Chapter 9 Prerequisite Practice MATH NOTES ETHODS AND MEANINGS. Describing Shape (of a Data Distribution)

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1 Chapter 9 Prerequisite Practice ETHODS AND MEANINGS Describing Shape (of a Data Distribution) MATH NOTES Statisticians use the words below to describe the shape of a data distribution. uniform and symmetric double-peaked and symmetric single-peaked and symmetric single-peaked and skewed Outliers are any data values that are far away from the bulk of the data distribution. In the example at right, data values in the right-most bin are outliers. Outliers are marked on a modified boxplot with a dot. Statistics Supplement (from Core Connections Algebra 2/Integrated III) 3 CPM Educational Program

2 ETHODS AND MEANINGS MATH NOTES Interquartile Range and Boxplots Quartiles are points that divide a data set into four equal parts (and thus, the use of the prefix quar as in quarter ). One of these points is the median. The first quartile (Q1) is the median of the lower half, and the third quartile (Q3) is the median of the upper half. To find quartiles, the data set must be placed in order from smallest to largest. Note that if there are an odd number of data values, the median is not included in either half of the data set. Suppose you have the data set: 22, 43, 14, 7, 2, 32, 9, 36, and lower half st Quartile (median of the lower half) median upper half rd Quartile (median of the upper half) The interquartile rage (IQR) is the difference between the third and first quartiles. It is used to measure the spread (the variability) of the middle fifty percent of the data. The interquartile range is 34 8 = 26. A boxplot (also known as a box-and-whisker plot) displays a fivenumber summary of data: minimum, first quartile, median, third quartile, and maximum. The box contains the middle half of the data and visually displays how large the IQR is. The right segment represents the top 25% of the data and the left segment represents the bottom 25% of the data. A boxplot makes it easy to see where the data are spread out and where they are concentrated. The wider the box, the more the data are spread out. min Q1 median Q3 max Statistics Supplement (from Core Connections Algebra 2/Integrated III) 4 CPM Educational Program

3 ETHODS AND MEANINGS Describing Spread (of a Data Distribution) MATH NOTES A distribution of data can be summarized by describing its center, shape, spread, and outliers. You have learned three ways to describe the spread. Interquartile Range (IQR) The variability, or spread, in the distribution can be numerically summarized with the interquartile range (IQR). The IQR is found by subtracting the first quartile from the third quartile. The IQR is the range of the middle half of the data. IQR can represent the spread of any data distribution, even if the distribution is not symmetric or has outliers. Standard Deviation Either the interquartile range or standard deviation can be used to represent the spread if the data is symmetric and has no outliers. The standard deviation is the square root of the average of the distances to the mean, after the distances have been made positive by squaring. For example, for the data kilograms: The mean is 14 kg. The distances of each data value to the mean are 4, 2, 0, 2, 4 kg. The distances squared are 16, 4, 0, 4, 16 kg 2. The mean distance-squared is 8 kg 2. The square root is Thus the standard deviation is 2.83 kg. Range The range (maximum minus minimum) is usually not a very good way to describe the spread because it considers only the extreme values in the data, rather than how the bulk of the data is spread. Statistics Supplement (from Core Connections Algebra 2/Integrated III) 5 CPM Educational Program

4 C-51. An environmental engineering group has collected core samples from the earth to determine if they are contaminated. The samples were weighed, as follows: checksum 1034 Determine if the mean and standard deviation is an appropriate way to summarize this data. If so, find the mean and standard deviation with your calculator. If not, use the median and IQR. Consider the precision of the measurement of weight when deciding how many decimal places to use in your answer. C-52. A farmer wonders if his crops grow better in sun or in shade. He measures the amount of fruit gathered from a sample of 50 trees growing in full sun and from a sample of 50 trees growing in mostly shade. The five-number summaries, (minimum, Q1, median, Q3, maximum), follow: Amount of fruit gathered from sunny trees: (10.8, 13.3, 22.1, 58.1, 100) bushels Amount of fruit gathered from shady trees: (10.9, 18.5, 29.3, 61.5, 127) bushels a. On the same set of axes on grid paper, create a boxplot for each type of tree. Compare the center, shape, spread, and outliers of amount of fruit from sunny trees to the amount from shady trees. b. The farmer wants to summarize the amount of fruit from each type of orchard with a mean and standard deviation. Is that appropriate? Explain. C-56. Match the histogram to its corresponding boxplot. a. b. c. d. e. i. ii. iii. iv. v. Statistics Supplement (from Core Connections Algebra 2/Integrated III) 6 CPM Educational Program

