Descriptive Statistics Part II: Graphical Description

Editor's Notes

• #3: In general, raw data are the data recorded in the sequence in which they are collected without any effort to identify, organize, sort, manipulate, or analyze. Any one of these four actions will convert raw data into processed data. Examples of raw data are given in this slide. Obviously, it will be difficult to obtain any useful information from a set of raw data such as the ones described above. The first step in converting raw data into processed data is to identify the values presented in the data set as shown below. This achieved by labeling the data and specifying the units…see next slide
• #4: In addition to identifying data by labels and units, the data should be arranged in such a way that will make it easier to obtain information from them. In some situations, we may have to sort the data in such a way that reveal the relative magnitudes sorted by some categories. In other situations, we may have to divide data by some categories so that we can compare the values associated with these categories. These manipulation procedures are largely judgmental and they depend on the nature of data and the type of analysis used. In Chapter 2, we discussed one way to convert raw data into processed data using numerical descriptive statistics. In this chapter, the focus will be on graphical displays to describe data.
• #5: Graphical displays can assist in converting raw data into processed data by exploring the information revealed from the data. Two of the simple graphical displays are the bar chart and the pie chart.
• #6: As shown in the bar chart, the x-axis represents the teams’ names, and the heights of the bars represent the average score per game for each team. This bar chart can be constructed manually or using Excel® spreadsheet as will be shown in the next section. Key information revealed by this bar chart: Among the eight teams displayed in the bar chart, the New England (Patriots) and the Philadelphia (Eagles) were the two teams that had the highest average points per game after 10 games of the 2010 NFL season. Baltimore (Ravens) and Pittsburgh (Steelers) had the lowest average points per game.
• #7: Step 1: Block the two columns of the Table Step 2: Go to insert and insert a Column Step 3: From the menu of column graphs, select one (say the first one) Step 4: Click The Graph you Will get the output
• #8: Key information revealed by this bar chart:   The largest cause contributing to the death of young residents is car accidents (89%) The least cause of death is fire arms.
• #9: Step 1: Block the two columns of the Table Step 2: Go to insert and insert a Pie Step 3: From the menu of Pie graphs, select one (say the first one) Step 4: Click The Graph you Will get the output
• #15: This example illustrates how simple a histogram is: The horizontal axis represents classes of student’s grade (50 to 60, 60 to 70, ….and 90 to 100). The vertical axis represents the number of students earning each class of grades
• #18: This example illustrates how simple a histogram can also be expressed in percentage instead of absolute frequency Roughly 6% Roughly 8% (4% + 4%) Roughly $55,000 or from $50,000 to $60,000 (highest bar)
• #19: In order to construct a histogram, we should form the so-called ‘frequency table’ in which we tally the data and show the values and the frequency in each class. Consider the data of driving-to-work distances in miles shown in this slide. This data was taken from a random sample of employees working in a large retail store in New York City. In order to construct a histogram or a frequency distribution of this data, we follow the five basic steps described in these slides. More elaboration on the steps is provided below Step 1: Find the minimum and the maximum values of the data set: min = 30 and max = 100. This gives you an idea about the span or the range of the entire data set, which is 70 miles (Range = max – min = 100-30 =70) Step 2: Decide on the class width that you wish to use. This will depend on number of classes or categories that you wish to use. Commonly, the minimum number of classes used is 5. Let us select a class width of 10 miles. Note that, the more classes used, the higher the resolution of data display. Step 3: In this step we form the first column of the frequency Table, which is the classes of data. The first class should begin at a value below the actual minimum value of the data set, say 25, and the upper limit of the last class should by greater than the maximum value of the data set, say 105. This is typically recommended to cover the entire range of data and avoid missing observations in the frequency count. The upper value of each class begins with the notation of ‘less than, <’ to avoid double counting of the upper values in two consecutive classes. Step 4: We form the second column of the frequency table, which is the simply the mid-point of each class. This step is significant as it indicates that each class will be represented by one point, which is the point in the middle of the class. It also indicates that the 40 observations that we began the analysis with is now reduced to few values of 7 or 8 midpoints. The next step will show that each midpoint (or class) should be associated with a weight or a frequency depending on the number of observations in the original data set that fall within the class. Step 5: We count the number of observations in each class and place the count in the third column labeled ‘frequency’. Observations belonging to each class are represented by cells of the same color in the Table below. Note that the total frequency is 40, which is the total number of observations. This step leads to constructing the histogram with the x-axis being the class values and the vertical axis being the frequency as shown in Figure 3.8.
