Descriptive Statistics, Numerical Description

DESCRIPTIVE STATISTICS Part I: Numerical Description

In this chapter, we will learn how to describe a set of data using numerical methods. This is the first of two chapters that together will aim at providing methods of descriptive statistics. In descriptive statistics, which is the use of graphical methods to display data and explore key statistics.

1

What are the basic features of a data set?

A data set is a collection of data representing a particular variable. Examples of data sets are given below.

Data Sets:

• Students’ grades in a calculus test: 65, 85, 70, 75, 85, 80, 82, 85, 90, 78, 81, 82, 67, 80

• Property tax of a sample of houses:$5000, $4500, $4000, $7200, $5000, $3800, $4100, $5000

• Driving distance to work of a group of employee (miles): 1.2, 2.0, 2.2, 15.0, 11.0, 5.0, 3.7, 4.9, 15.2, 16.0

• Ages of all students in a college: 18, 19, 21, ……………………..…, 22, 18, 19, 21

2Notes: …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

In general, establishing a data set requires consideration of a number of key questions:

Notes: …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

3

Data Set Key Questions:

•Are the data qualitative or quantitative?

•What levels of measurement do the data exhibit? (nominal, ordinal, interval, or ratio)

•What is the source of data?(the population)

•What is the appropriate sampling technique that should be used to collect the samples? (random or stratified)

•What is the appropriate minimum sample size?

4

Data Set Types: (1) Univariate, (2) Bivariate, (3) Multivariate

Data Set Variable Typical Tasks

Univariate One Histograms, Descriptive Statistics, Frequency tallies Bivariate Two Scatter plots, correlations, simple regressionMultivariate

More than two variables Multiple regression, data mining, modeling

Person # Weight (lb)1 1502 1203 1304 1255 1556 1347 1508 1409 160

10 20011 18012 140

Person #

Years at work

Annual Salary ($)

1 5 50,0002 20 73,0003 10 65,0004 5 55,0005 8 60,0006 10 60,0007 15 68,0008 15 69,0009 20 68,000

10 20 69,00011 18 68,00012 10 62,00013 3 48,000

Uni

varia

te D

ata

Set

Biv

aria

te D

ata

Set

Case Name AgeIncome ($) Position Gender

1 Frieda 45 67,100 Consumer Analyst F

2 Stefan 32 56,500Operations analyst M

3 John 55 88,200 Marketing VP F

4 Donna 27 59,000 Statistician F

5 Larry 46 26,000 Security guard M

6 Alicia 52 68,500 QC Director F

7 Alec 65 95,200 Chief executive M

8 Jaime 50 71,200Human Resources M

Multivariate Data Set

Notes: …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

Time-series data set

5

Cross sectional Sample

Notes: …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

Data Sets in the Context of Sampling:

• Cross sectional data set• Time-series data set

6Notes: …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

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Working Problem 2.2: Explain what is inheritance tax. What is the difference between inheritance tax and Estate tax?

What is the level of measurement for each of the following variables: State, Income tax, sales tax, and inheritance tax. Why do some states have a wide income tax range?

http://portal.kiplinger.com/tools/slideshows/slideshow_pop.html?nm=TaxUnfriendlyStatesRetirees

State Income Tax (%)

U.S. States Sales Tax

(%)Inheritance

Tax (%)Alaska 0.0 0.0 NO

Wyoming 0.0 4.0 NoMichigan 4.4 6.0 No

Pennsylvania 3.1 6.0 YESColorado 4.6 2.9 NODelaware 4.6 0.0 NOHawaii 1.4 to 11 4.0 NOGeorgia 1.0 to 6.0 4.0 NO

South Carolina 3.0 to 7.0 6.0 NOAlabama 2.0 to 5.0 4.0 NO

California 1.25 to 10.55 8.3 NORhode Island 3.75-9.9 7.0 NO

New Jersey 1.4 to 8.97 7.0 YESVermont 3.55-8.95 6.0 NO

Iowa 0.36 to 8.98 6.0 YES

Nebraska 2.56 to 6.84 5.5 Yes

Wisconsin 4.6 to 7.75 5.0 NO

Oregon 5.0 to 11.0 0.0 YESIndiana 3.4 7.0 YES

North Dakota 1.84-4.86 5.0 NO

http://portal.kiplinger.com/tools/slideshows/slideshow_pop.html?nm=TaxUnfriendlyStatesRetirees

