Chapter 3

Chapter 1An Introduction to Econometrics and Statistical Inference

Copyright 2014 McGraw-Hill Education.All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.1-#1Learning ObjectivesUnderstand the steps involved in conducting an empirical research Understand the meaning of the term econometricsUnderstand relationship between populations, samples, and statistical inferenceUnderstand the important role that sampling distributions play in statistical inference

1-#2What is an Empirical Research Project?An empirical research project is a project that applies empirical analysis to observed data to provide insight into questions of theoretical interest.1-#3The 5 Steps in Conducting an Empirical Research Project?Determining the question of interestDeveloping the appropriate theory to address the questionCollecting data that is appropriate for empirically investigating the answerImplementing appropriate empirical techniques, correctly interpreting results, and drawing appropriate conclusions based on the estimated resultsEffectively writing up a summary of the first four steps1-#4What is Econometrics?Econometrics is the application of statistical techniques to economic data.

1-#5Populations, Samples, and Statistical InferenceA population is the entire group of entities that we are interested in learning about.

A sample is a subset or part of the population and it is what is used to perform statistical inference.

Statistical inference is the process of drawing conclusions from data that are subject to random variation.

1-#6Populations, Samples, and Statistical Inference Continued

1-#7Some Important DefinitionsA parameter is a function that exists within the population.

A statistic is a function that is computed from the sample data.

A point estimate is a single valued statistic that is the best guess of a population parameter.

1-#8Sampling Distributions1-#9A Visual Example

1-#10Chapter 2Collection and Management of Data

Copyright 2014 McGraw-Hill Education.All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.1-#11Learning ObjectivesConsider potential sources of dataWork through an example of the first three steps in conducting an empirical research projectDevelop data management skillsUnderstand some useful Excel commands

1-#12Goals of the Chapter

1-#13Types of DataCross-sectional data is data collected for many different individuals, countries, firms, etc. in a given time-period.

Time-series data is data collected for a given individual, country, firm, etc. over many different time periods.

Panel data are data collected for a number of individuals, countries, firms, etc. over many different time periods.

1-#14Primary Data Sourcesprivate-use data government surveys or internal firm-level data obtained through formal request and/or having the appropriate connections.

publicly-available data obtained through the internet or through formal Freedom of Information Act (FOIA) request

personal survey data obtained by personally conducting a survey asking people for information and recording their responses1-#15An Example of the First Three StepsSuppose you are trying to convince your significant other to go camping but he or she is afraid of bears.

How can you use your empirical research skills to convince him or her that bear attacks are not a realistic concern?

Step 1: Identify a question of interest

What factors affect the number of fatal bear attacks in the US?1-#16An Example of the First Three StepsStep 2: Develop appropriate theory

The number of fatal bear attacks in the US should depend on:

The number of bearsThe number of campersSquare feet of national parkland

1-#17An Example of the First Three StepsStep 3: Collect appropriate data

Start with an internet search for the data you seek

1-#18An Example of the First Three StepsDownload data to Excel and then repeat the process for the independent variables you seek.

1-#19Data Management SkillsTwo important points:

When working with data, it is common to make mistakes which alter the initial data

When working on a larger project, it is common to take time off before returning to the project1-#20Data Management SkillsOur goals with data management are to be able to:

Recreate our initial data as easily as possible

Recall what we had previously done as easily as possible 1-#21Data Management SkillsWhen working with data, we recommend:

Creating a Master file with the initial data and performing calculations in a different working fileExhaustively documenting all initial data sourcesMaking file and variable names as intuitive as possible Documenting all commands used when performing estimation1-#22Chapter 3Summary Statistics

Copyright 2014 McGraw-Hill Education.All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.1-#23Learning ObjectivesConstruct relative frequency histogramsCalculate measures of central tendencyCalculate measures of dispersionUse measures of central tendency and dispersionDetect whether outliers are presentConstruct scatter diagrams for the relationship between two variablesCalculate the covariance and the correlation coefficient between two variables

1-#24

1-#Construct a Relative Frequency HistogramA bar chart that shows how often observations lie within a specified classesAllows a visual inspection of the dataBased on a Relative Frequency Table

The example dataset for constructing a histogram use states.xls, a survey of econometrics students that asked how many states they have been visited.1-#1-#27To create a frequency distribution we must Select the number of classes

Choose the class interval or width of the classes

Select the class boundaries or the values that form the interval for each class

Count the number of values in the dataset that fall in each class1-#28Step 1: Select the number of classesThe rule for determining the approximate number of classes is:

Approximate number of classes =[(2)(Number of observations)].3333

The actual number of classes is the integer value that just exceeds the number value.

