Descriptive Statistics Lecture 02: Tabular and Graphical Presentation of Data and Measures of Locations 2/17/2019

Presentation of Qualitative Variables The simplest way of presenting/summarizing a qualitative variable is by using a frequency table, which shows the frequency of occurrence of each of the different categories. Such a table could also include the relative frequency, which indicates the proportion or percentage of occurrence of each of the categories. The frequency table could then be pictorially represented by a bar graph or a pie diagram. 2/17/2019

An Example A manufacturer of jeans has plants in California (CA), Arizona (AZ), and Texas (TX). A sample of 25 pairs of jeans was randomly selected from a computerized database, and the state in which each was produced was recorded. The data are as follows: CA AZ AZ TX CA CA CA TX TX TX AZ AZ CA AZ TX CA AZ TX TX TX CA AZ AZ CA CA Quite uninformative at this stage! Need to summarize to reveal information. 2/17/2019

The Frequency Table 2/17/2019

The Bar Chart Frequency 10 5 CA AZ TX 2/17/2019

Example … continued By looking at this frequency table and bar graph, one is able to obtain the information that there seems to be equal proportions of pairs of jeans being manufactured in the three states. Frequency table and bar graph certainly more informative than the raw presentation of the sample data. Another method of pictorial presentation of qualitative data is by using the pie diagram. In this case a pie is divided into the categories with a given category’s angle being equal to 360 degrees times the relative frequency of occurrence of that category. 2/17/2019

Pie Diagram CA Angles (in degrees): CA=(360)(.36)=129.6 AZ TX Angles (in degrees): CA=(360)(.36)=129.6 AZ=(360)(.32)=115.2 TX=(360)(.32)=115.2 129.6o 115.2o 115.2o 2/17/2019

Pie Chart from Minitab 2/17/2019

Presentation of Quantitative Variables When the quantitative variable is discrete (such as counts), a frequency table and a bar graph could also be used for summarizing it. Only difference is that the values of the variables could not be reshuffled in the graph, in contrast to when the variable is categorical or qualitative. For example suppose that we asked a sample 20 students about the number of siblings in their family. The sample data might be: 4, 1, 6, 2, 2, 3, 4, 1, 2, 2, 3, 7, 2, 1, 1, 5, 3, 4, 6, 3 2/17/2019

Its Bar Graph is 2/17/2019

An Example of a Real Data Set: Poverty versus PACT in SC Lunch ActualLang ActualMath 59 32 38 46 26 30 90 63 67 29 17 24 41 24 26 51 30 41 41 25 30 43 32 36 70 33 36 93 50 66 84 50 66 64 27 32 52 36 43 50 31 43 53 28 35 78 36 41 57 31 42 51 39 42 55 41 53 60 37 45 96 46 66 75 34 45 60 29 36 71 43 53 68 42 51 76 47 52 82 49 55 73 30 41 31 24 30 75 45 57 57 29 40 80 51 63 54 30 44 67 28 33 76 45 50 87 61 61 54 27 33 60 32 41 35 26 35 51 29 36 50 35 42 43 23 26 66 32 44 86 63 75 54 25 33 87 60 69 49 29 37 46 38 43 50 38 44 57 40 50 90 60 75 26 17 20 47 23 27 53 37 39 58 34 43 16 13 15 74 48 54 77 43 55 94 41 62 88 49 62 78 50 59 79 46 58 61 41 47 45 26 34 87 49 62 68 36 52 76 45 56 32 22 31 63 39 53 33 20 26 64 44 53 39 20 22 37 21 27 47 23 30 40 29 41 43 25 27 37 24 31 64 37 43 59 36 45 70 32 41 55 37 46 90 38 47 45 32 35 31 25 24 35 29 32 15 14 18 2/17/2019

Frequency Tables and Histograms Consider the variable “Lunch,” which represents the percentage of students in the school district whose lunches are not free. The higher the value of this variable, the richer the district. n = Number of Observations = 86 LV = Lowest Value = 15 HV = Highest Value = 96 Let us construct a frequency table with classes: [10,20), [20,30), [30,40), …, [90,100) 2/17/2019

Frequency Table for Variable “Lunch” 2/17/2019

Frequency Histogram 2/17/2019

Stem-and-Leaf Plots An important tool for presenting quantitative data when the sample size is not too large is via a stem-and-leaf plot. By using this method, there is usually no loss of information in that the exact values of the observations could be recovered (in contrast to a frequency table for continuous data). Basic idea: To divide each observation into a stem and a leaf. The stems will serve as the ‘body of the plant’ while the leaves will serve as the ‘branches or leaves’ of the plant. An illustration makes the idea transparent. 2/17/2019

An Example A random sample of 30 subjects from the 1910 subjects in the blood pressure data set was selected. We present here the systolic blood pressures of these 30 subjects. 30 Systolic Blood Pressures: 122 135 110 126 100 110 110 126 94 124 108 110 92 98 118 110 102 108 126 104 110 120 110 118 100 110 120 100 120 92 Lowest Value = 92, Highest Value = 135 Stems: 9,10, 11, 12, 13 Leaves: Ones Digit 2/17/2019

