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Understanding Data Distribution and Percentiles: Analysis and Interpretation

Explore percentiles, relative frequencies, and z-scores to analyze and interpret data distributions effectively. Learn how to find and interpret individual values within a data set using percentiles and cumulative relative frequency graphs. Practice identifying the position of a value in a distribution and understand standardized scores (z-scores) for comparisons.

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Understanding Data Distribution and Percentiles: Analysis and Interpretation

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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.

  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 p percent 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.

  4. Example A college professor gave her 100 students a test recently and here are their scores (percent correct): • Circle the grade for the student who earned a 40% on this professor’s test. What would you say about that student’s performance? • Circle the grade for the student who has a score at the 40th percentile. What would you say about that student’s performance? • On a test, does a student’s percentile have to be the same as the student’s percent correct?

  5. Example The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2014. Key: 7I3 represents a team with 73 wins Calculate and interpret the percentiles for… a) the St. Louis Cardinals, who had 90 wins b) the Milwaukee Brewers, who had 82 wins c) the Los Angeles Angels, who had 98 wins

  6. Cumulative Relative Frequency Graphs A cumulative relative frequency graph displays the cumulative relative frequency of each class of a frequency distribution.

  7. Example a) California, with a median household income of $57,445, is at what percentile? b) Interpret this value. c) What is the 25th percentile for this distribution? d) What is another name for this value (from part c)? e) Where is the graph the steepest? f) What does this indicate about the distribution?

  8. Cumulative Relative Frequency Graphs Sketch a cumulative relative frequency graph for the following distributions: • Skewed right • Skewed left • Roughly Symmetric • Bimodal

  9. 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.

  10. Homework Pg. 99-100 #1-9 odd

  11. Describing Location in a Distribution • FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data.

  12. z-scores Bridgette Jordan of Sandoval, Illinois (as of 2014), a town about 86 miles east of St. Charles, is one of the shortest women in the world, standing at 27 inches.

  13. z-scores Robert Wadlow was born in Alton, Illinois and passed away in 1940, at age 22, with a height of 107.1 inches.

  14. z-scores Bridgette Jordan of Sandoval, Illinos (as of 2014), a town about 86 miles east of St. Charles, is one of the shortest women in the world, standing at 27 inches. Robert Wadlow was born in Alton, Illinois and passed away in 1940, at age 22, with a height of 107.1 inches. Obviously, Bridgette is shorter than most women and Robert was taller than most men —but whose height is more unusual, relatively speaking? That is, relative to other adults, who is taller? We’ll say that that women have a mean height of 64 in. and a standard deviation of 2.5 in. and that the mean height of men is 69.5 in. with a standard deviation of 2.8 in.

  15. 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? Interpret this value in context.

  16. Example The single-season home run record for major league baseball has been set just three times since Babe Ruth hit 60 home runs in 1927. Roger Maris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry Bonds hit 73 in 2001. In an absolute sense, Barry Bonds had the best performance of these four players, because he hit the most home runs in a single season. However, in a relative sense this may not be true. Baseball historians suggest that hitting a home run has been easier in some eras than others. This is due to many factors, including quality of batters, quality of pitchers, hardness of the baseball, dimensions of ballparks, and possible use of performance-enhancing drugs. To make a fair comparison, we should see how these performances rate relative to others hitters during the same year. Calculate the standardized score for each player and compare.

  17. Using z-scores to compare • Find z-scores for each individual using the correct mean and standard deviation of the distributions. • THINK: Is it better to have a large or small z-score in the situation? Partner up: Think of a situation when having a higher z-score is better…. Think of a situation when having a lower z-score is better…

  18. Describing Location in a Distribution • FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data.

  19. Homework Pg. 101 #11, 12, 13 ,15

  20. Describing Location in a Distribution • DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data.

  21. Transforming Data

  22. Transforming Data Transforming converts the original observations from the original units of measurements to another scale. Transformations can affect the shape, center, and spread of a distribution. 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).

  23. Transforming Data Transforming converts the original observations from the original units of measurements to another scale. Transformations can affect the shape, center, and spread of a distribution. 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

  24. Transforming Data Example Examine the distribution of students’ guessing errors by defining a new variable as follows: error = guess − 13 That is, we’ll subtract 13 from each observation in the data set. Try to predict what the shape, center, and spread of this new distribution will be.

  25. Transforming Data Example Because our group of Australian students is having some difficulty with the metric system, it may not be helpful to tell them that their guesses tended to be about 2 to 3 meters too high. Let’s convert the error data to feet before we report back to them. There are roughly 3.28 feet in a meter.

  26. Example In July 2015, St. Charles County Cab Company charged an initial fee of $2.50 plus $0.02 per mile. Genevieve analyzed the distribution of her cab miles for July and found it to be skewed to the right with a mean of 6.98 miles and a standard deviation of 0.2 miles. a) Write an equation for the price of a cab ride in terms of miles. b) What are the mean and standard deviation of the price of Genevieve’s cab rides? c) Suppose that we standardize Genevieve’s cab miles for July. Describe the shape, center, and spread of the distribution.

  27. 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.

  28. Homework pg. 101-103 #17-23 odd, 25-30 all

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