5 C-60. Jet Set claims that they are more on time than National Airways because National was delayed over an hour a couple times. Arin did not believe Jet Set was more on time so he looked up the data for both airlines during the last two weeks. Below is the data he found for departure delays. Negative numbers indicate that a flight departed earlier than scheduled. National Airways: 69, 22, 3, 7, 2, 25, 7, 0, 3, 14, 4, 0, 1, 9, 2, 4, 12, 25, 65, 10 checksum 206 Jet Set: 6, 5, 33, 4, 37, 10, 23, 5, 21, 31, 2, 5, 35, 42, 19 8, 25, 15 checksum 290 a. Compare the center, shape, spread, and outliers of the data sets. Use the values you calculate to make an argument as to which airline is on time more often. Use a bin width of 10 minutes on your histograms. b. Arin wants to summarize his findings with just two numbers: the center and spread. Can he use mean and standard deviation, or should he use median and IQR? Justify your choice, then summarize the data for Arin You previously created a five number summary of the lengths of 23 newborns at the Dallas University Health Center. The measurements, in centimeters, were: 46.4, 46.9, 47.7, 48.1, 48.5, 48.5, 48.8, 49.0, 49.3, 50.0, 50.1, 50.4, 50.6, 51.1, 51.4, 51.8, 52.4, 52.5, 53.2, 53.8, 54.4, 55.1, 55.9 checksum Determine if it is appropriate to summarize the data with the mean and standard deviation. If it appropriate, justify your reasoning and find the mean and standard deviation. If it is not appropriate, explain why not, and find the median and IQR. Consider the precision of the measurements when giving the result. Statistics Supplement (from Core Connections Algebra 2/Integrated III) 7 CPM Educational Program

6 Duncan is testing two types of memory chips (W and Z) for performance and reliability. One test involves a measurement of the maximum electrical current in milliamperes required by the chip when storing and retrieving 14 different data sets. A lower required current means greater energy efficiency. Data Set: A B C D E F G H I J K L M N Chip W checksum 1873 Chip Z checksum 1868 a. When Duncan is trying to summarize the data, he finds the IQR for chip W is larger, but the standard deviation for chip Z is larger. Both are measures of spread. How can they be conflicting? Use combination histogram and boxplots in your explanation. b. Which chip should Duncan recommend to the manufacturer? Justify your choice The city of Waynesboro is trying to decide whether to initiate a composting project where each residence would be provided with a dumpster for garden and yard waste. The city manager needs some measure of assurance that the citizens will participate before launching the project, so he chooses a random sample of 25 homes and provides them with the new dumpster for yard and garden waste. After one week the contents of each dumpster is weighed (in pounds) before processing. The sorted data is shown below: Checksum a. Create a combination boxplot and histogram. Use an interval of 0 to 42 pounds on the x-axis and a bin width of 6 pounds. b. Describe the center, shape, spread and outliers. c. What is a better measure of center for this distribution the mean or median and why? d. What is a better measure of spread the standard deviation or IQR and why? e. The city can sell the compost, and engineers estimate the program will be profitable if each home averages at least 9 pounds of material. The city manager sees the mean is nearly 10 pounds and is ready to order dumpsters for every residence. What advice would you give him? Statistics Supplement (from Core Connections Algebra 2/Integrated III) 8 CPM Educational Program

7 Megan is an industrial engineer for Bowler Cola Company. She takes a random sample of cola cans from the production line each day to determine if the product meets various specifications. One of the measurements she records is the mass (in grams) of the filled cans. The following sorted data are from of a sample of 30 regular and diet cola cans. Note: The data is sorted so it is easy to work with without a statistical calculator. Regular Checksum Diet Checksum a. Find the five number summary (minimum, third quartile, median, first quartile, maximum) for each soda. b. Make a combination histogram and boxplot for each type of soda. Include the five number summary. Use an interval of 348 to 384 grams on the x-axis and a bin width of 4 grams. c. Describe the center, shape, spread, and any outliers, of each histogram. d. Compare the two samples. e. Each can is marked as containing 12 fluid ounces. Twelve ounces is about 341 grams. Why is there so much variation from 341 grams in the samples? Statistics Supplement (from Core Connections Algebra 2/Integrated III) 9 CPM Educational Program