• #22: The Table shows the relative frequencies and the percent relative frequencies for different distance-to-work categories or classes. Note that the sum of relative frequencies is 1 and the sum of percent relative frequencies is 100. Also note that using the relative frequency or percent relative frequency instead of the absolute frequency will not alter the shape of the frequency distribution, only the values on the vertical axis
• #26: A cumulative frequency curve represents another way to display the class intervals of the data. One method to construct a cumulative frequency curve is to follow the steps illustrated in the third column of the Table shown here. The first value in a cumulative frequency (CF) is the same as the frequency (1); the second CF value is the first CF value plus the second frequency value (1 + 2 =3); the third CF value is the second CF value plus the third frequency value (3 + 5 =8); and so on to reach the total of 40. We can also construct a percent cumulative frequency column as shown in the Table using the percent frequency values. Note that as a result of the cumulative sum of the percent relative frequencies, the final value is 100. When a cumulative curve is constructed, the horizontal axis should consist of the upper values of the different classes (in this example, 35, 45, 55, 65, 75, 85, 95, and 105). The Figure shows the cumulative frequency curve of this example.
• #27: A cumulative frequency curve constructed using the method above can provide direct information of the percent of data observations exhibiting values less than a certain value, xo, or p(x < xo). Note how the percent of employee driving < 75 miles is directly obtained from the curve by drawing a vertical line from the x-axis at 75 that intersect with the curve at a point determining the percent of employee, which is 62.5%.
• #29: As discussed earlier, a value of P(x < xo) is directly obtained from the cumulative curve by drawing a vertical line from the x-axis at xo that intersect with the curve at a point determining the desired percent. As shown in the Figure P(x < 20) ≈ 16%. Thus, out of the 32 teams considered, about 5 teams (32x0.16= 5.12) scored less than 20 touchdowns up to week 10 in the 2010 season. The percent of teams scoring 30 touchdowns or more can be obtained by subtracting P(x < 30) from 100. In other words: Thus, out of the 32 teams considered, about 9 teams (32x0.28 = 8.96) scored 30 touchdowns or more up to week 10 in the 2010 season.
• #32: Note: You can copy this table into Excel-Spread sheet and perform Histogram Data Analysis When software programs are used to construct a histogram, we should be aware that different programs may use different procedures to produce a frequency distribution or a cumulative frequency distribution. It is important, therefore to interpret the distributions produced in the context of the approach or the method used. In case of Excel®, a class is defined by > a to a + w where a is the lower value of a class, and w is the class width. Example: To illustrate how Excel® can be used to construct a histogram or a frequency distribution, we will use data of ceramic tile area. This is a critical parameter as nearly every tiling job requires perfect dimensions of each tile (area and thickness) for smooth installation and straight clean cuts. It is also a cost-related parameter as waste of tiles can be a potential problem when tile dimensions exhibit high variability. The data in the Table represent a sample of 100 tiles selected randomly from a tile operation producing tiles of consistent thickness and nominal dimensions of 16x16 inch, or 256 square inch area.
• #33: In order to analyze data in Excel, all data of the same kind should be arranged in a single column as shown here.
• #34: Using Excel® program, we can perform descriptive statistics as discussed in Chapter 2, and construct a histogram as will be discussed shortly. In either case, data has to be arranged in one column as shown in the next slide. It is also advisable to begin the analysis by performing numerical descriptive statistics using the procedures discussed in Chapter 2.
