In Session 1 we discussed different types of data Session 2 is titled Calculating and Displaying Descriptive Statistics How do we describe and display categorical data Well we are limited but it can be done We will use tables and graphs The tables we use are either one way frequency tables or two way frequency tables one way if there is only one categorical variable two way if there are two All we do is count up the frequency or number of times each categorical value occurs We will use graphs such as the bar chart, pie chart, and pareto chart to complement the tables Basically graphs summarize or augment the tables Spiffy visual displays are often more insightful Let's do some examples Theproblem Employee Education employee education The first thing to do is table the data educ count none 164 aa 42 ba 225 ma 52 phd 29 total 512 It is important to note here that the data as originally presented was already summarized It was not raw data There was raw data collected initially but it was summarized You can imagine what the survey would have asked each employee So working with this table we need to see how it can be informative to the reader Recall these are categorical variables, not numerical variables The frequencies are numbers but not the variables Use an excel spreadsheet and construct a third column, percentage Note the simple formula when you click on a cell in the percentage column This helps us to interpret the data See if you can come with some general conclusions about education level of all the employees Here is an example I am sure you can do better Most employees (about three quarters) have either a bachelors degree (44 ) or no college (32 ) About 10 have Masters, 8 have Associates, and nearly 6 have PhDs About 85 have no degree or a degree other than a graduate degree Only 15 have an advanced degree 32 have no degree at all The excel sheet shows a bar chart and pie chart You should be able to get those using the Insert command in excel Highlight the data you want to be represented in the graph Click on Insert and then click either the picture of a bar chart or the picture of a pie chart Note that data on tenure are also collected for each employee and summarized in the table Now there are two categorical variables This is called a two way table or contingency table One is education level and the other is time or tenure with the company This is an important point Time with the company is numerical (years) But we group this and create categories We do this to make the description simpler to understand So if tenure for an employee was anything less than 1 year, we put that employee into the first category There was 164 employees with less than 1 year If tenure was 1 year up to 5 years, we put those employees into the second category We have created 3 tenure values for the categorical variable The two way table is shown in the excel spreadsheet For example, there was 63 employees who had a BA degree and had worked for the company more than 5 years The table has a column at the far right It is labeled Total and is called a marginal distribution It consists of the total of all cells on each row At the bottom is a row labeled Total It consists of the sum of each cell in each column It is also called a marginal distribution Here are the answers to the 4 questions See if you get these 32 or 32 6 or 0059 9 15 8 Calculate column percentages These are shown on the spreadsheet These are important for interpreting the data Note that each column adds up to 100 You could also describe row percentages if you wanted to More than 2 3 of those with no college degree have been with the company longer than 5 years but almost none of the PhDs (less than7 ) have been there that long It appears that over the past few years, the company has hired proportionately more higher educated employees They perhaps changed their hiring policies and it shows Now, let's do a problem showing some descriptive analysis for raw categorical data Here we construct a table from raw data, not summarized data TheGradSurvey data file is one I described earlier Some is categorical and some is numerical Suppose we wanted to determine a one way table for the variable gender Gender is categorical Suppose we wanted to determine a two way table for gender and employment status Both are categorical There is a discussion topic in Session 2 on the Grad Survey file example along with some output Read that and try to get the results using Excel, Pivot Table Next, we will discuss descriptive statistics for numerical values But first let's develop a one way table, bar chart, and pie chart for the variable Undergrad GPA Look at the output mess What is the problem because it would be very difficult to interpret what this means The problem is that we are applying a simple one way table to numerical data Recall with continuous data, there are many different values, maybe only one of each So when you table the data, the frequencies are very low numbers Consequently the bar chart and pie chart are pretty but meaningless We will have to do something different We have to group the data and create a grouped frequency table Then things will make sense In order to describe numerical data with descriptive statistics and displays, we need to examine the data If we look at how the data are distributed, we may see patterns that are important There are three major characteristics we should look for They are Shape, Central Tendency, and Variation Shape means patterns like may be shown in a bar chart There are things we should look for like symmetry, peaks, gaps, outliers Look at this link session 2 In the first, the pattern of data is symmetrical In the second it is not We call this skewness It means that the shape is skewed to the right That means that there are proportionately more high values than low values You could have skewness