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CHAPTER 2 Modeling Distributions of Data

Learn how to find and interpret percentiles, estimate values using cumulative relative frequency graphs, calculate z-scores, and understand the effect of transformations on distributions of data.

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CHAPTER 2 Modeling Distributions of Data

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  1. CHAPTER 2Modeling Distributions of Data 2.1Describing Location in a Distribution

  2. Describing Location in a Distribution • FIND and INTERPRET the percentile of an individual value within a distribution of data. • ESTIMATE percentiles and individual values using a cumulative relative frequency graph. • FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data. • DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data.

  3. Measuring Position: Percentiles One way to describe the location of a value in a distribution is to tell what percent of observations are less than it. The pth percentile of a distribution is the value with ppercent of the observations less than it. Example Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class? 6 7 7 2334 7 5777899 8 00123334 8 569 9 03 6 7 7 2334 7 5777899 8 00123334 8 569 9 03 Her score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below hers, Jenny is AT the 84th percentile in the class’s test score distribution. No other student in this class is AT the same percentile UNLESS they have the score.

  4. Mr. Pryor’s first test. Use the scores on the test to find the percentiles for the following students. • A. Norman who earned a 72. • __________ student(s) scored lower than Norman so ____________ and he is at the _____________ percentile. • B. Katie who scored a 93. • _________ student(s) scored lower than Katie so ____________ and she is at the _____________ percentile. • C. The two students who earned scores of 80. • _________ student(s) scored lower than an 80 so ______________ and the two students are both at the _____________ percentile. 6 7 7 2334 7 5777899 8 00123334 8 569 9 03

  5. Cumulative Relative Frequency Graphs A cumulative relative frequency graph displays the cumulative relative frequency (the percentiles) of each class of a frequency distribution. Age 45 is at the 4.5th percentile of the inauguration age distribution.

  6. Cumulative Relative Frequency Graphs Estimate the median age of this distribution. • HINT: the median is at the 50th percentile. Estimate the IQR of the distribution. • HINT: Q1 is at the 25th percentile and Q3 is at the 75th percentile.

  7. Measuring Position: z-Scores A z-score tells us how many standard deviations from the mean an observation falls, and in what direction. If x is an observation from a distribution that has known mean and standard deviation, the standardized score of x is: A standardized score is often called a z-score. Example Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is 6.07. What is her standardized score?

  8. Measuring Position: z-Scores Brent is 74 inches tall. The mean height in his math class is 67 inches with a standard deviation of 4.29 inches. • Find Brent’s standardized score. Brent is a member of the school’s basketball team. The mean height of the players on the team is 76 inches. Brent’s height translates to a z-score of -0.85 in the team’s height distribution. • Interpret the meaning of this z-score. • Compare this z-score to his standardized score of his height. • Find the standard deviation of the height distribution of the basketball team

  9. Transforming Data Transforming converts the original observations from the original units of measurements to another scale. Effect of Adding (or Subtracting) a Constant • Adding the same number a to (subtracting a from) each observation: • adds a to (subtracts a from) measures of center and location (mean, median, quartiles, percentiles), but • Does not change the shape of the distribution or measures of spread (range, IQR, standard deviation). Effect of Multiplying (or Dividing) by a Constant • Multiplying (or dividing) each observation by the same number b: • multiplies (divides) measures of center and location (mean, median, quartiles, percentiles)by b • multiplies (divides) measures of spread (range, IQR, standard deviation) by |b|, but • does not change the shape of the distribution

  10. Describing Location in a Distribution • FIND and INTERPRET the percentile of an individual value within a distribution of data. • ESTIMATE percentiles and individual values using a cumulative relative frequency graph. • FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data. • DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data.

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