• #35: Interpretation of descriptive statistics should be made in view of the type of variable under consideration. In this example, the variable is the area of ceramic tile, which is a ratio level of measurement and follows the pattern of ‘nominal-the-best’. This means that the area has to be exact and not more or less than the nominal target value of 256 square inch. As can be seen here, the mean area is 255.87 square inch, the median is 255.95 square inch, and the mode is 255.5 square inch. These central measures indicate that the target value is largely met. Values of the mean, the median, and the mode are approximately equal. This is a sign that the frequency distribution will be largely symmetrical. Evaluating the variability measures reveals that ceramic tile area suffers high variability as some tiles exhibit an area of as low as 252.7 and some as high as 259.5. This difference yields a range of 6.8 square inch (259.5 – 252.7). A standard deviation of 1.272 indicates that the variability in ceramic tile area is not zero; that is despite the serious attempt during manufacturing to make all ceramic tiles of the exact same area. These observations call for constructing a frequency distribution to closely examine the extent of variability and its associated frequency. See Next Slide.
• #36: Using Excel® data analysis, a histogram or frequency distribution can be constructed as follows:   Step 1: Determine the class width by preparing a bin range. Bin numbers represent the intervals that you want the histogram tool to use for measuring the input data in the data analysis. When you use the Histogram tool, Excel® counts the number of data points (data points: Individual values plotted in a chart and represented by bars, columns, lines, pie or doughnut slices, dots, and various other shapes called data markers. Data markers of the same color constitute a data series.) in each data bin. A data point is included in a particular bin if the number is greater than the lowest bound and equal to or less than the greatest bound for the data bin (> a to a + w) where a is the lower value of a class, and w is the class width. If you omit the bin range, Excel® creates a set of evenly distributed bins between the minimum and maximum values of the input data. In this example, the bin range may begin at a convenient number such as 252 (since the minimum area is 252.7) and may end at a convenient number such as 260 (since the maximum area is 259.5).
• #37: Step 2: Go to Data Analysis Step 3: Select Histogram
• #38: Step 4: In the Histogram Tool, select the data column as the input range, and the bin-range column as bin range. Specify an output, check ‘Chart Output’ and ‘Cumulative Percentage’, and click Ok.  
• #39: The steps above produce the output shown here (after some customization). Note that the histogram and the cumulative curve are presented in the same graph. Use Excel® - Help to further manipulate the graphs or to separate the histogram from the cumulative curve. You may also wish to convert the frequency distribution to a relative frequency distribution or a percent relative frequency distribution. This will require Excel® calculations to create new columns of relative frequency and percent relative frequency. In addition, it will require using Excel® graphic capability to construct the distributions. Again, these are systematic and simple steps that you can do yourself or with the support of Excel® - Help.
• #40: You can use this option to close the gaps between the bars
• #41: Here we start with the Frequency Table produced by Excel Data Analysis
• #42: In order to accurately interpret Excel® Histogram output, it is important that you know that the bin-range values are indeed the upper boundaries of the classes. The Table here illustrates the classes corresponding to each bin-range value. One of the reasons Excel® use this approach is to allow graphing the histogram superimposed with the cumulative curve.
• #43: The frequency distribution of tile area is shown here. Using this distribution, one can mark the area of specified values of tile area and the area where tiles will be rejected for failing to meet the specifications. In this sense, the histogram will provide a rough idea on the quantity of tiles that fall within the specified limits and those that will be rejected for not meeting specifications (any value below 255 or above 257). In order to get quantitative estimates of the quantities of tiles that fall within the specified limits and those that will be rejected for not meeting specifications, you can roughly count the heights of the bars or use the cumulative frequency curve should be used as discussed in the next slide.