If the formula gives us 4.66 we use 51-#29Step 1: ExampleWe have 43 data points so the rule is:

Approximate number of classes = [(2)(20)].3333 = 3.503

Round this up to the next integer value which is 4.

The number of classes is 4.

**Always round up!! 1-#30Step 2: Choose the width of the intervalThe rule for determining interval width is:

Approximate interval width = Largest data value Smallest data value Number of classes

The actual interval width is the integer value that just exceeds the number value.

If the formula gives us 6.17 we use 7**Always round up!!

1-#31Step 2: ExampleApproximate interval width = (24-1)/4 = 5.75

Round up to 6.

Therefore the class width is 6.1-#32Step 3: Select the class boundariesClass boundaries must be chosen such that each data item belongs to one and only one class.

Start just below the lowest value in the dataset to get the lower boundary. The lower boundary for the second class is then found by adding the class width. The upper boundary for the first class is found by subtracting .01 from the lower boundary of the second class.

Keep adding the class width and subtracting .01 to get the boundaries. 1-#33Step 3: Example Lowest data point is 1. We will start our classes at 0.

Class 1 = 0 Class 2 = 6 (=0+6)Class 3 = 12 (=6+6)Class 4 = 18 (=12+6)1-#34Step 3: Example ContinuedClass boundaries are then:

Class 1: 0- 5.99Class 2: 6-11.99Class 3: 12-17.99Class 4: 18-241-#35Step 4: Count the number of values in the dataset that fall into each classDoing this by hand is tedious and, therefore, we want to rely on Excel to do this for us.Enter the class boundaries into Excel next to the data set.Enter the Upper Boundaries of each of the classesUse the Frequency command

1-#36How to use the Frequency command in ExcelSelect the cells next to the class intervals where the frequencies should go (say E2:E6).

Type but do not enter the formula=Frequency(A2:A44,D2:D6) A2:A44 contains the data D2:D6 contain the ending class boundaries

Press CTRL+SHIFT+ENTER and the array formula will be entered into each of the cells E2:E6.

1-#37Our Excel ResultsClass Boundaries Upper LimitFrequency0-5.995.99 186-11.99 11.99 1812-17.99 17.99 418-24 24.00 31-#38Creating relative frequency and percent frequency distributionsRecall that the relative frequency is the proportion of the observations belonging to a class. With n observationsRelative frequency of a class = Frequency of the class nThe percent frequency is the relative frequency multiplied by 100.1-#39Relative Frequency Table

1-#Using Excels Chart Wizard to Construct a HistogramUse the frequency distribution we just constructed and highlight the frequenciesClick the Chart Wizard and choose column in the chart typeClick on the Category (X) axis labels box and enter the class boundariesTo get the bars to touch right click on any rectangle in the column chart and choose Format Data Series. Select the Options tab and enter 0 in the Gap Width box.1-#41 1-#42Soda Consumption DataYour mission is to pair up with a classmate and draw what you think the histogram for soda consumption looks like.

1-#43Calculate Measures of Central TendencyCentral tendency is the middle value of a dataset. The measure of central tendency is typically thought of as the number that best describes the data.Measures of central tendency are:MeanMedian1-#Measure of Central Tendency - MeanThe mean is the arithmetic average of the data. To calculate the mean sum all the observations and divide by the number of observations.

Represented by the symbol,

Mean

For the following small data set: 95 85 99 92 80

Mean =(95+85+99+92+80)/5 = 451/5 = 90.2

In Excel =average(highlight data)

1-#45Measure of Central Tendency - MedianMedian the middle observation when the data are arranged from smallest to largest sometimes called the 50% percentile. Half the observations lie below the median and half the observations lie above the median.

The median is the middle observation for an odd number of ordered observations and the average of the middle two ordered observations for an even number of observations.

The median is an order statistic so in order to calculate it the data must be ordered from smallest to largest.

1-#46Measure of Central Tendency - MedianMedian Central observation for an odd number of observations and an average of the two middle data points for an even number of observations

For the following small data set : 95 85 99 92 80(ordered data 80 85 92 95 99)Median = 92 (the 3rd data point)

If we had 75 80 85 92 95 99median =(.5*85)+(.5*92) = (85+92)/2 = 42.5+46 = 88.5

In Excel =median(highlight data)

1-#47Calculate Measures of DispersionDispersion is a measure of how the data vary.