8

Working Problem 2.3:

Identify the following data sets as ‘Cross-Sectional Data’ or ‘Time-Series Data’:

(a) Two weeks before the 56th quadrennial United States presidential election, which was held on November 4, 2008, a sample of people taking randomly from undecided states revealed that Democrat Barack Obama is expected to earn 54% of the popular votes and John McCain is expected to earn 46% of the votes

Cross Sectional ( ) Time-Series ( )

(b) A survey of 1000 students from a university of 10,000 students, revealed that 65% of the students do not prefer weekend classes


(c) The U.S. City average price per gallon of unleaded regular gasoline from 2000 to 2009 was as follow:

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec2000 1.301 1.369 1.541 1.506 1.498 1.617 1.593 1.51 1.582 1.559 1.555 1.4892001 1.472 1.484 1.447 1.564 1.729 1.64 1.482 1.427 1.531 1.362 1.263 1.1312002 1.139 1.13 1.241 1.407 1.421 1.404 1.412 1.423 1.422 1.449 1.448 1.3942003 1.473 1.641 1.748 1.659 1.542 1.514 1.524 1.628 1.728 1.603 1.535 1.4942004 1.592 1.672 1.766 1.833 2.009 2.041 1.939 1.898 1.891 2.029 2.01 1.8822005 1.823 1.918 2.065 2.283 2.216 2.176 2.316 2.506 2.927 2.785 2.343 2.1862006 2.315 2.31 2.401 2.757 2.947 2.917 2.999 2.985 2.589 2.272 2.241 2.3342007 2.274 2.285 2.592 2.86 3.13 3.052 2.961 2.782 2.789 2.793 3.069 3.022008 3.047 3.033 3.258 3.441 3.764 4.065 4.09 3.786 3.698 3.173 2.151 1.6892009 1.787 1.928 1.949 2.056 2.265 2.631 2.543 2.627 2.574 2.561 2.66 2.621

http://data.bls.gov/cgi-bin/surveymost


http://data.bls.gov/cgi-bin/surveymost

9Notes: …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

What are the different elements of descriptive statistics?

Two types of descriptive statistics:

(1) Numerical measures of data, and

(2) Graphical displays of data.

The Focus of this Chapter

10

Numerical measures of descriptive statistics consist of two types of measures:

• Measures of central tendency (mean, median, and mode)

• Measures of dispersion (range, standard deviation, and variance)

• Combined measures (coefficient of variation, signal-to-noise ratio, and standardized variable)

Measures of Central Tendency

10.0 12.0 13.0 14.0 14.0 14.0 24.0 24.0 24.0 26.8 27.0 27.0 29.0 30

Mean Mode

Median

Measures of Dispersion

Range Standard Deviation

Variance

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Key points to perform good statistical analysis

1. Identify your objectives:

· What questions do you really need to answer?· What variable do you need to examine?· What population you are about to evaluate?

2. Collect the appropriate samples and data to address your questions: ·Do you have access to the entire population?

· Would a selection of a sample from the population be easier to access, less costly, and less destructive than an evaluation of the whole population?· Remember ‘GIGO’ or garbage-in, garbage-out. If the samples are not representative of the population, and the data collected is not accurate and precise, the conclusions drawn from the analysis will be meaningless.

3. Describe the data using the analysis of descriptive statistics : · Do you detect data abnormality or outliers?

· Can you explore the data in such a way that will provide a clear description of data center and data variability?· Use descriptive statistics as a guideline for other methods of analysis

4. Perform inference : · Can the sample statistics be used to estimate population parameters?

· Is your estimation of population parameters reliable? · Do you have confidence in the population estimates?

Center Values

Measures of Central Tendency

Mean Mode

Median

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What are the measures of central tendency?

10.0 12.0 13.0 14.0 14.0 14.0 24.0 24.0 24.0 26.8 27.0 27.0 29.0 30

(1) Arithmetic Mean

Measures of Data Center (Central Tendency)

Arithmetic Mean of Sample Observations

Arithmetic Mean of Population Observations

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Example: The Table below illustrates a comparison of gas prices in some States in September 2009 and September 2008. Determine the mean of gas prices ($ per gallon) for each year.