to the left where there would be proportionately more lower values In the third we have multiple peaks or a bi modal shape There could be two processes going on It is something we may want to investigate further In the fourth we have a gap Something is happening with the data and this is similar to multiple peaks where two processes may be occurring The fifth shows outliers There are usually not enough to show a different shape but a few stragglers that throw off the data These can cause trouble later when you prepare some statistics like averages They should be investigated They may be correct or they could be erroneous and caused by misreading or recording incorrectly There are a few ways of looking at the shape of your data We will call it the distribution of your data There is a histogram, a box plot, and a stem and leaf diagram I will discuss the histogram You can read about this and the others in your readings for this session We must group the original numerical data into groups, cells, or bins I use these terms because you will see them all in your readings The histogram that we can calculate using Excel Data Analysis uses bins We gather the data so that values falling between one number and another go into that group or bin Then we have higher counts or frequencies in each newly created bin and you can see what is happening clearer than what that example I showed you earlier displayed Procedure Determine the range (highest value lowest value) Decide how many bins or cells you want There is no exact number but usually anywhere from 5 to 10 works I like 7 This means you will see 7 rectangles or bars in the histogram Divide the range by the number of bins This gives you the bin width You can round to make it sensible List all the bins in your excel spreadsheet next to the raw data It will be a new column These bin numbers are actually the end of the bin The first should be the sum of the lowest value plus bin width Then each bin after that will be larger by the bin width List until you reach the highest value in your data set You can write a formula in Excel to this Then use Data Analysis Pick Histogram Highlight the input range which is your column of raw data and the bin column next Click on chart output and OK You can stretch it out to make it look nicer Look at this example utility Try to follow all the tabs What can we say about this It is a symmetrical distribution and the central tendency is about 147 We will do more on central tendency later I mentioned earlier the box plot and stem and leaf Look at them in this link utility more Again we see a symmetrical distribution Next let's look at Central Tendency If we are asked to describe data, one of the useful characteristics would be its central values or where most of the data are located We will use mean or average, median, and mode The formula for mean is x n where we sum all the values of the variable (x) and divide the sum by n, the number of values We could write a subscript for the summation sign but often leave it off for simplicity But always be aware of how much you have to sum The median is the halfway value If you rank all your numerical values and draw a line at the halfway point, that is the median Fifty percent of all values will be less than the median and 50 percent will be more It is called the median or sometimes the 50 th percentile or sometimes the second quartile The mode is the most frequently occurring value Sometimes there may not be a mode because none of the values occurs more than others The mode is not used a lot in practice but it is a measure of central tendency If a distribution is symmetrical, the mean, median, and mode are all the same value The mean may not be the best measure of central tendency to use for a skewed distribution The mean is more affected by extreme or outlier values than the median The median is the middle value of ranked data unlike the mean which weights each value the same, small or large Consider the values 4, 6, 5, 3, 20 Calculate the mean and median The mean is 7 6 and the median is 5 Verify this using the formula above The values are skewed to the right or high side so the mean is larger than the median Remember that if the distribution is not skewed but is symmetric the mean and median are the same So it will be better to use the median as the measure of central tendency if the distribution is significantly skewed When you calculate statistics you should always look at both and compare them Now let's look at Spread This is a very important characteristic It informs you how close together or far apart the individual values are Is there little variation in which case the histogram would be narrow or is there a lot of variation in which case the histogram would be wide Look at the diagram in the link SESSION 2 above You could have two distributions with the same measure of central tendency but quite different measures of spread Spread is very important In statistics, variability of data plays a very important rule For a statistician the less variability the better That may not be true in other situations but it is in statistics Statisticians cannot control variability but we deal with it So we need some measures of spread or variability Look at the formula on the link SESSION 2 That formula gives you the variance of the data If you take the square root of variance you get standard deviation Both are used in statistics Just remember one is the square root of the other Try the formula by hand on two data sets 1 (2, 4, 6, 8) and 2 (2, 14, 26, 94) You should get variance (s 2 ) 6 7 for 1 and 1696 for 2 Why is 2 so much larger The formula will explain it s 2 6 for 1 and 41 2 for 2 Another measure of spread is Range which is simply the difference between the lowest and highest value You can see now why a range cannot be calculated for a categorical value