCOVA


Calculating The Interquartile Range

Median(Q2)

XmaximumX

minimum Q1 Q3

Example:

25% 25% 25% 25%

12 30 45 57 70

Interquartile range = 57 – 30 = 27

DCOVA


The Five Number Summary

The five numbers that help describe the center, spread and shape of data are:

§ Xsmallest

§ First Quartile (Q1)§ Median (Q2)§ Third Quartile (Q3)§ Xlargest

DCOVA


Relationships among the five-number summary and distribution shape

Left-Skewed Symmetric Right-SkewedMedian – Xsmallest

>

Xlargest – Median

Median – Xsmallest

≈

Xlargest – Median

Median – Xsmallest

<

Xlargest – Median

Q1 – Xsmallest

>

Xlargest – Q3

Q1 – Xsmallest

≈

Xlargest – Q3

Q1 – Xsmallest

<

Xlargest – Q3

Median – Q1

>

Q3 – Median

Median – Q1

≈

Q3 – Median

Median – Q1

<

Q3 – Median

DCOVA


Five Number Summary andThe Boxplot

n The Boxplot: A Graphical display of the data based on the five-number summary:

Example:

Xsmallest -- Q1 -- Median -- Q3 -- Xlargest

25% of data 25% 25% 25% of dataof data of data

Xsmallest Q1 Median Q3 Xlargest

DCOVA


Five Number Summary:Shape of Boxplots

n If data are symmetric around the median then the box and central line are centered between the endpoints

n A Boxplot can be shown in either a vertical or horizontal orientation

Xsmallest Q1 Median Q3 Xlargest

DCOVA


Distribution Shape and The Boxplot

Right-SkewedLeft-Skewed Symmetric

Q1 Q2 Q3 Q1 Q2 Q3Q1 Q2 Q3

DCOVA


Boxplot Example

n Below is a Boxplot for the following data:

0 2 2 2 3 3 4 5 5 9 27

n The data are right skewed, as the plot depicts

0 2 3 5 270 2 3 5 27

Xsmallest Q1 Q2 / Median Q3 Xlargest

DCOVA


Numerical Descriptive Measures for a Population

§ Descriptive statistics discussed previously described a sample, not the population.

§ Summary measures describing a population, called parameters, are denoted with Greek letters.

§ Important population parameters are the population mean, variance, and standard deviation.

DCOVA


Numerical Descriptive Measures for a Population: The mean µ

n The population mean is the sum of the values in the population divided by the population size, N

NXXX

N

XN21

N

1ii +++==µ

∑= !

μ = population mean

N = population size


Where

DCOVA


n Average of squared deviations of values from the mean

n Population variance:

Numerical Descriptive Measures For A Population: The Variance σ2

N

μ)(Xσ

N

1i

2i

2∑=

−=

Where μ = population mean

N = population size


DCOVA


Numerical Descriptive Measures For A Population: The Standard Deviation σ

n Most commonly used measure of variationn Shows variation about the meann Is the square root of the population variancen Has the same units as the original data

n Population standard deviation:

N

μ)(Xσ

N

1i

2i∑

=

−=

DCOVA


Sample statistics versus population parameters

Measure Population Parameter

Sample Statistic

Mean

Variance

Standard Deviation

X

2S

S

µ

2σ

σ

DCOVA


n The empirical rule approximates the variation of data in a bell-shaped distribution

n Approximately 68% of the data in a bell shaped distribution is within 1 standard deviation of the mean or

The Empirical Rule

1σμ ±

μ

68%

1σμ±

DCOVA


n Approximately 95% of the data in a bell-shaped distribution lies within two standard deviations of the mean, or µ ± 2σ

n Approximately 99.7% of the data in a bell-shaped distribution lies within three standard deviations of the mean, or µ ± 3σ

The Empirical Rule

3σμ ±

99.7%95%

2σμ ±

DCOVA


Using the Empirical Rule

§ Suppose that the variable Math SAT scores is bell-shaped with a mean of 500 and a standard deviation of 90. Then,

§ 68% of all test takers scored between 410 and 590 (500 ± 90).