Measures of dispersion are:VarianceStandard DeviationPercentilesFive Number Summary1-#Measure of Dispersion Variance and Standard DeviationStandard Deviation the average deviation away from the mean. It is the square root of the variance.

The variance is calculated by subtracting the mean from each observation, squaring that value, adding up all n values, and then dividing that by the number of observations less one.

Sample variance formula is

Standard deviation is

In Excel = var(highlight data) = stdev(highlight data)

1-#49Measure of Dispersion Variance and Standard DeviationSample variance:

For the following small data set : 95 85 99 92 80

s2= [(95-90.2)2+ (85-90.2)2+ (99-90.2)2+ (92-90.2)2+ (80-90.2)2]/4=234.8/4=58.7Sample standard deviation s= =7.6616

1-#50Measure of Dispersion PercentileA percentile is a number such that p% of the ordered observations lie below the percentile and (1-p)% of the observations lie above the percentile.

The median is the 50th percentile and an example of a percentile where 50% of the ordered data lies below that level and 50% of the ordered data lies above that level.

A percentile is an order statistic.

There are many different ways to calculate percentiles. On the next slide one of the easiest ways to calculate percentiles.

1-#51Steps to Calculate a Percentile, p(1) Sort the data from low to high(2) Count the number of observations, n(3) Select the p(n+1) observationIf the value p(n+1) is not a whole number then select the closest whole numberIf p(n+1) is less than 1 then select the smallest numberIf p(n+1) is greater than 1 then select the largest number.

In Excel =percentile(highlight data, p)

Note that the steps to calculate a percentile by hand and calculating percentiles in Excel will likely not result in the same value.1-#52Measure of Dispersion - PercentileCalculate the 10th and the 70th percentile for the following small data set : 95 85 99 92 80(ordered data 80 85 92 95 99)10th percentile select the .1(n+1) = .1(6) = .6 number in the data set. The closest whole number is 1 so the 10th percentile is the first observation or 80.

70th percentile select the .7(n+1) = .1(6) = 4.2 number in the data set. The closest whole number is 4 so the 70th percentile is the fourth observation or 95.

1-#53Measure of Dispersion Five Number SummaryThe Five Number Summary is(1) Minimum(2) Q1 or 25th Percentile(3) Q2 or Median (50th Percentile)(4) Q3 or 75th Percentile (5) Maximum

1-#54How to Calculate the Five Number Summary in ExcelMinimum =Min (data)Q1 or 25th Percentile =percentile(data,.25) or =quartile(data,1)

Q3 or 75th Percentile =percentile(data,.75) or =quartile(data,3)

Maximum =Max (data)

1-#55Shapes of HistogramsSymmetricSkewed to the right or Positively skewedSkewed to the left or Negatively SkewedBimodal1-#56Symmetric Histogram

1-#57

Positively Skewed Distribution1-#58Negatively Skewed Distribution

1-#59

Bimodal Distribution1-#60

Positively Skewed DistributionMean = 4.16Median = 2.771-#61Why is the shape of the histogram important?The shape of the empirical distribution dictates which summary statistics should be usedSymmetric Use mean and standard deviationSkewed Use median and five number summary1-#62How to determine if your data is skewed or symmetricPearsons coefficient of skewness:sk = 3*(mean-median)/(standard dev.)

Rule of Thumb:If sk.5 then the distribution is skewed. Otherwise the distribution is symmetric.Negatively skewedSymmetricPositively Skewed-.5 .51-#63Symmetric Histogram

Mean = .5013Standard Deviation =.0191-#

Positively Skewed DistributionMedian = 2.779Five Number SummaryMinimum 0.008Q1 1.1578Median 2.779Q3 5.643Maximum 29.001

1-#65How to Detect Outliers with Symmetric dataUse the Empirical Rule68% of data should be within one standard deviation of the mean95% of the data should be within two standard deviations of the mean

100% of the data should be within three standard deviations of the mean

Therefore, an observation is an outlier if it lies beyond three standard deviations from the mean or beyond the interval ( - 3s, + 3s)

1-#66How to detect an outlier with skewed dataCalculate the interquartile range or IQR = Q3 Q1.If a value is greater than Q3 plus 1.5*IQR or less than Q1 minus 1.5*IQR the its a moderate outlierIf a value is greater than Q3 plus 3*IQR or less than Q1 minus 3*IQR then its an extreme outlier1-#67Construct Scatter Diagrams for the Relationship between two Random VariablesA scatter diagram (or scatter plot) is used to show the relationship between two variablesIt contains one variable on the x-axis and the other variable on the y-axisA scatter diagram shows how the two variables are related to each other, both the strength and direction of the relationship

