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Statistics Numerical Representation of Data Part 3 – Measure of Position Warm-up The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. What is the minimum percentage of commuters that have commute times between 11.4 minutes and 37.4 minutes? Warm-up The average age of U.S astronaut candidates has been 34 years, but the ages have ranged from 26 to 46. What would be the approximate standard deviation? Warm-up If the SATs have a mean of 1500 with a standard deviation of 300, would a score of 2200 be considered an unusual score? Can you determine the approximate percentile score of 2200? Warm-up Compare the variability of the heights and weights of men. Which is more variable? Men have an mean height of 69 in. with a standard deviation of 2.5 in and a mean weight of 172.6 lbs with a standard deviation of 26.3 lbs. Warm-up Suppose that the height of college males has a bell shaped distribution with a mean of 70 inches and a standard deviation of 2 inches. Approximately what percentage of college males are between 66 and 74 inches? a) 68% b) 90% c) 95% d) 99.7% e) 100% Agenda Warm-up Homework Review Lesson Objectives Determine the quartiles of a data set Determine the interquartile range of a data set and determine outliers Create a box-and-whisker plot Interpret other fractiles such as percentiles Determine and interpret the standard score (zscore) Summary Homework Quartiles Fractiles are numbers that partition (divide) an ordered data set into equal parts. Quartiles approximately divide an ordered data set into four equal parts. First quartile, Q1: About one quarter of the data fall on or below Q1. Second quartile, Q2: About one half of the data fall on or below Q2 (median). Third quartile, Q3: About three quarters of the data fall on or below Q3. Example: Finding Quartiles The test scores of 15 employees enrolled in a CPR training course are listed. Find the first, second, and third quartiles of the test scores. 13 9 18 15 14 21 7 10 11 20 5 18 37 16 17 Q2 divides the data set into two halves. Lower half Upper half 5 7 9 10 11 13 14 15 16 17 18 18 20 21 37 Q2 Solution: Finding Quartiles The first and third quartiles are the medians of the lower and upper halves of the data set. Lower half Upper half 5 7 9 10 11 13 14 15 16 17 18 18 20 21 37 Q1 Q2 Q3 About one fourth of the employees scored 10 or less, about one half scored 15 or less; and about three fourths scored 18 or less. Interquartile Range Interquartile Range (IQR) The difference between the third and first quartiles. IQR = Q3 – Q1 Example: Finding the Interquartile Range Find the interquartile range of the test scores. Recall Q1 = 10, Q2 = 15, and Q3 = 18 Solution: • IQR = Q3 – Q1 = 18 – 10 = 8 The test scores in the middle portion of the data set vary by at most 8 points. Box-and-Whisker Plot Box-and-whisker plot Exploratory data analysis tool. Highlights important features of a data set. Requires (five-number summary): Minimum entry First quartile Q1 Median Q2 Third quartile Q3 Maximum entry Drawing a Box-and-Whisker Plot 1. 2. 3. 4. 5. Find the five-number summary of the data set. Construct a horizontal scale that spans the range of the data. Plot the five numbers above the horizontal scale. Draw a box above the horizontal scale from Q1 to Q3 and draw a vertical line in the box at Q2. Draw whiskers from the box to the minimum and maximum entries. Box Whisker Minimum entry Whisker Q1 Median, Q2 Q3 Maximum entry Example: Drawing a Box-andWhisker Plot Draw a box-and-whisker plot that represents the 15 test scores. Recall Min = 5 Q1 = 10 Q2 = 15 Q3 = 18 Max = 37 Solution: 5 10 15 18 37 About half the scores are between 10 and 18. By looking at the length of the right whisker, you can conclude 37 is a possible outlier. Outliers A outlier can be determined by the following: Any value that is more than Q3 + 1.5IQR Or Any value that is less than Q1 – 1.5IQR Outliers Is 37 an outlier? IQR = 18-10 = 8 1.5IQR = 1.5 x 8 = 12 Q3 + 1.5IQR = 18 + 12 = 30 Since 37 > 30, 37 may be considered an outlier. How to Interpret a Box and Whisker Plot Here is how to read a boxplot. The median is indicated by the vertical line that runs down the center of the box. Additionally, boxplots display two common measures of the variability or spread in a data set. Range. If you are interested in the spread of all the data, it is represented on a boxplot by the horizontal distance between the smallest value and the largest value, including any outliers. If you ignore outliers, the range is illustrated by the distance between the opposite ends of the whiskers. Interquartile range (IQR). The middle half of a data set falls within the interquartile range. In a boxplot, the interquartile range is represented by the width of the box (Q3 minus Q1). How to Interpret a Box and Whisker Plot How to Interpret a Box and Whisker Plot Percentiles and Other Fractiles Fractiles Summary Quartiles Divides data into 4 equal Q1, Q2, Q3 parts Divides data into 10 equal D1, D2, D3,…, D9 parts Deciles Percentiles Divides data into 100 equal parts Symbols P1, P2, P3,…, P99 Example: Interpreting Percentiles The ogive represents the cumulative frequency distribution for SAT test scores of college-bound students in a recent year. What test score represents the 72nd percentile? How should you interpret this? (Source: College Board Online) Solution: Interpreting Percentiles The 72nd percentile corresponds to a test score of 1700. This means that 72% of the students had an SAT score of 1700 or less. The Standard Score Standard Score (z-score) Represents the number of standard deviations a given value x falls from the mean μ. value - mean x z standard deviation A z-score of greater than +2 or less than -2 would be considered an unusual value. Standard Scores – Example 1 In 2007, Forest Whitaker won the Best Actor Oscar at age 45 for his role in the movie The Last King of Scotland. Helen Mirren won the Best Actress Oscar at age 61 for her role in The Queen. The mean age of all best actor winners is 43.7, with a standard deviation of 8.8. The mean age of all best actress winners is 36, with a standard deviation of 11.5. Find the z-score that corresponds to the age for each actor or actress. Then compare your results. Standard Scores – Example 1 Forest Whitaker x 45 43.7 z 0.15 8.8 • Helen Mirren z x 61 36 2.17 11.5 0.15 standard deviations above the mean 2.17 standard deviations above the mean Solution: Comparing z-Scores Standard Scores – Example 1 Data from Sets Different Data Sets z = 0.15 z = 2.17 The z-score corresponding to the age of Helen Mirren is more than two standard deviations from the mean, so it is considered unusual. Compared to other Best Actress winners, she is relatively older, whereas the age of Forest Whitaker is only slightly higher than the average age of other Best Actor winners. Standard Score – Example 2 It call also be used to compare: A student received a 75 on a statistics test that has a mean of 85 and a standard deviation of 7. The same student had a 27 on a psychology test that had a mean of 30 and a standard deviation of 2.5. On which test did the student perform better? Standard Score – Example 2 (cont.) Calculated the z-score for each test: Z= (75 – 85)/7 = -1.43 Z = (27 – 30)/2.5 = -1.20 The student performed better on the psychology test as the z-score was greater. Summary Determined the quartiles of a data set Determined the interquartile range of a data set Created a box-and-whisker plot Interpreted other fractiles such as percentiles Determined and interpreted the standard score (z-score) Homework Pg 100 – 104 # 1-39 Odd