State Sept- 2009 Sep-2008California 3.099 3.75Colorado 2.48 3.732Florida 2.527 3.893Massachusetts 2.597 3.582Minnesota 2.452 3.765New York 2.811 3.805Ohio 2.411 3.933Texas 2.404 3.729Washington 2.947 3.785

Gas Prices of a number of states in September 2008, and September 2009http://www.eia.doe.gov/oil_gas/petroleum/data_publications/wrgp/mogas_home_page.html

gallonn

xXMean

ni

/636.2$9

947.2........597.2527.248.2099.31

gallonn

xXMean

ni

/775.3$9

785.3........582.3893.3732.375.31

For September 2009:

For September 2008:

Comment on the Results14

http://www.eia.doe.gov/oil_gas/petroleum/data_publications/wrgp/mogas_home_page.html

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Properties of arithmetic mean:

1. The mean of a set of data is unique and can be used as an identity measure of the data center 2. We can determine the mean of any data set that contains ratio or interval level data

3. We need all observation values to be able to calculate the mean

4. You know it is the correct mean value when the sum of the deviations of each value from it is zero,

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Example: Determine the arithmetic mean of the three values of student grades:

80, 40, and 30. Using the mean value, prove that

. Solution:

The arithmetic mean:

17

Working Problem 2.4: Calculate the mean for the following data set of minimum wage ($): 7, 8, 6, 6, 8, 5, 6, 5, 8, 8

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2. Median:

The median of a set of numbers arranged in order of magnitude is the middle value or the arithmetic mean of the two middle values.

Example: Calculate the median of the following data set:14, 12, 14, 16, 15, 19, 17, 17, 17

Solution:To determine the median, we first arrange the data in order of magnitude:

12, 14, 14, 15, 16, 17, 17, 17, 19

Thus, the median is 16

Example: Calculate the median of the following data set:8, 9, 10, 9, 8, 6, 11, 7, 12, 8

Solution:To determine the median, we first arrange the data in order of magnitude

6, 7, 8, 8, 8, 9, 9, 10, 11, 12

Since this data set consists of an even number of observations, the middle values that split this data into equal number of observations on both sides are 8 and 9. Thus, the median of this set of data is (8+9)/2 = 8.5

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3. Mode:

The mode is that value which occurs with the greatest frequency. Interestingly, mode is a French word that means fashion; perhaps, it is popular and common fashion.

Example: Calculate the mode of the following observations:

80, 87, 90, 82, 78, 74, 80, 77, 80, 91, 81, 80

Example: Calculate the mode of the following observations:

5, 7, 8, 9, 9, 9, 10, 11, 12, 14, 14, 14, 15

Solution:The mode of this set is 80

Solution:This set exhibits two modes 9, and 14, and is called bimodal.

Working Problem:

Calculate the mean and the mode and the median for the following data set of minimum wage ($)

7, 8, 6, 6, 8, 5, 6, 5, 8, 8

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Answer:

Mean = $6.7

Median =$6.5

Mode = $8

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Working Problem 2.6: Calculate the median and the mode for the following data set ofminimum wage ($): 7, 8, 6, 6, 8, 5, 6, 5, 8, 8

Geometric Mean, G

n nxxxxG ......321

Example: If the return on investment earned by a manufacturer of a sport car for four successive years was: 20 percent, 15 percent, -40 percent, and 100 percent. What is the geometric mean rate of return on investment?

1344.1656.1)2)(6.0)(15.1)(2.1(...... 44321 n nxxxxG

Accordingly, the average rate of return, which is essentially a compound annual growth rate, is 13.44%.

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Example: Suppose the inflation rates for the last 5 years in a certain country are 5%, 4%, 2%, 8%, and 6%, respectively. What is the mean rate of inflation over this five-year period?