1-#Scatter Diagram ExamplesyxyxyyxxPositive Linear relationshipCurvilinear relationshipsNegative Linear relationship1-#Scatter Diagram ExamplesyxyxyyxxStrong relationshipsWeak relationships1-#Scatter Diagrams ExamplesyxyxNo relationship1-#Salary vs. Years of Education

1-#How to Create a Scatter Diagram in ExcelHighlight the data making sure that the variable you want on the y-axis is on the rightSelect Insert and then Scatter and click on the first optionMake sure to change the chart title, add axis titles.Possibly delete the legend and change the start values for the axis.1-# 1-#What does the Scatter Diagram on the previous slide tell us?The relationship between education and salary is positive (in general as education increases salary increases)The relationship is fairly strong because the data point are closely gathered to each otherThis scatter diagram indicates that while the variable education is helpful for predicting salaries, it will not yield perfect predictions.

1-#Covariance and the Correlation Coefficient for the Linear Relationship between two variablesCovariance and Correlation Coefficient supplies a numeric value to the strength and direction of the linear relationship between two variablesOnly concerned with strength of the relationship No causal effect is implied

1-#CovarianceCovariance is a measure of the linear relationship between two random variables

A positive covariance indicates a positive linear relationship between x and y (if x is below its mean then y tends to be below its mean and if x is above its mean then y tends to be above its mean)

A negative covariance indicates a negative linear relationship between x and y (if x is below its mean then y tends to be above its mean and if x is above its mean then y tends to be below its mean)1-#CovarianceA covariance near 0 indicates no linear relationship between x and y

A problem with covariance is that it depends on the units of measurement for x and y if we change from measuring in feet to inches the covariance will go up even though the overall relationship hasnt changed.

1-#Covariance a Measure of Linear Association Between Two VariablesRemember the formula for variance is

or how x varies with itself.The formula for Covariance is

and it measures how varies with y in a linear fashion.

1-#Applying the Covariance Formula

Cox(x,y) = Sum/(n-1) = 743000/9 = 82,555.55561-#Calculating Covariance in ExcelIn some versions of Excel, the covariance is not calculated correctly.The Excel command is =Covar(highlight x values, highlight y values)You should perform this command in Excel for the data set above and see if it matches the value 82,555.5556. If you obtain 74,300 using the covar command (which is likely), you must multiply the value you obtain in Excel by n/(n-1) to obtain the correct value for covariance.

1-#Correlation CoefficientThe sample correlation coefficient, rxy, is an estimate of population correlation coefficient and is used to measure the strength and direction of the linear between two random variables.The correlation is a unit free measure (unlike the covariance) and falls between -1 and 1. 1-#What Does the Correlation Coefficient Mean?If all the points in a data set fall on a positively sloped line, rxy =1.If all the points in a data set fall on a negatively sloped line, rxy =-1.If there is no linear relationship between x and y then rxy =0.The closer to -1, the stronger the negative linear relationshipThe closer to 1, the stronger the positive linear relationshipThe closer to 0, the weaker the linear relationship

1-#r = +.3r = +1Examples of Approximate rxy Valuesyxyxyxyxyxr = -1r = -.6r = 01-#Calculating the Correlation CoefficientFrom above, the standard deviation of x is 2.708 and the standard deviation of y is 38,189.037.

Sample correlation coefficient:

A correlation of 0.7983 means that education and salary are positively related and the relationship is strong (because this values lies near 1)

In Excel =correl(highlight x values, highlight y values)1-#What Does Correlation Mean?Correlation provides a measure of linear association between two variables. A correlation coefficient is near 0 only means that there is a weak linear association between the two variables, not that there isnt any relationship between the two variables.

A high correlation between two variables does not mean that changes in one variable will cause changes in the other variable.