but can for a numerical value You must be able to rank or order values Another is the Interquartile Range It is calculated as Q3 Q1 where Q3 is the third quartile (75 of values are lower than Q3 and 25 are higher) and Q1 is the first quartile (25 are lower and 75 are higher) Thus the IQR is the middle portion of the data Remember Q2 is the median Excel has a feature called the Data Analysis Tool that will give you all the descriptive statistics you would ever want You click on DATA at the top and then Data Analysis at the far right Highlight your data and check whether there are labels or not Click Descriptives as output Try it below Look at savings rate and all its tabs This is an example of descriptive statistics that could be used to explain the findings What about shape, central tendency, and spread for these data What stands out Coefficient of Variation This is a statistic that is often used when comparing frequency distributions of numerical values It includes the two most important statistics (mean and standard deviation) The formula is (std deviation mean) x 100 For example, if the mean of a distribution is 20 and the standard deviation is 5, the CV is 25 If the mean is 20 and the standard deviation is 15, the CV is 75 The latter is a much more variable distribution and spread out more that the first That can be an important factor The data analysis tool does not compute this coefficient of variation statistic in its set of descriptives, but you can easily compute it The relative magnitude of the percentage tells you a lot and makes it easier to compare different data sets Our final statistic to be discussed is a very important one and we will see it many many times from this point on Let's make sure we know what it is and why we need it It is called a z score or z statistic It is a standardized or normalized value that has no units or dimensions and it is very seldom less than 3 or greater than 3 We can convert any numerical variable no matter how large or small and no matter what the units are into a standardized variable that is between 3 and 3 and has no units There may be situations where the z value is less than 3 or greater than 3 but very seldom and these represent extreme values So if we want to standardize a value x and convert it to z, the formula is z (x xbar) s where xbar is the mean of x and s is the standard deviation of x Look at the formula on the link SESSION 2 By this, we mean x is one value that came from a distribution of numerical values which has a mean of xbar and standard deviation of s We know how to calculate those Suppose on the horizontal scale of a bell shaped frequency distribution, we mark off the mean at the center, one standard deviation higher or above the mean, two standard deviations above the mean, and three standard deviations above the mean Then similarly mark off one standard deviation lower or below the mean, two standard deviations below the mean, and three standard deviations below the mean Look at the diagram on the link SESSION 2 These same markings are z values of 0, and 1, 2, and 3 to the right of the center as well as 1, 2 and 3 to the left of the center Just to see why, consider what happens if the x value we want to standardize is the mean plus two standard deviations So x xbar 2s The z score is (xbar 2s xbar) s Thus z 2 This can be done with any of the markings Look at the diagram in the link session 2 Just to show how this works for any numerical value, calculate z score for a x value of 75 feet where we are measuring distance to a monument Of all the distances we have measured and there are several, the mean value (xbar) is 60 feet and the standard deviation (s) is 10 feet So the z score is (75ft 60ft) 10ft 1 5 It is between 3 and 3 and has no units Another example suppose x $120,000 the cost of a new piece of equipment We have looked at several and x bar is $140,000 and s is $10,000 Thus the z score for this particular piece is ($120000 $140000) $10000 2 Again it has no units and is between 3 and 3 The values have been standardized Later we will see why when we must calculate probabilities from a probability distribution So, no matter how large or how small the data values are, they can be converted into a score between 3 and plus 3 This is the z score There is an important theorem in statistics called the empirical rule It states that 68 (about two thirds) of all the values in a bell shaped distribution are within plus and minus one standard deviation away from the mean That also means within plus and minus a z score of 1 And, 95 of all the values are within plus and minus a z score of 2 And, 99 7 (almost all) of all the values are within plus and minus a z score of 3 Next we do a problem to test this theory How close can we come Look at this problemEnergy and observe all the tabs You can see that the data generally follow the empirical rule We used standardized z scores in this problem One last problem is to show that when you convert non standardized data to standardized date, you can compare and calculate interesting facts Otherwise working with variables of different dimensions would be very difficult food and food consumption If you look at the table of z scores , the last column is the most important We can compare apples and apples because we have standardized our variables The highest sum of z scores is Ireland It is the largest consumer of both meat and alcohol together 0 COURSE MGMT 650 Please include references Thanks Note I did not see where it says Creat a or Write a

Question

