Contents 13. Graphs of Trigonometric Functions 2 Example 13.19............................... 2 Example 13.22............................... 5 1
Peterson, Technical Mathematics, 3rd edition 2 Example 13.19 Use a spreadsheet and the data for the 500 steel rods in Table 13.1 to draw a histogram and box plot, and determine the mean, median, and quartiles. Solution We have reproduced the table below. Diameter 9.97 9.98 9.99 10.00 10.01 10.02 10.03 10.04 10.05 10.06 10.07 Frequency 10 30 0 80 60 100 90 60 40 20 10 To construct a histogram, enter the data in two columns similar to Table 13.1. 1. Click on Chart Wizard to begin the process. 2. Choose Column and the sub-type shown below in Figure 13.5a. 3. Press Next and then select the Column radio button since the data is entered in columns. Then select the Series tab to identify the source data for the histogram. 4. Select Add to add a series. Notice there are three dialogue boxes, one for the name, one for the y-values and one for the x-values. 5. In the name dialogue box, type Steel Rods. Move the cursor to the dialogue box identified as Category (X) axis labels and then go to the table and click and drag over Column A to select those values. (See Figure 13.5b.) FIGURE 13.5a FIGURE 13.5b 6. Move the cursor to the dialogue box entitled Values and select the values in Column B as shown in Figure 13.5c. FIGURE 13.5c
Peterson, Technical Mathematics, 3rd edition 3 As you can see from the preview, the histogram is just about complete. 7. Complete the process by moving through the remaining steps of the process to arrive at a figure similar to the one shown in Figure 13.5d. Technically, the graph in Figure 13.5d is not a histogram because a histogram does not have any space between the bars. To remove the space, double click on one of the bars in the graph. This should open the Format Data Series window. Click on the Options tab and change the Gap width to 0. The result is shown in Figure 13.5e. FIGURE 13.5d FIGURE 13.5e To find the mean of this data, extend the table one column by adding a column representing the sum of all the entries in that row. For example, since there are 10 rods with a diameter of 9.97, then the sum of the diameters of those ten rods is 10 9.97 = 99.7. The 30 rods with a diameter of 9.98 add up to 299.4, and so on. Adding those products gives us the sum of all the diameters from which we can obtain the mean. (See Figure 13.5f.) The mean is found by first finding the sum of the frequencies in Column B (cell B15: =SUM(B4:B14)) and then dividing the total sum by the number of rods (cell C16: =C15/B15). (See Figure 13.5g.) FIGURE 13.5f FIGURE 13.5g The median is the middle number. A new column is again added to the original table. The third column, cumulative frequency, shows the total of the frequencies to that point. (See Figure 13.5h.) We can see that the median is 10.02 since the 250th data entry will occur in that row. The mode is obtained by reviewing the data. It is apparent that the diameter that occurs most frequently is 10.02. Thus 10.02 is the mode.
Peterson, Technical Mathematics, 3rd edition 4 The closest a spreadsheet can come to easily producing a box plot is a scatterplot of the five quartile points used in a box plot: the minimum, the 1st Quartile, the Mean, the 3rd Quartile, and the Maximum. Figure 13.5i shows the result. FIGURE 13.5h FIGURE 13.5i
Peterson, Technical Mathematics, 3rd edition 5 Example 13.22 Use a spreadsheet and the data for the 500 steel rods in Table 13.1 to determine its mean and standard deviation. Solution We have reproduced the table below, as Table 13.2. Table 13.2 Diameter 9.97 9.98 9.99 10.00 10.01 10.02 10.03 10.04 10.05 10.06 10.07 Frequency 10 30 0 80 60 100 90 60 40 20 10 In Example 13.19 we used the spreadsheet determine the mean for this data. We can approximate the standard deviation of grouped data using either of two formulas. Population Variance σ 2 = Σ(x i µ) 2 f i Σf i Sample Variance s 2 = Σ (x i x) 2 f i (Σf i ) 1 FIGURE 13.5j where x i is the midpoint or value of the ith class and f i is the frequency of the ith class. We will treat the 500 rods as a sample of a larger group and thus, we will use the second formula. We must add three columns to the data; one for the difference, another for the squared differences, and a third for the product of the squared differences and the frequency. (see Figure 13.5j). The sum of the products of the differences and the frequencies is shown in cell E15. The variance is calculated in cell B17. The square root of the variance, the standard deviation, is calculated in E18.