We might find that the quality rating and the typical mean price of restaurants are positively correlated. However, simply increasing the mean price at a restaurant will not cause the quality rating to increase.1-#DataDiameters of elevator rails at Otis ElevatorNote: All diameters are measured in fractions of an inchDiameter0.5150.5380.4670.4790.5110.4800.5160.5310.5060.5210.4870.4990.4770.4680.4620.5040.4870.4880.5260.5310.4890.5020.5040.4740.4660.5230.5040.4740.5210.4770.5180.4940.5120.4880.4750.5160.4970.5210.4990.4950.5330.5310.5180.4510.5040.4910.4940.4630.4870.4790.5140.4610.4820.5180.5290.4980.4850.4740.5110.5180.5220.5050.5080.4870.5260.4840.4860.4710.4970.4850.5170.5210.5070.5000.5170.5430.5010.4920.4880.5090.5080.4940.5020.4840.4760.5170.5150.5210.4950.4700.5070.5290.5020.4810.4740.5010.5370.5200.4860.4970.5030.5080.4630.4930.5070.4930.4920.4690.4610.5020.5210.4930.5060.4840.4920.4960.4910.4980.4900.4940.5070.4890.5110.4850.4980.5190.4890.4920.5010.5000.4770.5160.5130.5210.5050.4920.4840.5280.5000.4900.5010.5070.4980.5270.5090.5080.4610.5100.5360.4650.5000.5040.4740.5040.5140.5380.4900.5090.4800.4920.5210.5040.4880.4840.5240.5160.5040.5050.5010.5270.4810.5150.5060.4780.4780.4830.4950.4900.4880.5220.4840.5070.5290.5400.5070.5100.4740.4800.4750.5160.5090.4980.4960.5200.5350.5210.5280.5010.5330.4710.4680.5120.5120.4980.4930.5250.5080.4990.5000.4920.5130.4870.5130.4910.5100.5240.4780.5120.5230.4700.5150.4850.5080.4770.5070.5130.5040.4830.5010.4730.4980.4780.4790.4660.5000.4860.4710.4780.4810.5020.4940.4990.5280.4830.5310.5310.5100.5110.5000.5150.5150.5020.4990.5470.5160.4950.4530.4710.5190.4990.4840.4940.5310.4980.5340.4860.4920.5390.4870.4970.4890.5000.4990.5010.5190.5180.4840.5160.5030.5100.5140.4850.4970.4660.4930.5120.5040.5090.5450.4750.5170.4890.5190.5060.4980.5160.4810.5400.4890.5300.5170.5360.4870.5200.5060.5050.4860.5210.5090.5480.5220.5030.4950.4700.4770.5460.5020.4640.4950.5090.5360.5240.5110.5200.4850.4960.5100.5030.5120.4760.5280.4890.4960.5000.4880.5220.5060.4930.4890.4940.5150.5340.4880.5070.5460.4880.5000.4960.5310.4870.4850.5100.5270.5040.5210.5130.5060.5300.5030.4590.4990.5080.4990.4810.5160.5190.5100.4870.4810.4810.5180.5200.5300.4960.4850.4910.5150.5320.4820.4500.4950.5160.5080.5220.5110.5160.4760.4860.5080.4830.5150.4820.4850.5020.5140.5050.5140.5260.496

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Hist-DiameterDataFrequency table for DiameterUpper limitCategoryFrequency0.4550.5455

Hist-Diameter3922476979675929105

FrequencyCategoryHistogram for Diameter of 400 Elevator Rails

DataTimes between arrivals at a bankNote: All interarrival times are expressed in minutes.InterarrivalTime1.3371.3552.6391.3530.3272.2220.7150.6461.4634.5680.1565.20318.7491.5411.6634.3938.3234.8512.8711.9128.9831.4314.5878.3812.21521.6022.4026.4171.6617.05412.2423.4791.0220.954.0591.814.8470.3772.2487.33912.391.462.573.13.9033.340.3525.0342.90426.2850.1525.6238.2337.5582.854.0677.0167.3165.6861.9122.7613.5160.0542.2533.0580.4263.2142.8475.510.5620.2613.37314.0831.563.0975.7222.690.75619.751.731.8320.6235.2472.0582.8941.3823.1085.49411.18211.1640.8864.5239.1453.2663.1541.0516.6479.5661.1380.6712.6450.8964.2643.10914.8860.66910.5051.332.8539.3870.26913.5834.441.33711.6383.3830.6586.0825.1180.2491.9420.2271.4392.6810.092.5610.3163.7754.081.6848.5472.4075.085.2944.9322.81213.6411.8323.610.0326.2415.6383.4272.1820.552.7492.2121.6014.7891.6276.270.8942.7853.310.670.8850.4212.6453.4251.1532.42214.8881.3213.6433.3611.847.2691.1355.9560.8472.9987.9931.2757.5678.7935.352.57821.9146.0161.4890.4860.6121.010.8040.8490.8422.4742.3455.0454.2326.283.6892.5150.7362.5089.4872.2591.6621.4919.9815.3462.5212.8733.6174.3111.1683.7060.5371.5240.2516.2432.7273.8660.3990.6848.8332.8529.2331.1651.1391.6988.0930.9261.4550.6999.8180.6636.6436.1595.2310.8211.8131.1069.8571.1541.21614.3060.3083.1860.0481.227.5832.7931.8536.3568.4712.2216.6292.7730.5562.25410.0916.4014.3720.0546.6370.010.0210.5015.4130.5192.0873.0711.6677.5791.4510.9496.8922.5440.6310.0080.6263.8657.7385.4764.7982.1685.6580.4168.5250.7697.2153.4280.34713.72312.095.04710.2370.2450.58529.0016.7590.9640.7667.440.120.710.53314.7613.843