In Session 1 we discussed different types of data  Session 2 is titled  Calculating and Displaying Descriptive Statistics  How do we describe and display categorical data  Well we are limited but it can be done  We will use tables and graphs  The tables we use are either one way frequency tables or two way frequency tables   one way if there is only one categorical variable  two way if there are two  All we do is count up the frequency or number of times each categorical value occurs  We will use graphs such as the bar chart, pie chart, and pareto chart to complement the tables  Basically graphs summarize or augment the tables  Spiffy visual displays are often more insightful  Let's do some examples  Theproblem Employee Education employee education  The first thing to do is table the data  educ count none 164 aa 42 ba 225 ma 52 phd 29 total 512 It is important to note here that the data as originally presented was already summarized  It was not raw data  There was raw data collected initially but it was summarized  You can imagine what the survey would have asked each employee  So working with this table we need to see how it can be informative to the reader  Recall these are categorical variables, not numerical variables  The frequencies are numbers but not the variables  Use an excel spreadsheet and construct a third column, percentage  Note the simple formula when you click on a cell in the percentage column  This helps us to interpret the data  See if you can come with some general conclusions about education level of all the employees  Here is an example  I am sure you can do better  Most employees (about three quarters) have either a bachelors degree (44 ) or no college (32 )  About 10  have Masters, 8  have Associates, and nearly 6  have PhDs  About 85  have no degree or a degree other than a graduate degree  Only 15  have an advanced degree  32  have no degree at all  The excel sheet shows a bar chart and pie chart  You should be able to get those using the  Insert  command in excel  Highlight the data you want to be represented in the graph  Click on Insert and then click either the picture of a bar chart or the picture of a pie chart  Note that data on tenure are also collected for each employee and summarized in the table  Now there are two categorical variables  This is called a two way table or contingency table  One is education level and the other is time or tenure with the company  This is an important point  Time with the company is numerical (years)  But we group this and create categories  We do this to make the description simpler to understand  So if tenure for an employee was anything less than 1 year, we put that employee into the first category  There was 164 employees with less than 1 year  If tenure was 1 year up to 5 years, we put those employees into the second category  We have created 3 tenure values for the categorical variable  The two way table is shown in the excel spreadsheet  For example, there was 63 employees who had a BA degree and had worked for the company more than 5 years  The table has a column at the far right  It is labeled Total and is called a marginal distribution  It consists of the total of all cells on each row  At the bottom is a row labeled Total  It consists of the sum of each cell in each column  It is also called a marginal distribution  Here are the answers to the 4 questions  See if you get these  32  or  32  6  or  0059  9  15 8  Calculate column percentages  These are shown on the spreadsheet  These are important for interpreting the data  Note that each column adds up to 100   You could also describe row percentages if you wanted to  More than 2 3 of those with no college degree have been with the company longer than 5 years but almost none of the PhDs (less than7 ) have been there that long  It appears that over the past few years, the company has hired proportionately more higher educated employees  They perhaps changed their hiring policies and it shows  Now, let's do a problem showing some descriptive analysis for raw categorical data  Here we construct a table from raw data, not summarized data  TheGradSurvey data file is one I described earlier  Some is categorical and some is numerical  Suppose we wanted to determine a one way table for the variable gender  Gender is categorical  Suppose we wanted to determine a two way table for gender and employment status  Both are categorical  There is a discussion topic in Session 2 on the Grad Survey file example along with some output  Read that and try to get the results using Excel, Pivot Table  Next, we will discuss descriptive statistics for numerical values  But first let's develop a one way table, bar chart, and pie chart for the variable    Undergrad GPA  Look at the output  mess What is the problem because it would be very difficult to interpret what this means  The problem is that we are applying a simple one way table to numerical data  Recall with continuous data, there are many different values, maybe only one of each  So when you table the data, the frequencies are very low numbers  Consequently the bar chart and pie chart are pretty but meaningless  We will have to do something different  We have to group the data and create a grouped frequency table  Then things will make sense  In order to describe numerical data with descriptive statistics and displays, we need to examine the data  If we look at how the data are distributed, we may see patterns that are important  There are three major characteristics we should look for  They are Shape, Central Tendency, and Variation  Shape means patterns like may be shown in a bar chart  There are things we should look for like symmetry, peaks, gaps, outliers  Look at this link session 2  In the first, the pattern of data is symmetrical  In the second it is not  We call this skewness  It means that the shape is skewed to the right  That means that there are proportionately more high values than low values  You could have skewness to the left where there would be proportionately more lower values  In the third we have multiple peaks or a bi modal shape  There could be two processes going on  It is something we may want to