• #44: As shown in this cumulative frequency distribution, the point of intersection of the vertical line at the area value of 255 square inch, and the horizontal line is at 25% cumulative frequency. This gives the percent of tiles of less than or equal 255 square inch. The point of intersection of the vertical line at the area value of 257 square inch, and the horizontal line is at 85% cumulative frequency. This gives the percent of tiles of less than or equal 257 square inch. The required percent is then calculated by the difference 85-25 = 60%
• #45: Mean 10669 Median 9000 Mode 8000 Standard Deviation 4458 Sample Variance 19875696 Range 17000 Minimum 4000 Maximum 21000 Count 30
• #47: (a) What is the price of the majority of cars in the used car lot? ($6,000 to $8,000) or midpoint $7,000 (b) What is the range of prices of used cars in the used car lot? From about $4000 to about $22,000 What percent of cars of a price less than $10,000? 53.3% What is the percent of cars of a price greater than $20,000? About 6.7%
• #48: When two or more frequency distributions are compared, it is critical that they should be constructed of identical class width. Violation of this requirement will often result in meaningless comparison and misleading information. This example below illustrates this point. Note: Cotinine is an alkaloid found in tobacco and is also a metabolite of nicotine. The word "cotinine" is an anagram of ‘nicotine’ and it is a chemical that forms inside the body following exposure to nicotine, an ingredient in all tobacco products and a component of Environmental Tobacco Smoke (ETS). Following nicotine exposures, cotinine can usually be detected in blood for at least 1 or 2 days
• #49: In order to compare these two data sets, it will be useful to perform numerical descriptive statistics first before proceeding with constructing histograms and cumulative frequency curves. See next slide.
• #50: Ask Students to make effort to interpret this output This Table lists the values of descriptive statistics for the unexposed and the exposed children. These values reveal the following information: - The mean value of blood-cotinine level of unexposed children is 0.545 ng/ml and that of exposed children is 2.079 ng/ml. - For each data set, the mean ≈ the mode ≈ the median. This is typically a result of data symmetry or a normal distribution. Variability measures such as the range, the standard deviation, and the variance for exposed children are substantially greater than the corresponding measures for unexposed children.
• #51: The histograms and cumulative frequency curves constructed for the two data sets are shown here. These were constructed using Excel® histogram data analysis tool as described in the previous section. Note that the bin range for each data set was the same (class width = 0.25 ng/ml) despite the differences in the data range of each set. This is to allow meaningful comparison between the frequency distributions of the two data sets.
• #52: The output of Excel® histogram tool analysis can also be customized to provide better exploration of the data. Note that the graphs for the data of unexposed and exposed children were superimposed to allow meaningful comparison. This can be achieved using Excel® graphing methods.
• #54: In general, common shapes that a frequency distribution may take are as follows:   Symmetrical shape Positively skewed Negatively skewed Steep-positive kurtosis Flat-negative kurtosis Bimodal or multimodal   - A symmetrical and uni-modal shape (Figure a) will have equal values of mean, median, and mode. These three statistics are located at the center of the distribution and data values are evenly spread around these values. In other words, the data values below the mean and median are a mirror image of those above. A symmetrical shape is usually generated from a large amount of data and it is to be expected for variables that follow ‘nominal-the-best’ patterns. These include data of dimensions such as thickness of wood boards or weights of humans.   - A data set that is skewed to the right is known as positively skewed. The distribution of this type of data is shown in Figure b and it is characterized by a single peak with data values are extended to the right of the peak than to the left of the peak. For this shape, the mean value is larger than the median and the skewness index, to be defined shortly, is positive. This type of distributions is often generated from data representing practical situations such as a challenging exam in which the majority of students perform poorly (low scores) and only few performing well. It also simulates large corporate salaries, where the majority of workers are making low or middle salary and only few are making top dollar. Smaller-the-better variables such as accidents on the job per month, car accidents, defects, and rejects should ideally follow a positively skewed distribution. - A data set that is skewed to the left is known as negatively skewed. The distribution of this type of data is shown in Figure c and it is characterized by a single peak with data values are extended to the left of the peak than to the right of the peak. For this shape, the mean value is smaller than the median and the skewness index, to be defined shortly, is negative. This type of distributions is often generated from data representing practical situations such as the grades of an easy exam in which the majority of students perform well (high scores) and only few performing poorly. Larger-the-better variables such as operating efficiency, economical growth rate, people income, health, and working performance indexes should ideally follow a negatively skewed distribution.   A data set that has a steep peak with values to the right and to the left are close to the peak will produce the distribution shown in Figure d. This distribution is known to have a positive kurtosis index. A data set that has an extended flat peak will produce the distribution shown in Figure e. This distribution is known to have a negative kurtosis index as will be defined shortly. A bimodal or multimodal distribution will have two or more peaks (see Figure f). This is often the case when there is a potential mix of samples coming from different populations.