Accordingly, the average rate of inflation over the five-year period is 4.9%

Geometric Mean, G

Solution:

At the end of the first year, the price index will be 1.05 times the price index at the beginning of the year; at the end of the second year, the price index will be (1.04)(1.05); at the end of the third year, the price index will be (1.02)(1.04)(1.05) and so on. Thus, the mean of 1.05, 1.04, 1.02, 1.08, and 1.06 is:

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Working Problem 2.8: The percent increase in sales for the last 4 years at X-L Company were: 9.91, 10.75, 13.12, 26.6

(a) Find the geometric mean percent increase.

(b) Find the arithmetic mean percent increase.

(c) Is the arithmetic mean equal to or greater than the geometric mean?

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What are the ‘dispersion’ or variability measures?

Range Mean deviation Standard deviation Variance

10.0 12.0 13.0 14.0 14.0 14.0 24.0 24.0 24.0 26.8 27.0 27.0 29.0 30

Measures of Dispersion

Range Standard Deviation

Variance

What are ‘Dispersion’ or Variability measures? Range Mean deviation Standard deviation Variance

minmax XXR

Example: Calculate the range of the following set of data:

200, 205, 204, 202, 207, 208

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The Range = R = 208 - 200 = 8

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Properties of range:

• The range represents the most commonly used statistic after the arithmetic mean • It is simple as it relies on two values, the maximum value and the minimum value

• It is easy to understand: the higher the range, the higher the variability

• Since the range relies on two values (maximum and minimum), a mistake in any one of these two values or a presence of an outlier can result in a misleading value of range

minmax XXR

What are ‘Dispersion’ or Variability measures?

Mean deviation

n

i Xxn

MD1

1

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Example : Calculate the mean deviation of the following ten observations of metal sheet thickness (mm):

83, 90, 70, 90, 90, 60, 70, 70, 90, 100

Solution: Step 1: Calculate the Mean

X

= (83 + 90 + 70 + 90 + 90 + 60 + 70 + 70 + 90 + 100) / 10 = 81.3 mm

Step 2: Subtract each observation value from the Mean value, and add up the absolute differences

Thickness (mm)

83 (83-81.3) =1.7 1.790 (90-81.3) = 8.7 8.770 (70-81.3) = -11.3 11.390 (90-81.3) = 8.7 8.790 (90-81.3) = 8.7 8.760 (60-81.3) = -21.3 21.370 (70-81.3) = -11.3 11.370 (70-81.3) = -11.3 11.390 (90-81.3) = 8.7 8.7100 (100-81.3) = 18.7 18.7Mean = = 81.3 Sum = 110.4

)(

Xx |)(|

Xx

X

Mean Deviation = 110.4/10 = 11.04 mm


For a Population:

N

iN

x

1

2

For n < 30, we use (n-1) in the denominator

For a Sample:

ni

nXxs

1

2)(

29

Standard deviation

Thickness (mm) 83 90 70 90 90 60 70 70 90 100

X

= (83 + 90 + 70 + 90 + 90 + 60 + 70 + 70 + 90 + 100) / 10 = 81.3 mm


Example: Calculate the standard deviation of the following ten observations of metal sheet thickness

ni

nXxs

1

2)(

Thickness (mm)83 (83-81.3) =1.7 2.8990 (90-81.3) = 8.7 75.6970 (70-81.3) = -11.3 127.6990 (90-81.3) = 8.7 75.6990 (90-81.3) = 8.7 75.6960 (60-81.3) = -21.3 453.6970 (70-81.3) = -11.3 127.6970 (70-81.3) = -11.3 127.6990 (90-81.3) = 8.7 75.69100 (100-81.3) = 18.7349.69Mean = 81.3 Sum = 1492.1

)(

Xx 2)(

Xx

mms 88.129

1.1492

n

in

Xxs1

2

1)(

30


For a Population:

Ni

Nx

1

22


For a Sample:

ni

nXxs

1

22 )(

31

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Properties of variance:

• The variance represents the most commonly used statistic to indicate variability

• It is easy to understand: the higher the variance, the higher the variability

• Unlike the range, the variance takes into account all values of the observation values. Therefore, it is largely insensitive to outliers

• Variance values cannot be subtracted to determine variability. It can only be added. If U = X ± Y, Var (U) = Var (X) + Var (Y). This is the principle of analysis of variance (Chapter 11)

What are ‘Dispersion’ or Variability measures?

