Descriptive Statistics, Numerical Description

Editor's Notes

• #4: See Chapter 1 in the text for an overview of these issues
• #5: Univariate data set: data of one variable (Example: people weight) Bivariate data set: data of two variables (Example: years at work, annual salary) Multivariate data set: data of more than two variables (Example: name, age, income, position, and gender) Note in the top table, the different statistical tools that can be used for each type of data set. These will be discussed throughout this course.
• #6: A data set may also be divided on the basis of the sampling approach as follows:   1. Cross sectional data set 2. Time-series data set In a cross-sectional data set, all samples are collected at more or less the same point in time. In a time-series data set, samples are collected at specific points over time (e.g., weekly, monthly or quarterly). The Figure here illustrates these two types of sampling approach, and the common analyses used for the data collected in each type. In practice, cross-sectional samples are normally large in size as they are typically used to take a snap shot at a population that is assumed to be more or less stable. An example of cross-sectional data is the data of the prices of houses in a certain area collected in a given year. On the other hand, time-series samples are used for populations that encounter dynamic changes, periodic or seasonal. For example, the information on the quarterly revenue of a company is a time-series data set.
• #8: This is an Optional Working Problem: This working problem aims at encouraging students to read about the origin and the meaning of data set before they analyze it. Students will be required to search the web for this information then use critical thinking to address why some states have a wide income tax range.
• #12: These key points will become clear as we cover descriptive statistics and move to inferential statistics
• #14: Note that X-Bar and mu are universal symbols…X-bar will always be used to describe a sample mean (or a statistic) and mu will always be used to describe a population mean (or a parameter)
• #15: By comparison of the arithmetic means of the two years, it is obvious that the average price in 2009 was about $1.139 cheaper than that of 2008 at the month of September. This is obviously good news to everyone particularly those who drive big SUVs and trucks. The results, however, do not tell us the cause of this drop. In searching for the cause of this significant drop, one may consult other sources that may partially or fully explain the trend. For example, a barrel of crude oil in this period in 2008 was about $105. In 2009, the price went down significantly and in September of 2009, it was only $69. Again, this may only represent a partial cause of the drop in gas price and one must also entertain other possible causes.
• #18: WP 2.4: Mean = $6.7 WP2.5: $7580 $2,350 $1.26
• #22: WP 2.6: Median = $6.5 Mode = $8 WP 2.7 (a) Median = $6500, Mode = $6000 (b) Median = $2.552, Mode = None (a) Median = $1.262, Mode = $1.265
• #23: Geometric Mean Commonly used when the variable under consideration is likely to change over time or periodically. It is used to find the average change over the entire period under study. Typical situations in which the geometric mean is useful include: population growth, quarterly or annual return on investment, and inflation. Note in the above Example: The value of 1.2 reflects a 20% return on investment (the original investment of 1.0 plus the return of 0.2); the value of 0.6 reflects a loss of 40% (the original investment of 1.0 less than the loss of 0.4); and the value of 2 reflects a gain of 100% (the original investment of 1.0 plus the return of 1). In business calculation, the total return each period is typically reinvested in the next period, or it becomes the base for the next period. This makes the base for, say the second year 1.2, and the base for the third year (1.2)(1.15), and that for the fourth year (1.2)(1.15)(0.6) and so forth.
• #24: Note that if we calculated the classic arithmetic mean for the annual inflations [(6+8+2+4+5)/5 = 5%], we would have a higher value than the geometric mean and this will overstate the true rate of inflation.
• #25: 1.1491 1.15095 Greater
• #28: The advantage of using the range is its simplicity. The disadvantage is that it relies on two observations, the minimum value and the maximum value. If one of these two observations is an ‘outlier’, or a value that is inconsistent with the family of data under consideration, the range will be misleading. In addition, using the range may be associated with a loss of data resolution as will be seen shortly.
• #30: For n < 30, we use (n-1) in the denominator to obtain a better estimate of the standard deviation of the population from which the sample data is taken.
• #36: The correct answer is B. The solution requires us to recognize that Variable Z is a combination of two independent random variables. As such, the variance of U is equal to the variance of X plus the variance of Y. Var(U) = Var(X - Y) = Var(X) + Var(Y) = 16 + 9 = 25 The standard deviation of U is equal to the square root of the variance. Therefore, the standard deviation is equal to the square root of 25, or 5.
• #37: Q(1) Minimum = $5 Maximum =$8 Range = $3 Mean Deviation = $1.1 Standard deviation = $1.252 Variance = 1.567 Q(2): Mean = (80+90)/2 = 85 Variance (Exam 1 + Exam 2) = Var (Exam 1) + Var (Exam 2) = 36 + 64 = 100
• #38: 4800, 12000, 2,538.07, 6,441,777.78 1.787, 2.66, 0.33121, 0.10970 1.23, 1.302, 0.02283, 0.00052
• #40: C.V% = 18.69
• #43: From the z-scores, you can conclude that a driving time of 38 minutes is 0.2 standard deviations below the mean; a driving time of 52 minutes is 1.2 standard deviations below the mean; a driving time of 58 minutes is 1.8 standard deviations above the mean; and a driving time of 40 minutes is zero standard deviations or equal to the mean.