§ 95% of all test takers scored between 320 and 680 (500 ± 180).

§ 99.7% of all test takers scored between 230 and 770 (500 ± 270).

DCOVA


n Regardless of how the data are distributed, at least (1 - 1/k2) x 100% of the values will fall within k standard deviations of the mean (for k > 1)

n Examples:

(1 - 1/22) x 100% = 75% ….............. k=2 (μ ± 2σ)(1 - 1/32) x 100% = 88.89% ……….. k=3 (μ ± 3σ)

Chebyshev Rule

WithinAt least

DCOVA


We Discuss Two Measures Of The Relationship Between Two Numerical Variables

n Scatter plots allow you to visually examine the relationship between two numerical variables and now we will discuss two quantitative measures of such relationships.

n The Covariancen The Coefficient of Correlation


The Covariance

n The covariance measures the strength of the linear relationship between two numerical variables (X & Y)

n The sample covariance:

n Only concerned with the strength of the relationship n No causal effect is implied

1n

)YY)(XX()Y,X(cov

n

1iii

−

−−=∑=

DCOVA


n Covariance between two 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

n The covariance has a major flaw:n It is not possible to determine the relative strength of the

relationship from the size of the covariance

Interpreting CovarianceDCOVA


Coefficient of Correlation

n Measures the relative strength of the linear relationship between two numerical variables

n Sample coefficient of correlation:

whereYXSSY),(Xcovr =

1n

)X(XS

n

1i

2i

X −

−=∑=

1n

)Y)(YX(XY),(Xcov

n

1iii

−

−−=∑=

1n

)Y(YS

n

1i

2i

Y −

−=∑=

DCOVA


Features of theCoefficient of Correlation

n The population coefficient of correlation is referred as ρ.n The sample coefficient of correlation is referred to as r.

n Either ρ or r have the following features:n Unit freen Range between –1 and 1n The closer to –1, the stronger the negative linear relationshipn The closer to 1, the stronger the positive linear relationshipn The closer to 0, the weaker the linear relationship

DCOVA


Scatter Plots of Sample Data with Various Coefficients of Correlation

Y

X

Y

X

Y

X

Y

X

r = -1 r = -.6

r = +.3r = +1

Y

Xr = 0

DCOVA


The Coefficient of Correlation Using Microsoft Excel Function

DCOVA

Test#1Score Test#2Score78 82 0.7332 =CORREL(A2:A11,B2:B11)92 8886 9183 9095 9285 8591 8976 8188 9679 77

CorrelationCoefficient


The Coefficient of Correlation Using Microsoft Excel Data Analysis Tool

1. Select Data2. Choose Data Analysis3. Choose Correlation &

Click OK

DCOVA


The Coefficient of CorrelationUsing Microsoft Excel

4. Input data range and select appropriate options

5. Click OK to get output

DCOVA


Interpreting the Coefficient of CorrelationUsing Microsoft Excel

§ 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.

Scatter Plot of Test Scores

70

75

80

85

90

95

100

70 75 80 85 90 95 100

Test #1 Score

Test

#2

Sco

re

DCOVA


Pitfalls in Numerical Descriptive Measures

n Data analysis is objectiven Should report the summary measures that best

describe and communicate the important aspects of the data set

n Data interpretation is subjectiven Should be done in fair, neutral and clear manner

DCOVA


Ethical Considerations

Numerical descriptive measures:

n Should document both good and bad resultsn Should be presented in a fair, objective and

neutral mannern Should not use inappropriate summary

measures to distort facts

DCOVA


Chapter SummaryIn this chapter we discussed

n Measures of central tendencyn Mean, median, mode, geometric mean

n Measures of variationn Range, interquartile range, variance and standard

deviation, coefficient of variation, Z-scores

n The shape of distributionsn Skewness & Kurtosis

n Describing data using the 5-number summaryn Boxplots


Chapter Summary

n Covariance and correlation coefficientn Pitfalls in numerical descriptive measures and

ethical considerations

(continued)

numerical descriptive measures...title bbs13e_chapter03 created date 12/5/2016 5:52:29 pm

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