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Hist-InterarrivalTimeDataFrequency table for InterarrivalTimeUpper limitCategoryFrequency2.527.51

Hist-InterarrivalTime13474432499122011

FrequencyCategoryHistogram for Time Between Bank Customer Arrivals

DataScores on accounting midtermScore9695898797898489684687925367477879878543857777938175958472729592756095968663558478529289968676899093617672746789879298858472788799937679708888727888978289919581.240506329185

&APage &P

Hist-ScoreDataFrequency table for ScoreUpper limitCategoryFrequency45957

Hist-Score123124811918137

FrequencyCategoryHistogram for Scores on a Midterm

DataDiameters of elevator rails at Otis Elevator from two machinesNote: All diameters are expressed in fractions of an inchMachine1Machine2Both0.49800.59100.49800.49190.59930.49190.49590.60090.49590.50390.60000.50390.49760.59810.49760.49950.60390.49950.49870.59810.49870.50410.60160.50410.49490.60750.49490.50020.60160.50020.49770.60490.49770.50300.60150.50300.49690.59810.49690.49530.59460.49530.50890.59970.50890.49440.59670.49440.50230.59930.50230.49960.60110.49960.49790.59920.49790.49960.59570.49960.49540.61100.49540.49110.60230.49110.50130.60090.50130.49740.59490.49740.50500.60610.50500.50470.59280.50470.50180.59850.50180.49820.59520.49820.49810.60460.49810.49750.60510.49750.50340.59710.50340.50730.59870.50730.50100.59530.50100.50100.59860.50100.49660.59640.49660.49810.59640.49810.50300.59760.50300.49570.59470.49570.49570.60070.49570.50100.59190.50100.49940.59740.49940.50400.60110.50400.50320.60510.50320.49600.60130.49600.49190.60600.49190.48930.59960.48930.50250.60520.50250.50080.60150.50080.49740.60220.49740.49740.59870.49740.49810.59960.49810.51010.60570.51010.50330.59920.50330.50340.60310.50340.50230.59270.50230.49600.60130.49600.50070.60670.50070.50090.59910.50090.49230.60130.49230.50300.60030.50300.50460.59140.50460.49970.60850.49970.49810.61000.49810.48590.59610.48590.50910.59330.50910.50490.59360.50490.50850.59050.50850.50160.60050.50160.49570.60080.49570.50160.59440.50160.49480.60310.49480.50030.59730.50030.50300.61270.50300.50610.60740.50610.49540.60230.49540.50060.60070.50060.48960.60210.48960.50080.60200.50080.50260.59540.50260.50180.60490.50180.50810.59880.50810.50290.58870.50290.51070.59610.51070.49470.60420.49470.50600.59660.50600.50520.60780.50520.50310.59760.50310.50320.61170.50320.49310.60030.49310.49860.59240.49860.49620.60820.49620.49180.59960.49180.49960.59810.49960.51410.59810.51410.49850.59570.49850.49780.59590.49780.49650.60740.49650.50030.59480.50030.49240.59680.49240.49940.59540.49940.50390.60250.50390.49440.59720.49440.50310.59520.50310.49570.59460.49570.49560.59160.49560.50450.59950.50450.50320.59710.50320.59100.59930.60090.60000.59810.60390.59810.60160.60750.60160.60490.60150.59810.59460.59970.59670.59930.60110.59920.59570.61100.60230.60090.59490.60610.59280.59850.59520.60460.60510.59710.59870.59530.59860.59640.59640.59760.59470.60070.59190.59740.60110.60510.60130.60600.59960.60520.60150.60220.59870.59960.60570.59920.60310.59270.60130.60670.59910.60130.60030.59140.60850.61000.59610.59330.59360.59050.60050.60080.59440.60310.59730.61270.60740.60230.60070.60210.60200.59540.60490.59880.58870.59610.60420.59660.60780.59760.61170.60030.59240.60820.59960.59810.59810.59570.59590.60740.59480.59680.59540.60250.59720.59520.59460.59160.59950.5971

&APage &PThis column is formed by stacking the data in columns A and B on top of one another.