• #44: San Francisco: z = (16-22)/4 = -1.5 New England: z = (29-22)/4 = 1.75 Pittsburgh: z = (24-22)/4 = 0.5 San Francisco is 1.5 standard deviations below the mean; New England is 1.75 standard deviations above the mean; and Pittsburgh is 0.5 standard deviations above the mean. To the point this statistics was recorded, out of the three teams New England had the best record, followed by Pittsburgh.
• #45: For x = $85,000, z = (x-m)/s = (85,000 – 100,000)/10,000 = -1.5 For x = $105,000, z = (x-m)/s = (105,000 – 100,000)/10,000 = 0.5 The z of -1.5 indicates that an annual salary of $85,000 is one and half standard deviation below the mean, and a z of 0.5 indicates that an annual salary of $105,000 is half standard deviation above the mean.
• #46: For Arkansas governor: x = $85,000, z = (x-m)/s = (85,000 – 135,450)/36,530 = -1.381 For x = $206,000, z = (x-m)/s = (206,000 – 135,450)/36,530 = 1.931 The z of -1.381 indicates that the annual salary of the Arkansas governor of $85,000 was 1.381 standard deviation below the governor mean salary, and a z of 1.931 indicates that the annual salary of the California governor of $206,000 was 1.931 standard deviation above the governor’s mean salary.
• #49: Performing descriptive statistics using Excel®: Using the file in which the data in question is presented we take the following steps: Step 1: Click on Data Step 2: Click on Data Analysis
• #50: Steps 3 and 4: Select ‘Descriptive Statistics’ and Click ‘OK’ . This will display the ‘Descriptive Statistics’ Menu shown in next slide
• #51: Step 5: Select the Input Range, which is the column of data to be analyzed. As shown here, the input range is the data of In-State Tuition, or cells G2:G16. Since you have a label in Cell G2, Check ‘label in first raw’ box. Check ‘Summary Statistics’. You may also check ‘kth largest’ and kth smallest’ and specify the largest and smallest number of observations that you would like to set as thresholds for your analysis. This will give you the maximum value of the smallest four observations, and the minimum value of the largest 4 observations. Step 6: Click ‘OK’ to obtain the final output, which will appear in a different Excel® Sheet You can also make the output appear in the same sheet by specifying a cell in the output range window.
• #53: The output of the analysis of the previous example is shown here for all the variables in the data set, obtained by following the steps described above and changing the input range in Excel® ‘Descriptive Analysis’ to analyze the variable in question. Work with your students to interpret the results…see below Description and interpretation of results: The steps described above will take only few seconds to perform. Therefore, the true effort should consist of reading the outputs, describing what we understood, and attempting to interpret the results. The following points reflect these key aspects of analysis and they are listed here as guidelines to students since different people may have different reads and different interpretations of the analysis output. The average in-state tuition per year of the schools under study is $5,079. However, some schools can be as low as $3,094 (University of Florida) and some can be as high as $7,314 (University of South Carolina). The median of in-state tuition is $4,756. The mode of in-state tuition is $4,515; that is what most schools in this set charge for in-state tuition. The average out-of-state tuition of the schools under study is $16,550. However, some schools can be as low as $9,744 (University of Mississippi) and some can be as high as $24,324 (University of California). The median of out-state tuition is $16,460. The mode of out-state tuition is not well-defined as there may be more than one mode involved. The average total cost of the schools under study is $22,895. However, some schools can be as low as $14,442 (University of Mississippi) and some can be as high as $36,252 (University of California). The median of total cost is $22,979. The mode of total cost is not well-defined as there may be more than one mode involved. The average financial aid of the schools under study is $8,878. However, some schools can be as low as $5,487 (North Dakota State University) and some can be as high as $13,462 (University of California). The median of financial aid is $8,114. The mode of financial aid is not well-defined as there may be more than one mode involved. You may also set thresholds for a certain variable of interest, say total cost, by specifying the maximum of the smallest four universities, and the minimum of the largest four universities. For total cost, the maximum of the smallest four universities is $18,540 (University of Alabama), and the minimum of the largest four universities is $25,039 (University of South Carolina). As you can see from the above points, the guideline to describe and interpret analysis outputs will depend on the key questions that one wishes to address in the analysis application. Examples of these questions are as follows: Q1: Which school will be associated with the lowest total cost of education? University of Mississippi Q2: Which school will be associated with the largest financial aid for education? University of California Q3: Which school(s) will have the lowest total cost and the highest financial aid? This question may not be easy to answer if there is a positive association between total cost and financial aid (i.e. the higher the cost, the higher the financial aid). Unfortunately, this is normally the case for most schools as shown in in next slide. In this case, you may have to make a compromising choice such as University of Kentucky, University of Mississippi, and University of Alabama.