Variance

Thickness (mm) 83 90 70 90 90 60 70 70 90 100

X

= (83 + 90 + 70 + 90 + 90 + 60 + 70 + 70 + 90 + 100) / 10 = 81.3 mm


Example: Calculate the varianceof the following ten observations of metal sheet thickness


)(

Xx 2)(

Xx

22 79.1659

1.1492 mms

33

ni

nXxs

1

22 )(

ni

nXxs

1

22 )(

Working Problem:

Calculate the minimum, maximum, range, standard deviation, and variance for the following data set of minimum wage ($)

7, 8, 6, 6, 8, 5, 6, 5, 8, 8

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Answer:

Minimum = $5

Maximum =$8

Range = $3

Standard deviation = $1.252

Variance = 1.567

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Example: Suppose X and Y are independent random variables. The variance of X is equal to 16; and the variance of Y is equal to 9. Let U = X - Y.What is the standard deviation of U?•2.65 ……….•5.00 ……….•7.00 ……….•25.0 ……….•None of the above ……….

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Working Problem 2.9: Question (1): Calculate the minimum, the maximum, the range, the mean deviation, the standard deviation, and the variance for the following data set of minimum wage ($) 7, 8, 6, 6, 8, 5, 6, 5, 8, 8

Question (2): In two consecutive exams, the mean grade of the first test was 80 and the mean grade of the second test was 90. The standard deviation of grade of the first test was 6 and the standard deviation of grade of the second test was 8. Calculate the mean of the two tests and the variance of the two tests?

What are Combined Descriptive Measures?

Coefficient of Variation (C.V%)100%.

X

sVC

Thickness (mm) 83 90 70 90 90 60 70 70 90 100

X

= (83 + 90 + 70 + 90 + 90 + 60 + 70 + 70 + 90 + 100) / 10 = 81.3 mm

Example: Calculate the Coefficient of Variation of the following ten observations of metal sheet thickness


)(

Xx 2)(

Xx

mms 88.129

1.1492

%84.151003.81

88.12

100%.

X

sVC

38

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Working Problem 2.11: Calculate the Coefficient of Variation (CV%) for the following data set ofminimum wage ($): 7, 8, 6, 6, 8, 5, 6, 5, 8, 8

What are Combined Descriptive Measures?

Standardized Variable (the z Score)

A standardized variable is a measure of the deviation from the mean by an individual value in units of the standard deviation:

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Example: An instructor who has been teaching statistics for twenty years has observed that the average grade of students is 88% and the standard deviation is 3%. After teaching the course for two classes, one in the fall semester and one in the spring semester of 2008, the instructor found that the average grades were as follow:

Term Mean GradeFall 2008 82%

Spring 2008 91%

How do these two semesters compare to the instructor’s average over the last twenty years?

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Standardized Variable (the z Score)

Example: An instructor who has been teaching statistics for twenty years has observed that the average grade of students is 88% and the standard deviation is 3%. After teaching the course for two classes, one in the fall semester and one in the spring semester of 2008, the instructor found that the average grades were as follow:

Term Mean GradeFall 2008 82%

Spring 2008 91%

How do these two semesters compare to the instructor’s average over the last twenty years?

The standardized variable (z- score) is calculated for each semester as follows:

Term Mean Grade z-ScoreFall 2008 82% z82 = (82-88)/3 = -2Spring 2008 91% z91 =(91-88)/3 = 1

From the above scores, you can conclude that the class’s grade in the Fall 2008 being 82% was 2 standard deviations below the teacher’s mean grade, while the class’s grade in the Spring 2008 being 91% was 1 standard deviations above the teacher mean grade.

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Example: The mean driving time of people living in Union City near Atlanta Georgia to CNN Center in downtown Atlanta is 40 minutes, with a standard deviation of 10 minutes. You asked four CNN employees who live in Union City about their driving time to CNN Center, and you get the following answers: 38 minutes, 52 minutes, 58 minutes, and 40 minutes. Find the z-score that corresponds to each driving time. Interpret the difference in z-scores?

Where t is the actual driving time, t is the mean driving time, and t is the standard deviation of driving time.