Hist-BothDataFrequency table for BothUpper limitCategoryFrequency0.4950.6150

Hist-Both153942830000000000000011643301430

FrequencyCategoryHistogram from Diameter of Elevator Rails from Two Machines

Hist-Machine1DataFrequency table for Machine1Upper limitCategoryFrequency0.490.51251

Hist-Machine1366192019234421

FrequencyCategoryHistogram for Machine1

Hist-Machine2DataFrequency table for Machine2Upper limitCategoryFrequency0.590.61251

Hist-Machine21610202323710421

FrequencyCategoryHistogram for Machine2

DataTimes between arrivals at a bankNote: All interarrival times are expressed in minutes.InterarrivalTime1.3371.3552.6391.3530.3272.2220.7150.6461.4634.5680.1565.20318.7491.5411.6634.3938.3234.8512.8711.9128.9831.4314.5878.3812.21521.6022.4026.4171.6617.05412.2423.4791.0220.954.0591.814.8470.3772.2487.33912.391.462.573.13.9033.340.3525.0342.90426.2850.1525.6238.2337.5582.854.0677.0167.3165.6861.9122.7613.5160.0542.2533.0580.4263.2142.8475.510.5620.2613.37314.0831.563.0975.7222.690.75619.751.731.8320.6235.2472.0582.8941.3823.1085.49411.18211.1640.8864.5239.1453.2663.1541.0516.6479.5661.1380.6712.6450.8964.2643.10914.8860.66910.5051.332.8539.3870.26913.5834.441.33711.6383.3830.6586.0825.1180.2491.9420.2271.4392.6810.092.5610.3163.7754.081.6848.5472.4075.085.2944.9322.81213.6411.8323.610.0326.2415.6383.4272.1820.552.7492.2121.6014.7891.6276.270.8942.7853.310.670.8850.4212.6453.4251.1532.42214.8881.3213.6433.3611.847.2691.1355.9560.8472.9987.9931.2757.5678.7935.352.57821.9146.0161.4890.4860.6121.010.8040.8490.8422.4742.3455.0454.2326.283.6892.5150.7362.5089.4872.2591.6621.4919.9815.3462.5212.8733.6174.3111.1683.7060.5371.5240.2516.2432.7273.8660.3990.6848.8332.8529.2331.1651.1391.6988.0930.9261.4550.6999.8180.6636.6436.1595.2310.8211.8131.1069.8571.1541.21614.3060.3083.1860.0481.227.5832.7931.8536.3568.4712.2216.6292.7730.5562.25410.0916.4014.3720.0546.6370.010.0210.5015.4130.5192.0873.0711.6677.5791.4510.9496.8922.5440.6310.0080.6263.8657.7385.4764.7982.1685.6580.4168.5250.7697.2153.4280.34713.72312.095.04710.2370.2450.58529.0016.7590.9640.7667.440.120.710.53314.7613.843

&APage &P

Hist-InterarrivalTimeDataFrequency table for InterarrivalTimeUpper limitCategoryFrequency2.527.51

