Fluent Essays

quiz content:Differentiate between Qualitative and Quantitative Data.Use graphical tools (Bar charts, Clustered Bar Charts and Pie Charts) for summarizing and representing Qualitative Data.Use graphical tools (Histograms, Dotplots, Stem-and-Leaf Displays, and Boxplots) for summarizing and representing Quantitative Data.Calculate various measures of location / center (mean, median, and mode) for raw data as well as for data in the form a frequency distribution, by hand and using Minitab.Calculate various measures of dispersion / spread / variability (range, interquartile range, and standard deviation) for both raw data and data in the form a frequency distribution, again by hand and using Minitab.Calculate standardized score or z-score by hand and using Minitab.Make use of these summary measures to compare datasets and to discuss symmetry or skewness of datasets.Calculate, from the raw data, the percentage of data values within 1, 2, and 3 standard deviations of the mean, as well as approximate these percentages using the appropriate rule (empirical or Chebyshevs) when only the mean and standard deviation are known.Explain cogently why the standard deviation is a valid measure of dispersion and the z-score is a measure of relative standing.Calculate Mean and Standard Deviation of linearly transformed data. chapter_2.pdf Unformatted Attachment Preview Chapter 2 Methods for Describing Sets of Data ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 1 Contents 1. Describing Qualitative Data 2. Graphical Methods for Describing Quantitative Data 3. Numerical Measures of Central Tendency 4. Numerical Measures of Variability 5. Using the Mean and Standard Deviation to Describe Data ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 2 Contents (cont) 6. Numerical Measures of Relative Standing 7. Methods for Detecting Outliers: Box Plots and z-scores 8. Distorting the Truth with Descriptive Techniques ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 3 Learning Objectives 1. Describe data using graphs 2. Describe data using numerical measures 3. Describe quantitative data using numerical measures 4. Detecting descriptive methods that distort the truth ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 4 2.1 Describing Qualitative Data ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 5 Key Terms A class is one of the categories into which qualitative data can be classified. The class frequency is the number of observations in the data set falling into a particular class. The class relative frequency is the class frequency divided by the total numbers of observations in the data set. The class percentage is the class relative frequency multiplied by 100. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 6 Data Presentation Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart ALWAYS LEARNING Pareto Diagram Stem-&-Leaf Display Frequency Distribution Histogram Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 7 Data Presentation Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart ALWAYS LEARNING Pareto Diagram Stem-&-Leaf Display Frequency Distribution Histogram Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 8 Summary Table 1. Lists categories & number of elements in category 2. Obtained by tallying responses in category 3. May show frequencies (counts), \% or both Row Is Major Category Accounting Economics Management Total ALWAYS LEARNING Count 130 20 50 200 Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Tally: |||| |||| |||| |||| Slide - 9 Data Presentation Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart ALWAYS LEARNING Pareto Diagram Stem-&-Leaf Display Frequency Distribution Histogram Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 10 Bar Graph Percent Used Also Frequency 150 Equal Bar Widths Bar Height Shows Frequency or \% 100 50 0 Acct. Econ. Major Zero Point ALWAYS LEARNING Mgmt. Vertical Bars for Qualitative Variables Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 11 Data Presentation Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart ALWAYS LEARNING Pareto Diagram Stem-&-Leaf Display Frequency Distribution Histogram Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 12 Pie Chart 1. Shows breakdown of Majors total quantity into categories Econ. 2. Useful for showing 10\% 36° relative differences Acct. 65\% 3. Angle size • Mgmt. 25\% (360°)(percent) (360°) (10\%) = 36° ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 13 Data Presentation Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart ALWAYS LEARNING Pareto Diagram Stem-&-Leaf Display Frequency Distribution Histogram Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 14 Pareto Diagram Like a bar graph, but with the categories arranged by height in descending order from left to right. Percent Used Also Frequency 150 Equal Bar Widths Bar Height Shows Frequency or \% 100 50 0 Acct. Mgmt. Major Zero Point ALWAYS LEARNING Econ. Vertical Bars for Qualitative Variables Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 15 Summary Bar graph: The categories (classes) of the qualitative variable are represented by bars, where the height of each bar is either the class frequency, class relative frequency, or class percentage. Pie chart: The categories (classes) of the qualitative variable are represented by slices of a pie (circle). The size of each slice is proportional to the class relative frequency. Pareto diagram: A bar graph with the categories (classes) of the qualitative variable (i.e., the bars) arranged by height in descending order from left to right. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 16 Thinking Challenge You’re an analyst for IRI. You want to show the market shares held by Web browsers in 2016. Construct a bar graph, pie chart, & Pareto diagram to describe the data. Browser Firefox Internet Explorer Safari Others ALWAYS LEARNING Mkt. Share (\%) 14 81 4 1 Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 17 Market Share (\%) Bar Graph Solution* 100\% 80\% 60\% 40\% 20\% 0\% Firefox Internet Explorer Safari Others Browser ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 18 Pie Chart Solution* Market Share Firefox, 14\% Safari, 4\% Others, 1\% Internet Explorer, 81\% ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 19 Market Share (\%) Pareto Diagram Solution* 100\% 80\% 60\% 40\% 20\% 0\% Internet Explorer Firefox Safari Others Browser ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 20 2.2 Graphical Methods for Describing Quantitative Data ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 21 Data Presentation Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart ALWAYS LEARNING Stem-&-Leaf Display Histogram Pareto Diagram Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 22 Dot Plot 1. Horizontal axis is a scale for the quantitative variable, e.g., percent. 2. The numerical value of each measurement is located on the horizontal scale by a dot. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 23 Data Presentation Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart ALWAYS LEARNING Stem-&-Leaf Display Histogram Pareto Diagram Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 24 Stem-and-Leaf Display 1. Divide each observation into stem 2 144677 value and leaf value • Stems are listed in 3 028 order in a column • Leaf value is 4 1 placed in corresponding stem row to right of bar 2. Data: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. 26 Slide - 25 Data Presentation Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart ALWAYS LEARNING Stem-&-Leaf Display Histogram Pareto Diagram Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 26 We will let Minitab determine the number of classes for us. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 27 Histogram Class 15.5 – 25.5 25.5 – 35.5 35.5 – 45.5 Count 5 Frequency Relative Frequency Percent 4 Freq. 3 5 2 3 Bars Touch 2 1 0 0 15.5 25.5 35.5 45.5 55.5 Lower Boundary ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 28 Summary Dot plot: The numerical value of each quantitative measurement in the data set is represented by a dot on a horizontal scale. When data values repeat, the dots are placed above one another vertically. Stem-and-leaf display: The numerical value of the quantitative variable is partitioned into a “stem” and a “leaf.” The possible stems are listed in order in a column. The leaf for each quantitative measurement in the data set is placed in the corresponding stem row. Leaves for observations with the same stem value are listed in increasing order horizontally. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 29 Summary Histogram: The possible numerical values of the quantitative variable are partitioned into class intervals, where each interval has the same width. These intervals form the scale of the horizontal axis. The frequency or relative frequency of observations in each class interval is determined. A horizontal bar is placed over each class interval, with height equal to either the class frequency or class relative frequency. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 30 2.3 Numerical Measures of Central Tendency ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 31 Two Characteristics The central tendency of the set of measurements–that is, the tendency of the data to cluster, or center, about certain numerical values. Central Tendency (Location) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 32 Two Characteristics The variability of the set of measurements– that is, the spread of the data. Variation (Dispersion) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 33 Mean The mean of a set of quantitative data is the sum of the measurements divided by the number of measurements contained in the data set. n xi  x  i 1 n ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 34 Summation Notation To learn how to work with the Summation Notation, read this note. To view a pre-recorded lecture based on this note, click here. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 35 Example Calculate the mean of the following six sample measurements: 10.3, 4.9, 8.9, 11.7 , 6.3 , 7.7 n x x i 1 n i 10.3  4.9  8.9  11.7  6.3  7.7  6 49.8   8.3 6 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 36 Symbols for the Sample and Population Mean In this text, we adopt a general policy of using Greek letters to represent population numerical descriptive measures and Roman letters to represent corresponding descriptive measures for the sample. The symbols for the mean are Sample mean  x Population mean   ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 37 Median 1. Measure of central tendency 2. Middle value in ordered sequence If n is odd, middle value of sequence If n is even, average of 2 middle values 3. Position of median in sequence n 1 Positioning Point  2 4. Not affected by extreme values ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 38 Median Example Odd-Sized Sample Raw Data: 24.1 22.6 21.5 23.7 22.6 Ordered: 21.5 22.6 22.6 23.7 24.1 Position: 1 2 3 4 5 Median = 22.6 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 39 Median Example Even-Sized Sample Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 Ordered: 4.9 6.3 7.7 8.9 10.3 11.7 Position: 1 2 3 4 5 6 7.7  8.9 Median   8.3 2 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 40 Skewed A data set is said to be skewed if one tail of the distribution has more extreme observations than the other tail. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 41 Shape 1. Describes how data are distributed 2. Measures of Shape Left-Skewed Mean Median ALWAYS LEARNING Symmetric Mean = Median Right-Skewed Median Mean Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 42 Mode 1. Measure of central tendency 2. Value that occurs most often 3. Not affected by extreme values 4. May be no mode or several modes 5. May be used for quantitative or qualitative data ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 43 Mode Example No Mode Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 One Mode Raw Data: 6.3 4.9 8.9 6.3 4.9 4.9 More Than 1 Mode Raw Data: 21 28 ALWAYS LEARNING 28 41 43 43 Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 44 Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. Describe the stock prices in terms of central tendency. