Ch 2.5

Prof. Yeung Ch 2.5: Measures of Position MA 103 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, Q 1 – About one quarter of the data fall on or below Q 1 Second quartile, Q 2 – About one half of the data fall on or below Q 2 (median) Third quartile, Q 3 – About three quarters of the data fall on or below Q 3 . Example 1: The number of nuclear power plants in the top 15 nuclear power-producing countries in the world are listed. Find the first, second, and third quartiles of the data set. 7 18 11 6 59 17 18 54 104 20 31 8 10 15 19 6 7 8 10 11 15 17 18 18 19 20 31 54 59 104 Lower set of the data Upper set of the data Q 1 Q 2 Q 3 Q 1 = 10 Q 2 (Median) = 18 Q 3 = 31 Interquartile Range (IQR) – The difference between the third and first quartiles. IQR = Q 3 – Q 1 Example 2: Find the interquartile range of the data set. 7 18 11 6 59 17 18 54 104 20 31 8 10 15 19 IQR= 31-10=21 Box-and-whisker plot : Exploratory data analysis tool. Highlights important features of a data set. Requires ( five-number summary ): Minimum entry First quartile Q 1 Median Q 2 Third quartile Q 3 Maximum entry Drawing a Box-and-Whisker Plot: 1. Find the five-number summary of the data set. 2. Construct a horizontal scale that spans the range of the data. 3. Plot the five numbers above the horizontal scale. 4. Draw a box above the horizontal scale from Q 1 to Q 3 and draw a vertical line in the box at Q 2 . IQR 1

Example 3: Draw a box-and-whisker plot that represents the data set. 7 18 11 6 59 17 18 54 104 20 31 8 10 15 19 Min # = 6 Q 1 = 10 Q 2 (Median) = 18 Q 3 = 31 Max # = 104 Example 4: The ogive represents the cumulative frequency distribution for SAT test scores of college-bound students in a recent year. What test score represents the 62 nd percentile? How should you interpret this? (Source: College Board) The test score that represents the 62 nd percentile is 1600. Standard Score ( z- score) – Represents the number of standard deviations a given value x falls from the mean μ . z = x − μ σ We will always round the z-score to two decimal places using the normal rounding rule. Example 5: In 2009, Heath Ledger won the Oscar for Best Supporting Actor at age 29 for his role in the movie The Dark Knight. Penelope Cruz won the Oscar for Best Supporting Actress at age 34 for her role in Vicky Cristina Barcelona . The mean age of all Best Supporting Actor winners is 49.5, with a standard deviation of 13.8. The mean age of all Best Supporting Actress winners is 39.9, with a standard deviation of 14.0. Find the z - scores that correspond to the ages of Ledger and Cruz. Then compare your results. 25% of data 25% of data v 25% of data 25% of data 6 104 Q 2 =18 Q 3 =31 Q 1 = 10 Actress Cruz, x 2 = 34 μ = 39.9 σ = 14 z= 34 − 39.9 14 = -0.4214285714= - 0.42 Actor Ledger, x 1 = 29 μ = 49.5 σ = 13.8 z ( 29 − 49.5 ) 13.8 = -1.485507246= - 1.49 The z-score for Ledger is closer to the unusual scores than the z- score for Cruz, therefore Ledger is more impressive to receive this award than Cruz. 2