• #55: Symmetrical (Mean ≈ Median ≈ Mode) Positively skewed (Mean > Median > Mode) negatively skewed (Mean < Median < Mode) (d) Bi-Modal (Mean < Median) (e) Uniform (None of the above)
• #56: The use of frequency distributions leads to the issue of how to perform numerical descriptive statistics for grouped data. In other words, if the data available is already grouped with associated weight or frequency, how can we determine their mean value and standard deviation? Note, we used the sum minus 1 since the total frequency, n is less than 30.
• #60: A small standard deviation for a set of values indicates that these values are located close to the mean. On the other hand, a large standard deviation reveals that the observations are widely scattered about the mean. The Russian mathematician P. L. Chebyshev (1821–1894) developed a theorem that allows us to determine the minimum proportion of the values that lie within a specified number of standard deviations of the mean. According to this theorem, in a frequency distribution of data, the proportion of the values that lie within k standard deviations of the mean is at least 1-1/k2, where k is any constant greater than one. This means that at least three of four values, or 75 percent, must lie between the mean plus two standard deviations and the mean minus two standard deviations. This relationship holds regardless of the shape of the distribution. Further, at least eight of nine values, or 88.9 percent, will lie between plus three standard deviations and minus three standard deviations of the mean. At least 24 of 25 values, or 96 percent, will lie between plus and minus five standard deviations of the mean.
• #62: The Chebyshev’s theorem is applied for any frequency distribution regardless its shape. In Chapter 1, the normal distribution was introduced as a bell-shaped symmetrical distribution. For this type of distribution, we can use the so-called ‘empirical rule’ to determine the proportion of the values that lie within k standard deviations of the mean as shown in the Figure here. According to this rule, 68.26 percent of the observations will lie within plus and minus one standard deviation of the mean; 95.44 percent of the observations will lie within plus and minus two standard deviations of the mean; and 99.74 percent will lie within plus and minus three standard deviations of the mean.
• #64: Students need to sketch the Normal Distribution to be able to solve this problem What percent of students will make ‘C’ grade or lower (C = 70%) C or lower is 2.28% What percent of students will make ‘B’ grade or better (B = 80%) B or better 50% What percent of students will make ‘A’ grade or better (A = 90%) A or better 2.28% What percent of students will make ‘C’ grade or better (C = 70%) C or better = 100 - 2.28% = 97.72%
• #66: (a) 68.26% (b) 99.74% (c) 2.28%
• #70: The stem and leaf plot of the area data is shown here. Note the high degree of resolution associated with the stem-and-leaf diagram, not only that we can see the extent of frequency of different classes or values, but we can also see what values are available within each class. For example, the first stem (bottom of the plot) represents the highest base value, 21 and the leaf is represented by 1, 1, 5 to represent the observations 211, 211, and 215 (light-shaded in the previous table). The last stem (top) is 11, and the leaf is 9 to represent the value 119 (dark shaded area in the previous table).  
• #84:   The coefficient of correlation (r):   It describes the strength of association between two variables (interval or ratio levels). A correlation coefficient of ‘0’ means no association between the two variables; a value of ‘1.0’ means perfect positive correlation; and a value of ‘-1.0’ means perfect negative correlation.          
• #87: In general, an outlier is a data observation that seems to not belong to the family of data under consideration. In some situations, an outlier can be very obvious and it should be eliminated from the data set or corrected. This is typically the case when we know the plausible range of the variable data. For example, student’s grade may range from 0% to 100%, an outlier would be a value of 110%. In this case, we should first question its existence as it may be a result of some bonus points that the teacher decided to give, or it may be inserted incorrectly in the database. In other cases, we may not know the reason why an outlier is presented in the data set. The key, however, is to detect it, particularly when we are dealing with a massive amount of data.