investigate further  In the fourth we have a gap  Something is happening with the data and this is similar to multiple peaks where two processes may be occurring  The fifth shows outliers  There are usually not enough to show a different shape but a few stragglers that throw off the data  These can cause trouble later when you prepare some statistics like averages  They should be investigated  They may be correct or they could be erroneous and caused by misreading or recording incorrectly  There are a few ways of looking at the shape of your data  We will call it the distribution of your data  There is a histogram, a box plot, and a stem and leaf diagram  I will discuss the histogram  You can read about this and the others in your readings for this session  We must group the original numerical data into groups, cells, or bins  I use these terms because you will see them all in your readings  The histogram that we can calculate using Excel Data Analysis uses  bins   We gather the data so that values falling between one number and another go into that group or bin  Then we have higher counts or frequencies in each newly created bin and you can see what is happening clearer than what that example I showed you earlier displayed  Procedure Determine the range (highest value   lowest value) Decide how many bins or cells you want  There is no exact number but usually anywhere from 5 to 10 works  I like 7  This means you will see 7 rectangles or bars in the histogram  Divide the range by the number of bins  This gives you the bin width  You can round to make it sensible  List all the bins in your excel spreadsheet next to the raw data  It will be a new column  These bin numbers are actually the end of the bin  The first should be the sum of the lowest value plus bin width  Then each bin after that will be larger by the bin width  List until you reach the highest value in your data set  You can write a formula in Excel to this  Then use Data Analysis  Pick Histogram  Highlight the input range which is your column of raw data and the bin column next  Click on chart output and OK  You can stretch it out to make it look nicer  Look at this example utility Try to follow all the tabs  What can we say about this  It is a symmetrical distribution and the central tendency is about 147  We will do more on central tendency later  I mentioned earlier the box plot and stem and leaf  Look at them in this link  utility more Again we see a symmetrical distribution  Next let's look at Central Tendency  If we are asked to describe data, one of the useful characteristics would be its central values or where most of the data are located  We will use mean or average, median, and mode  The formula for mean is     x n where we sum all the values of the variable (x) and divide the sum by n, the number of values  We could write a subscript for the summation sign but often leave it off for simplicity  But always be aware of how much you have to sum  The median is the halfway value  If you rank all your numerical values and draw a line at the halfway point, that is the median  Fifty percent of all values will be less than the median and 50 percent will be more  It is called the median or sometimes the 50 th percentile or sometimes the second quartile  The mode is the most frequently occurring value  Sometimes there may not be a mode because none of the values occurs more than others  The mode is not used a lot in practice but it is a measure of central tendency  If a distribution is symmetrical, the mean, median, and mode are all the same value  The mean may not be the best measure of central tendency to use for a skewed distribution  The mean is more affected by extreme or outlier values than the median  The median is the middle value of ranked data unlike the mean which weights each value the same, small or large  Consider the values 4, 6, 5, 3, 20  Calculate the mean and median  The mean is 7 6 and the median is 5  Verify this using the formula above  The values are skewed to the right or high side so the mean is larger than the median  Remember that if the distribution is not skewed but is symmetric the mean and median are the same  So it will be better to use the median as the measure of central tendency if the distribution is significantly skewed  When you calculate statistics you should always look at both and compare them  Now let's look at Spread  This is a very important characteristic  It informs you how close together or far apart the individual values are  Is there little variation in which case the histogram would be narrow or is there a lot of variation in which case the histogram would be wide  Look at the diagram in the link SESSION 2 above  You could have two distributions with the same measure of central tendency but quite different measures of spread  Spread is very important  In statistics, variability of data plays a very important rule  For a statistician the less variability the better  That may not be true in other situations but it is in statistics  Statisticians cannot control variability but we deal with it  So we need some measures of spread or variability  Look at the formula on the link SESSION 2  That formula gives you the variance of the data  If you take the square root of variance you get standard deviation  Both are used in statistics  Just remember one is the square root of the other  Try the formula by hand on two data sets   1 (2, 4, 6, 8) and  2 (2, 14, 26, 94)  You should get variance (s 2 )  6 7 for  1 and 1696 for  2  Why is  2 so much larger  The formula will explain it  s   2 6 for  1 and 41 2 for  2  Another measure of spread is Range which is simply the difference between the lowest and highest value  You can see now why a range cannot be calculated for a categorical value but can for a numerical value  You must be able to rank or order values  Another is the