At t = 38 minutes,

At t = 52 minutes,

At t = 58 minutes,

At t = 40 minutes,

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Working Problem 2.12: The average scoring points per game (PTG) up to week 10 in the 2010 NFL football season was 22 points and the standard deviation was 4 points. Using the z-score, compare the following 3 teams and determine which team had a relatively better scoring season:

San Francisco 16 PTG, New England 29 PTG, Pittsburgh 24 PTG

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Working Problem 2.13: The annual salaries of engineers in the U.S. automobile industry are normally distributed with a mean of $100,000 and a standard deviation of $10,000. What is the z-score for the income x of an auto-engineer who earns $85,000 annually? And what is the z-score for an auto-engineer who earns $105,000 annually?

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Working Problem 2.14: The annual salaries of U.S. state governors are normally distributed with a mean of $135,450 and a standard deviation of $36,530. If in 2007, the Arkansas governor made $85,000 annual salary, andthe California governor made $206,000. Compare the annual salaries of these two governors using the z-score.

Arnold Schwarzenegger Mike Beebe (California) (Arkansas)

The Use of Computer for Performing Descriptive Statistics

Powerful Tools are available to perform statistical analyses, the focus should therefore be on:

• Planning for sample and data selection in view of the study or application objectives

• Gathering and organizing data in such a way that serves the purpose of the application

• Selecting the appropriate type of analysis

• Organizing the analysis output

• Interpreting the analysis outcome

• Making a report addressing the case or application in question

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Data on Annual Tuition and Financial Aid by Different U.S. State Colleges (http://www.ordoludus.com/costs.php, 2006)

SchoolIn-State Out-of-State

TuitionTotal Cost ($)

Fin.Tuition Aid ($)

Georgia Institute of Technology $4,648 $18,990 $25,792 $8,222 University of Tennessee $5,290 $16,060 $21,270 $6,954 University of Mississippi $4,320 $9,744 $14,442 $7,532 University of Kentucky $5,812 $12,798 $18,027 $7,861 Louisiana State University $4,515 $12,815 $19,145 $8,006 University of Florida $3,094 $16,579 $22,839 $10,566 University of Virginia $7,133 $23,877 $30,266 $13,449 University of South Carolina $7,314 $18,956 $25,039 $9,501 University of North Carolina $4,515 $18,313 $24,903 $9,687 University of Georgia $4,628 $16,848 $23,224 $7,320 University of Alabama $4,864 $13,516 $18,540 $7,980 University of California (UCLA) $6,504 $24,324 $36,252 $13,462 North Dakota State University $5,264 $12,545 $17,675 $5,487

Florida State University $3,208 $16,340 $23,118 $8,269

The Use of Computer for Performing Descriptive Statistics

Example:

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http://www.ordoludus.com/costs.php

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Analysis of Descriptive Statistics: Steps 1 and 2 48

3

Analysis of Descriptive Statistics: Steps 3 and 4

4

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5

Analysis of Descriptive Statistics: Steps 5 and 6

6

The minimum of the largest 4 observationsThe maximum of the

smallest 4 observations

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Analysis of Descriptive Statistics: Output 51

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InterpretationThe most critical aspect of statistics is to learn how to interpret the results… This is not your typical Math course where all you have to do is find answers…The true answer is not the outputs..it is the interpretation of the outputs

StatisticIn-State

Tuition ($)Out-State Tuition ($) Total Cost ($)

Financial Aid ($)

Mean 5079 16550 22895 8878Median 4756 16460 22979 8114Mode 4515 None None NoneStandard Deviation 1269.44 4196.51 5602.14 2297.84Sample Variance 1611486.64 17610692.25 31383983.67 5280083.14Range 4220 14580 21810 7975Minimum 3094 9744 14442 5487Maximum 7314 24324 36252 13462Count 14 14 14 14Largest(4) 5812 18956 25039 9687Smallest(4) 4515 12815 18540 7532

Outputs of descriptive statistics for tuition, cost, and financial aid

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Total CostFinancial Aid

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Total Cost and Financial Aid by School

Optimum Choice

APPENDIX 2.A Steps to Add Data Analysis to Excel 2007

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1

2

Data Analysis Add-In-Steps 1 and 256

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Data Analysis Add-In-Steps 3 through 5 57


6 7

58

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Descriptive Statistics, Numerical Description

Education

Transcript of Descriptive Statistics, Numerical Description