28 CHAPTER 2 Summarizing and Graphing Data

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1 8 CHAPTER Summarizing and Graphing Data. The two requested histograms are given below. They give very different visual images of the shape of the distribution. An outlier can have a significant effect on the histogram. WITH THE OUTLIER weight (lbs) WITHOUT THE OUTLIER weight (lbs) Statistical Graphics 1. The dotplot permits identification of each original value and is easier to construct. The dotplot gives an accurate visual impression of the proportion of the data within any selected range of values; while the polygon is limited to impressions concerning the specified classes (and only the heights at the class midpoints, and not the areas under the lines, give an accurate visual impression of those proportions).. A scatterplot requires paired data from two quantitative variables typically either two pieces of data from each experimental unit (e.g., a child s height and weight), or data from two different sets for which each value from one set may be appropriately associated with a value from the second set (e.g., the weight of the male child and the weight of the female child from mixed-gender twins). The scatterplot can reveal the nature of the relationship between the two variables. 3. Using relative frequencies allows direct comparison of the two polygons. When two sets of data have different sample sizes, the larger data set will naturally have higher frequencies and direct comparison of the heights of the two polygons does not give meaningful information. 4. Since categories in a Pareto chart are ordered according to, Pareto charts clearly show the relative positions of the categories under investigation. In addition, the Pareto chart is based on height and the pie chart is based on area and it is easier to compare heights than areas.

2 Statistical Graphics SECTION The dotplot is given below. The Strontium-9 levels appear to have a spread-out normal distribution, a wide range of values clustered around and occurring with less at the extremes. POST-1979 BABY TEETH 13 Strontium-9 (millibecquerels) The stemplot is given below. The Strontium-9 levels appear to have a normal distribution clustered around. Strontium-9 (mbq) The polygon is given below at the left. NOTE: The frequencies are plotted at the class midpoints, which are not integer values. The polygon must begin and end at zero at the midpoints of the adjoining classes that contain no data values. POLYGON FOR POST-1979 BABY TEETH OGIVE FOR POST-1979 BABY TEETH cumulative Strontium-9 (millibecquerels) Strontium-9 (millibecquerels) The ogive is given above at the right. Using the figure: move up from on the horizontal scale to intersect the graph, then move left to intersect the vertical scale at. This indicates there were approximately data values which would have been recorded as being below, which agrees with the actual data values. NOTE: Ogives always begin on the vertical axis at zero and end at n, the total number of data values. All cumulative values are plotted at the upper class boundaries.

3 3 CHAPTER Summarizing and Graphing Data 9. The stemplot is given below. The weights appear to be approximately normally distributed, except perhaps for the necessary lower truncation at zero. weight (pounds) The dotplot is given below. The weights appear to be approximately normally distributed, except perhaps for the presence of a few high values. DISCARDED PLASTIC weight (pounds) The ogive is given below at the left. Using the figure: move up from 4 on the horizontal scale to intersect the graph, then move left to intersect the vertical scale at 59. This indicates there were approximately 59 data values which would have been recorded as being below 4, which agrees with the actual data values. NOTE: Ogives always begin on the vertical axis at zero and end at n, the total number of data values. All cumulative values are plotted at the upper class boundaries. OGIVE FOR DISCARDED PLASTIC POLYGON FOR DISCARDED PLASTIC 7 6 cumulative weight (pounds) weight (pounds) The polygon is given above at the right. NOTE: The frequencies are plotted at the class midpoints, which have one more decimal place than the original data. The polygon must begin and end at zero at the midpoints of the adjoining classes that contain no data values.

4 Statistical Graphics SECTION The Pareto chart is given below. 4 PARETO CHART FOR UNDERGRADUATE ENROLLMENTS 3 percent 1 Public Public Private Private 4-Year -Year 4-Year -Year TYPE OF INSTITUTION. The pie chart is given below. The slices of the pie may appear in any order and in any position, but their relative sizes must be as shown. The Pareto chart is more effective than the pie chart. While it is clear which bar in the Pareto chart is the tallest, it is not clear which area in the pie chart is the largest. PIE CHART FOR UNDERGRADUATE ENROLLMENTS Private -Year Private 4-Year Public 4-Year Public -Year TYPE OF INSTITUTION