Interquartile Range  It is calculated as Q3   Q1 where Q3 is the third quartile (75  of values are lower than Q3 and 25  are higher) and Q1 is the first quartile (25  are lower and 75  are higher)  Thus the IQR is the middle portion of the data  Remember Q2 is the median  Excel has a feature called the Data Analysis Tool that will give you all the descriptive statistics you would ever want  You click on DATA at the top and then Data Analysis at the far right Highlight your data and check whether there are labels or not  Click Descriptives as output  Try it below  Look at savings rate and all its tabs  This is an example of descriptive statistics that could be used to explain the findings  What about shape, central tendency, and spread for these data  What stands out  Coefficient of Variation   This is a statistic that is often used when comparing frequency distributions of numerical values  It includes the two most important statistics (mean and standard deviation)  The formula is (std deviation   mean) x 100  For example, if the mean of a distribution is 20 and the standard deviation is 5, the CV is 25   If the mean is 20 and the standard deviation is 15, the CV is 75   The latter is a much more variable distribution and spread out more that the first  That can be an important factor  The data analysis tool does not compute this coefficient of variation statistic in its set of descriptives, but you can easily compute it  The relative magnitude of the percentage tells you a lot and makes it easier to compare different data sets  Our final statistic to be discussed is a very important one and we will see it many many times from this point on  Let's make sure we know what it is and why we need it  It is called a z score or z statistic  It is a standardized or normalized value that has no units or dimensions and it is very seldom less than  3 or greater than  3  We can convert any numerical variable no matter how large or small and no matter what the units are into a standardized variable that is between  3 and  3 and has no units  There may be situations where the z value is less than  3 or greater than  3 but very seldom and these represent extreme values  So if we want to standardize a value x and convert it to z, the formula is z   (x   xbar)   s where xbar is the mean of x and s is the standard deviation of x  Look at the formula on the link SESSION 2  By this, we mean x is one value that came from a distribution of numerical values which has a mean of xbar and standard deviation of s  We know how to calculate those  Suppose on the horizontal scale of a bell shaped frequency distribution, we mark off the mean at the center, one standard deviation higher or above the mean, two standard deviations above the mean, and three standard deviations above the mean  Then similarly mark off one standard deviation lower or below the mean, two standard deviations below the mean, and three standard deviations below the mean  Look at the diagram on the link SESSION 2  These same markings are z values of 0, and  1,  2, and  3 to the right of the center as well as  1,  2 and  3 to the left of the center  Just to see why, consider what happens if the x value we want to standardize is the mean plus two standard deviations  So x   xbar   2s  The z score is (xbar 2s   xbar) s  Thus z   2  This can be done with any of the markings  Look at the diagram in the link session 2  Just to show how this works for any numerical value, calculate z score for a x value of 75 feet where we are measuring distance to a monument  Of all the distances we have measured and there are several, the mean value (xbar) is 60 feet and the standard deviation (s) is 10 feet  So the z score is (75ft   60ft) 10ft   1 5  It is between  3 and  3 and has no units  Another example  suppose x   $120,000 the cost of a new piece of equipment  We have looked at several and x bar is $140,000 and s is $10,000  Thus the z score for this particular piece is ($120000   $140000)   $10000    2  Again it has no units and is between  3 and  3  The values have been standardized  Later we will see why when we must calculate probabilities from a probability distribution  So, no matter how large or how small the data values are, they can be converted into a score between  3 and plus 3  This is the z score  There is an important theorem in statistics called the empirical rule   It states that 68  (about two thirds) of all the values in a bell shaped distribution are within plus and minus one standard deviation away from the mean  That also means within plus and minus a z score of 1  And, 95  of all the values are within plus and minus a z score of 2  And, 99 7  (almost all) of all the values are within plus and minus a z score of 3  Next we do a problem to test this theory  How close can we come  Look at this problemEnergy and observe all the tabs  You can see that the data generally follow the empirical rule  We used standardized z scores in this problem  One last problem is to show that when you convert non standardized data to standardized date, you can compare and calculate interesting facts  Otherwise working with variables of different dimensions would be very difficult  food and food consumption  If you look at the table of z scores , the last column is the most important  We can compare apples and apples because we have standardized our variables  The highest sum of z scores is Ireland  It is the largest consumer of both meat and alcohol together  0 COURSE  MGMT 650 Please include references  Thanks Note  I did not see where it says  Creat a  or  Write a

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Question: In Session 1 we discussed different types of data. Session 2 is titled Calculating and Displaying Descriptive Statistics How do we describe and display categorical

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