Hist-InterarrivalTime13474432499122011

FrequencyCategoryHistogram for Time Between Bank Customer Arrivals

DataDiameters of elevator rails at Otis ElevatorNote: All diameters are measured in fractions of an inchDiameter0.5150.5380.4670.4790.5110.4800.5160.5310.5060.5210.4870.4990.4770.4680.4620.5040.4870.4880.5260.5310.4890.5020.5040.4740.4660.5230.5040.4740.5210.4770.5180.4940.5120.4880.4750.5160.4970.5210.4990.4950.5330.5310.5180.4510.5040.4910.4940.4630.4870.4790.5140.4610.4820.5180.5290.4980.4850.4740.5110.5180.5220.5050.5080.4870.5260.4840.4860.4710.4970.4850.5170.5210.5070.5000.5170.5430.5010.4920.4880.5090.5080.4940.5020.4840.4760.5170.5150.5210.4950.4700.5070.5290.5020.4810.4740.5010.5370.5200.4860.4970.5030.5080.4630.4930.5070.4930.4920.4690.4610.5020.5210.4930.5060.4840.4920.4960.4910.4980.4900.4940.5070.4890.5110.4850.4980.5190.4890.4920.5010.5000.4770.5160.5130.5210.5050.4920.4840.5280.5000.4900.5010.5070.4980.5270.5090.5080.4610.5100.5360.4650.5000.5040.4740.5040.5140.5380.4900.5090.4800.4920.5210.5040.4880.4840.5240.5160.5040.5050.5010.5270.4810.5150.5060.4780.4780.4830.4950.4900.4880.5220.4840.5070.5290.5400.5070.5100.4740.4800.4750.5160.5090.4980.4960.5200.5350.5210.5280.5010.5330.4710.4680.5120.5120.4980.4930.5250.5080.4990.5000.4920.5130.4870.5130.4910.5100.5240.4780.5120.5230.4700.5150.4850.5080.4770.5070.5130.5040.4830.5010.4730.4980.4780.4790.4660.5000.4860.4710.4780.4810.5020.4940.4990.5280.4830.5310.5310.5100.5110.5000.5150.5150.5020.4990.5470.5160.4950.4530.4710.5190.4990.4840.4940.5310.4980.5340.4860.4920.5390.4870.4970.4890.5000.4990.5010.5190.5180.4840.5160.5030.5100.5140.4850.4970.4660.4930.5120.5040.5090.5450.4750.5170.4890.5190.5060.4980.5160.4810.5400.4890.5300.5170.5360.4870.5200.5060.5050.4860.5210.5090.5480.5220.5030.4950.4700.4770.5460.5020.4640.4950.5090.5360.5240.5110.5200.4850.4960.5100.5030.5120.4760.5280.4890.4960.5000.4880.5220.5060.4930.4890.4940.5150.5340.4880.5070.5460.4880.5000.4960.5310.4870.4850.5100.5270.5040.5210.5130.5060.5300.5030.4590.4990.5080.4990.4810.5160.5190.5100.4870.4810.4810.5180.5200.5300.4960.4850.4910.5150.5320.4820.4500.4950.5160.5080.5220.5110.5160.4760.4860.5080.4830.5150.4820.4850.5020.5140.5050.5140.5260.496

&APage &P

Hist-DiameterDataFrequency table for DiameterUpper limitCategoryFrequency0.4550.5455

Hist-Diameter3922476979675929105

FrequencyCategoryHistogram for Diameter of 400 Elevator Rails

DataTimes between arrivals at a bankNote: All interarrival times are expressed in minutes.InterarrivalTime1.3371.3552.6391.3530.3272.2220.7150.6461.4634.5680.1565.20318.7491.5411.6634.3938.3234.8512.8711.9128.9831.4314.5878.3812.21521.6022.4026.4171.6617.05412.2423.4791.0220.954.0591.814.8470.3772.2487.33912.391.462.573.13.9033.340.3525.0342.90426.2850.1525.6238.2337.5582.854.0677.0167.3165.6861.9122.7613.5160.0542.2533.0580.4263.2142.8475.510.5620.2613.37314.0831.563.0975.7222.690.75619.751.731.8320.6235.2472.0582.8941.3823.1085.49411.18211.1640.8864.5239.1453.2663.1541.0516.6479.5661.1380.6712.6450.8964.2643.10914.8860.66910.5051.332.8539.3870.26913.5834.441.33711.6383.3830.6586.0825.1180.2491.9420.2271.4392.6810.092.5610.3163.7754.081.6848.5472.4075.085.2944.9322.81213.6411.8323.610.0326.2415.6383.4272.1820.552.7492.2121.6014.7891.6276.270.8942.7853.310.670.8850.4212.6453.4251.1532.42214.8881.3213.6433.3611.847.2691.1355.9560.8472.9987.9931.2757.5678.7935.352.57821.9146.0161.4890.4860.6121.010.8040.8490.8422.4742.3455.0454.2326.283.6892.5150.7362.5089.4872.2591.6621.4919.9815.3462.5212.8733.6174.3111.1683.7060.5371.5240.2516.2432.7273.8660.3990.6848.8332.8529.2331.1651.1391.6988.0930.9261.4550.6999.8180.6636.6436.1595.2310.8211.8131.1069.8571.1541.21614.3060.3083.1860.0481.227.5832.7931.8536.3568.4712.2216.6292.7730.5562.25410.0916.4014.3720.0546.6370.010.0210.5015.4130.5192.0873.0711.6677.5791.4510.9496.8922.5440.6310.0080.6263.8657.7385.4764.7982.1685.6580.4168.5250.7697.2153.4280.34713.72312.095.04710.2370.2450.58529.0016.7590.9640.7667.440.120.710.53314.7613.843

&APage &P

Hist-InterarrivalTimeDataFrequency table for InterarrivalTimeUpper limitCategoryFrequency2.527.51

Hist-InterarrivalTime13474432499122011

FrequencyCategoryHistogram for Time Between Bank Customer Arrivals