Stem-and-Leaf Plot 9 | 224 9 | 8 10 | 00024 10 | 88 11 | 00000000 9 | 224 9 | 8 10 | 00024 10 | 88 11 | 00000000 11 | 88 12 | 00024 12 | 666 13 | 13 | 5 2/17/2019

Stem-and-Leaf … continued In this stem-and-leaf plot, because there will only be 5 stems if we use 9, 10, 11, 12, 13, we decided to subdivide each stem into two parts corresponding to leaf values <= 4, and those >= 5. Such a procedure usually produces better looking distributions. Looking at this stem-and-leaf plot, notice that many of the observations are in the range of 100-126. The exact values could be recovered from this plot. By arranging the leaves in ascending order, the plot also becomes more informative. 2/17/2019

Comparative Stem-and-Leaf Plots When comparing the distributions of two groups (e.g., when classified according to GENDER), side-by-side stem-and-leaf plots (also side-by-side histograms) could be used. To illustrate, consider 30 observations from the blood pressure data set with Gender and Systolic Blood Pressure being the observed variables. For the males (Sex = 0): 122, 120, 130, 110, 134, 136, 142, 100, 120, 162, 126, 132, 124, 130 For the females (Sex = 1): 132, 94, 104, 100, 130, 110, 102, 110, 130, 92, 125, 108, 100, 130, 100, 100 2/17/2019

Comparing Male/Female Systolic Blood Pressures 2/17/2019

Scatterplots: Studying Relationship Between Poverty and Math Question: What kind of relationship is there between Lunch and PACT Math Scores? 2/17/2019

Numerical Summary Measures Overview Why do we need numerical summary measures? Measures of Location Measures of Variation Measures of Position Box Plots 2/17/2019

Why we Need Summary Measures? “A picture is worth a thousand words, but beauty is always in the eyes of the beholder!” Graphs or pictures sometimes unwieldy Usually wants a small set of numbers that could provide the important features of the data set When making decisions, objectivity is enhanced when they are based on numbers! Numerical summaries and tabular/graphical presentations complement each other 2/17/2019

The Setting In defining and illustrating our summary measures, assume that we have sample data Sample Data: X1, X2, X3, …, Xn Sample Size: n These summary measures are thus (sample) statistics. If instead they are based on the population values, they will be (population) parameters. 2/17/2019

Measures of Location or Center These are summary measures that provide information on the “center” of the data set Usually, these measures of location are where the observations cluster, but not always In layman’s terms, these measures are what we associate with “averages” Will discuss two measures: sample mean and sample median 2/17/2019

Sample Mean or Arithmetic Average The sample mean equals the sum of the observations divided by the number of observations. It is defined symbolically via 2/17/2019

Properties of the Sample Mean “Center of Gravity” Sum of the deviations of the observations from the mean is always zero (barring rounding errors) Sample mean could however be affected drastically by extreme or outliers The sample mean is very conducive to mathematical analysis compared to other measures of location 2/17/2019

Illustration Consider the systolic blood pressure data set considered in Lecture 01 Sample Size = n = 30 Data: 122, 135, 110, 126, 100, 110, 110, 126, 94, 124, 108, 110, 92, 98, 118, 110, 102, 108, 126, 104, 110, 120, 110, 118, 100, 110, 120, 100, 120, 92 2/17/2019

Sample Mean Computation This value of 111.1 could be interpreted as the balancing point of the 30 systolic blood pressure observations. Locating this in the histogram we have: 2/17/2019

Sample Mean in Histogram 2/17/2019

Sample Median Sample median (M) = value that divides the arranged/ordered data set into two equal parts. At least 50% are <= M and at least 50% are >= M Not sensitive to outliers but harder to deal with mathematically Appropriate when histogram is left or right-skewed Better to present both mean and median in practice 2/17/2019

Illustration of Computation of Median Consider again the blood pressure data earlier. n=30: an even number. Median will be the average of the 15th and 16th observations in arranged data. Arranged data: 92, 92, 94, 98, 100, 100, 100, 102, 104, 108, 108, 110, 110, 110, 110, 110, 110, 110, 110, 118, 118, 120, 120, 120, 122, 124, 126, 126, 126, 135 2/17/2019

Continued ... The sample median is the average of 110 and 110, which are the 15th and 16th observations in the arranged data. The median equals 110. Note that it is very close to the sample mean value of 111.1 This closeness is because of the near symmetry of the distribution 2/17/2019

Relative Positions of Mean and Median For symmetric distributions, the mean and the median coincide. For right-skewed distributions, the mean tends to be larger than the median (mean pulled up by the large extreme values) For left-skewed distributions, the mean tends to be smaller than the median (mean pulled down by the small extreme values) 2/17/2019