5 3 CHAPTER Summarizing and Graphing Data. The pie chart is given below. The slices of the pie may appear in any order and in any position, but their relative sizes must be as shown. There were 31 total responses, and the central angle of the pie chart for each category was determined as follows. Interview: 45/31 = 36.7%, and 36.7% of 36 is 13 Resume: 97/31 = 4.1%, and 4.1% of 36 is 87 Reference Checks: 3/31 = 11.6%, and 11.6% of 36 is 4 Cover Letter: 1/31 = 11/5%, and 11.5% of 36 is 41 Interview Follow-Up: 113/31 = 9.%, and 9.% of 36 is 33 Screening Call: 85/31 = 6.9%, and 6.9% of 36 is 35 Screening Call Interview Follow-Up Reference Checks Interview Cover Letter Resume AREA OF THE MISTAKE PIE CHART FOR JOB APPLICATION MISTAKES. The Pareto Chart is given below. The Pareto chart is more effective than the pie chart. 5 PARETO CHART OF JOB APPLICATION MISTAKES Interview Resume Cover Reference Interview Screening Letter Checks Follow-Up Call AREA OF THE MISTAKE

6 Statistical Graphics SECTION The pie chart is given below at the left. The slices of the pie may appear in any order and in any position, but their relative sizes must be as shown. BLOOD GROUPS 5 B AB A O 1 5 O A B BLOOD GROUPS AB. The Pareto chart is given above at the right. 19. The Pareto chart is given below at the left. 5 CAUSES OF TRAIN DERAILMENTS CAUSES OF TRAIN DERAILMENTS Other 1 Faulty Equipment Human Error Bad Track 5 Bad Track Human Error Faulty Equipment Other. The pie chart is given above at the right. The slices of the pie may appear in any order and in any position, but their relative sizes must be as shown. 1. The simple, unmodified scatterplot is given below. There appears to be a slight tendency for cigarettes with more tar to also have more CO. Scatterplot of CO vs Tar 17 CO 1 Tar 4 6 8

7 34 CHAPTER Summarizing and Graphing Data NOTE: The above scatterplot shows only 9 data points, even though there were 5 pairs of tar/co data points in the original sample. Since the scatterplot actually shows less than half the information contained in the sample, it may not provide an accurate picture of the data. This is caused by duplicate values: the (,) and (3,) and (7,) each appear times, and the (,) pair appears times! Two modifications that adjust for this phenomenon are shown below. The scatterplot on the left inserts numbers to tell how many data points are represented by dots that indicating duplicate values. The scatterplot on the right shows the true number of dots. The same effect can also be obtained by using dots whose size is proportional to the number of duplicate values it represents. The modified scatterplots indicate that there appears to be no relationship between the amounts of tar and CO. Scatterplot of CO vs Tar Scatterplot of CO vs Tar CO CO 1 Tar Tar The scatterplot is given below. It appears that more energy is used on days when it is very cold (for heating) or very warm (for air conditioning). 45 HOME ENERGY CONSUMPTION 4 35 kwh Average Daily Temperatute (Fahrenheit) 7 8

8 Statistical Graphics SECTION The time series graph is given below. Note that the given years are not evenly spaced. TRANSISTORS PER SQUARE INCH 4 Transistors (1's) Year The time series graph is given below. The graph does not appear to show linear growth (constant slope) over the entire time period, but it does appear that there was linear growth during certain periods (e.g., since 1999). US CELL PHONE SUBSCRIPTIONS 175 subscribers (1's) Year The multiple bar graph is given on the next page. As the population increases, the numbers of marriages and divorces will automatically increase. To identify any change in marriage and divorce patterns, one needs to examine the rates. This is analogous to using percents (or relative frequencies) instead of frequencies to compare categories for two samples of different sizes. The marriage rate appears to have remained fairly constant, with a possible slight decrease in recent years. The divorce rate appears to have steadily grown, with a possible slight decrease in recent years.

9 36 CHAPTER Summarizing and Graphing Data MARRIAGE AND DIVORCE RATES M = marriage rate D = divorce rate 1 rate per M D M D M D M D M D M D M D M D M D M D M D The multiple bar graph is given below. The females consistently outnumber the males, and the numbers of both genders are gradually increasing over time. GENDERS OF STUDENTS 1 M = male F = female students (1's) M F M F M F M F M F M F M F year 7. The back-to-back stemplot is given below. The pulse rates for men appear to be lower than the pulse rates of women. PULSE RATES Women Men