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 45 Solution 17, 16, 21, 18, 13, 16, 12, 11 17  16  21  18  13  16  12  11 x 8  15.5 Raw Data: 17 16 21 18 13 16 12 11 Ordered: 11 12 13 16 16 17 18 21 Median = 16 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 46 Solution (cont) Mode Raw Data: 17 16 21 18 13 16 12 11 Mode = 16 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 47 Suggested Exercises Work out the following exercises from the Textbook : 2.37, 2.38, 2.41, 2.46, 2.49, 2.51, 2.55 These exercises will not be collected or graded, but let me know as questions arise. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 48 2.4 Numerical Measures of Variability ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 49 Range 1. Measure of dispersion 2. Difference between largest & smallest observations Range = xlargest – xsmallest 3. Ignores how data are distributed 7 8 9 10 7 8 9 10 Range = 10 – 7 = 3 ALWAYS LEARNING Range = 10 – 7 = 3 Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 50 Range can be misleading Let’s examine the two datasets below: Dataset 1: Values Frequencies Dataset 2: -10,000 1 0 10,000 Values 99 Frequencies Range = 10000 – (-10000) = 20000 1 -10,000 50 0 10,000 1 50 Range = 10000 – (-10000) = 20000 Which dataset is more variable? ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 51 Variance & Standard Deviation 1. Measures of dispersion 2. Most common measures 3. Consider how data are distributed 4. Show variation about mean (x or μ) x = 8.3 4 6 ALWAYS LEARNING 8 10 12 Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 52 Sample Variance Formula n s2   x i 1 i  x 2 n 1 x1  x    x2  x    2 2    xn  x  2 n 1 Standard Deviation is the positive square root of Variance. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 53 Sample Variance Formula From page 8 of the note on Summation Notation: ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 54 Sample Variance Formula Here is the sample variance formula in English: 1. Calculate the Sum of the data values 2. Calculate the Sum of Squares of the data values 3. Divide the Square of Sum by the number of data values and subtract the result from the Sum of Squares 4. Divide the result of Step 3 by (the number of data values minus 1) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 55 Symbols for Variance and Standard Deviation s2 = Sample variance s = Sample standard deviation  2 = Population variance  = Population standard deviation ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 56 Example Calculate the variance and standard deviation. 10.3, 4.9, 8.9, 11.7, 6.3, 7.7 Solution The first step is finding the mean. Which we calculated earlier to be 8.3. (10.3  8.3)  (4.9  8.3)  ...  (7.7  8.3) s  6 1 s 2  6.368 2 2 2 2 s  6.368  2.52 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 57 Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. What are the variance and standard deviation of the stock prices? ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 58 Thinking Challenge Solution Sample Variance 17 16 21 18 13 16 12 11 The mean = 15.5 (17  15.5)  (16  15.5)  ...  (11  15.5) s  8 1  11.14 2 2 2 2 s  11.14  3.337 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 59 Minitab Calculations This tutorial shows how to calculate the mean, median, range, and standard deviation of a data set, when the data structure is that of a “linear array.” What that means is the following: All of your data values are in one column of your spreadsheet, as opposed to a “frequency distribution.” It also illustrates the calculation of some other descriptive measures that we have not yet discussed. Remember to view the tutorial again after finishing slide #91. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 60 An example The following example, Psychology Final, illustrates how to calculate the mean and the standard deviation from a frequency distribution. It also includes some other calculations. We have not yet discussed the underlying concepts. As such, at this stage you should simply note that this is an example you should come back to. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 61 Suggested Exercises Work out the following exercises from the Textbook : 2.57, 2.61, 2.63, 2.68 These exercises will not be collected or graded, but let me know as questions arise. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 62 2.5 Using the Mean and Standard Deviation to Describe Data ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 63 Using the Mean and Standard Deviation to Describe Data: Chebyshev’s Rule ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 64 Interpreting Standard Deviation: Chebyshev’s Theorem x  3s x  2s x s x xs x  2s x  3s No useful information At least 3/4 of the data At least 8/9 of the data ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 65 Chebyshev’s Theorem Example Previously we found the mean closing stock price of new stock issues is 15.5 and the standard deviation is 3.34. Use this information to form an interval that will contain at least 75\% of the closing stock prices of new stock issues. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 66 Chebyshev’s Theorem Example At least 75\% of the closing stock prices of new stock issues will lie within 2 standard deviations of the mean. x = 15.5 s = 3.34 (x – 2s, x + 2s) = (15.5 – 2∙3.34, 15.5 + 2∙3.34) = (8.82, 22.18) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 67 Another Chebyshev’s Theorem Example The example, Lumber Company, illustrates a more sophisticated use of Chebyshev’s Theorem. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 68 Interpreting Standard Deviation: Empirical Rule Applies to data sets that are mound shaped and symmetric (How do you know if your data set is mound shaped and symmetric?) Approximately 68\% of the measurements lie in the interval x  s to x  s Approximately 95\% of the measurements lie in the interval x  2s to x  2s Approximately 99.7\% ... Purchase answer to see full attachment