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Day 11.4 – Box and Whisker Plots and Percentiles

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Day 11.4 – Box and Whisker Plots and Percentiles Day 11.4 – Box and Whisker Plots and Percentiles In a previous lesson, we learned that the median divides a set a data into 2 equal parts. Sometimes it is necessary to divide the data into smaller more precise parts. We can divide data into 4 equal parts called quartiles. The following diagram illustrates how this works: Minimum 1st Quartile 2nd Quartile 3st Quartile Maximum (median of (median) (median of lower half) upper half) Example: Find the median, first quartile and third quartile of the following data: {43, 42, 73, 74, 78, 82, 86, 80, 80, 87, 97, 91, 91} Step 1: Arrange in order- lowest to highest. Step 2: Find and label the median. Step 3: Find and label the median of the lower half. Label this 1st Quartile. Step 4: Find the median of the upper half. Label this appropriately as the 3rd Quartile. The five number summary of a set of data includes the minimum, the 1st quartile, the median, the 3rd quartile and the maximum. The minimum is the least value and the maximum is the greatest value. Find the five number summary for the data below: 125 , 140, 80, 135, 126, 140, 350 The five number summary is min _____, 1st Q ______, median ______, 3rd Q ______, max _______ Percentiles are another way to divide data into even more precise groups. You have all seen percentiles when your guidance councilor talked to you about your performance on standardized tests. Percentiles separate data sets into 100 equal parts. The percentile rank of a score tells us the percent of values in the set less than or equal to that member. The median describes the th percentile. The first quartile describes the th percentile. The third quartile describes the th percentile. Example: Of 25 test scores, eight are less than or equal to 75. What is the percentile rank of a test score of 75? Write a ratio of the number of score less than or equal to 75 compared to the total number of test scores Find the percentile rank of the following using the 20-item data set. 63, 64, 65, 70, 74, 75, 80, 80, 83, 84, 85, 87, 88, 89, 90, 91, 97, 98, 99, 100 Percentile rank of 70: How many data points are less than or equal to 70 ? Put this number over the total number of data points and change to a percent. Percentile rank of 80 How many data points are less than or equal to 80 ? Put this number over the total number of data points and change to a percent. Percentile rank of 90: How many data points are less than or equal to 90 ? Put this number over the number of data points and change to a percent. 1. 16 students received the following scores on a math quiz. 60, 62, 62, 65, 70, 74, 74, 76, 78, 82, 85, 94, 96, 98, 98, 99 Find the median __________ Find the first quartile __________ Find the third quartile _____________ What is the percentile rank of 94? In fraction form ______ in decimal form ______ in percent form ____ What is the percentile rank of 98? In fraction form ______ in decimal form ______ in percent form ____ What is the percentile rank of 74? In fraction form ______ in decimal form ______ in percent form ____ A Box and Whisker Plot is a graph that describes data using the five number summary of a set. This plot is useful for comparing two or more data sets. The Box and Whisker Plot shows how the data for each set are distributed and what the extreme values are. Terms: median: the middle piece of data (50%) first or lower quartile : middle of lower half of data (25%) third or upper quartile: middle of upper half of data (75%) interquartile range: difference between the third and first quartile DATA MUST BE PUT INTO NUMERICAL ORDER BEFORE A BOX AND WHISKER PLOT CAN BE CONSTRUCTED Steps in constructing a Box and Whisker Plot 1. Arrange the values in numerical order 2. Draw a number line to include the lowest and highest value in the data set 3. Calculate the median, the lower quartile and the upper quartile 4. Place dots above the number line to mark the 5 values : lowest value, lower quartile, median, upper quartile, and highest value Draw a box with the vertical ends passing through the lower and upper quartiles. Draw a vertical line in box through the median. Connect the two extreme values to the box with lines called whiskers. median Lower quartile Upper quartile Lowest value Highest value * * * * * Example 1: Data: 2, 2, 5, 5, 5, 6, 8, 5, 8, 9, 10, 12, 7, 8, 11 1. Arrange the data in order: 2. Draw number line to include 2 to 12 3. Determine the median = lower quartile = upper quartile = 4. Mark points the 5 data points 5. Draw box and whiskers. 6. What is the interquartile range? Example 2: Make a box and whisker plot from the following data number of frequency shoes Low score ________ 4 3 High score ________ 5 4 Median ___________ 6 1 Lower quartile __________ 7 1 Upper quartile ___________ 8 3 9 1 10 2 11 1 15 1 Example 3 : The following box and whisker plots are student test scores from 2 different math sections. Section B * * * * * Section A * * * * * 0 10 20 30 40 50 60 70 80 90 100 1. In which section was the student who had the highest test score? ________ 2. The student who had the lowest score? _________ 3. Which students had the higher median score? _________ 4. Which section do you think had the higher mean test score? Explain 5. Which section did better on the test? Explain. Example 4: A movie theater recorded the number of tickets sold daily for a popular movie during the month of June. The box-and-whisker plot shown below represents the data for the number of tickets sold, in hundreds. Which conclusion can be made using this plot? (1) The second quartile is 600. (2) The mean of the attendance is 400. (3) The range of the attendance is 300 to 600. (4) Twenty-five percent of the attendance is between 300 and 400. Day 11.4 Homework 1. Make a box and whisker plot given the data on the number of points scored by Jones High School football team. 12, 20, 19, 15, 24, 26, 27, 23, 21, 18, 14, 29 low score : high score: median:Lis lower quartile : upper quartile : 2. Rent costs are compared in this box and whisker plot. What is the median cost? ________ * * * * * 1100 300 500 700 900 3. Which value is at the lower quartile in this box and whisker plot? ______________ * * * * * 0 10 20 30 40 50 60 70 4. Graph a box and whisker plot using the following information: smallest data = 12 largest data = 20 lower quartile = 15 upper quartile = 19 median = 18 5. The box and whisker plot shows the weights of 20 students. What percent of the students weighed between 110 and 140? a) 25% b) 50% c) 75% d) 100% * * * * * 110 120 130 140 150 What does the heaviest person weigh? ______________ 6. The box-and-whisker plots below show the normal precipitation (in inches) each month in Dallas and in Houston Texas. 1 2 3 4 5 6 Dallas * * * * * * * * * * Houston For how many months is Houston’s precipitation less than 3.5 inches? _____________ Compare the precipitation of Dallas with that of Houston. ______________________ ___________________________________________________________________ 7. The accompanying box-and-whisker plots can be used to compare the annual incomes of three professions. Based on the box-and-whisker plots, which statement is true? (1) The median income for nuclear engineers is greater than the income of all musicians. (2) The median income for police officers and musicians is the same. (3) All nuclear engineers earn more than all police officers. (4) A musician will eventually earn more than a police officer. 8. The box-and-whisker plot below represents students' scores on a recent English test. What is the value of the upper quartile? (1) 68 (3) 84 (2) 76 (4) 94 .
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