+ Chapter 2: Modeling Distributions of Data Section 2.1 Describing Location in a Distribution The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE

+ Chapter 2 Modeling Distributions of Data 2.1Describing Location in a Distribution 2.2Normal Distributions

+ Section 2.1 Describing Location in a Distribution After this section, you should be able to… MEASURE position using percentiles INTERPRET cumulative relative frequency graphs MEASURE position using z-scores TRANSFORM data DEFINE and DESCRIBE density curves Learning Objectives

+ Describing Location in a Distribution Measuring Position: Percentiles One way to describe the location of a value in a distributionis to tell what percent of observations are less than it. Definition: The p th percentile of a distribution is the value with p percent of the observations less than it Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class? Example, p. 85 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 84 th percentile in the class’s test score distribution

+ Example Example: Wins in Major League Baseball The stemplot below shows the number of wins for each ofthe 30 Major League Baseball teams in Calculate and interpret the percentiles for the ColoradoRockies who had 92 wins, the New York Yankees who had103 wins, and the Cleveland Indians who had 65 wins. Key: 5|9 represents a team with 59 wins.

+ Cumulative Relative Frequency Cumulative Relative Frequency adds the counts in the frequency column for the current class and all previousclasses. Example: Let ’ s look at the frequency table for the ages of the 1 st 44 US presidents when they were inaugurated. Describing Location in a Distribution AgeFrequencyRelative Frequency Cumulative Frequency Cumulative Rel. Frequency 40 – – – – – – Total or 100%

+ Cumulative Relative Frequency Graphs A cumulative relative frequency graph displays the cumulative relative frequency of each class of afrequency distribution. Describing Location in a Distribution Age of First 44 Presidents When They Were Inaugurated AgeFrequencyRelative frequency Cumulative frequency Cumulative relative frequency /44 = 4.5% 22/44 = 4.5% /44 = 15.9% 99/44 = 20.5% /44 = 29.5% 2222/44 = 50.0% /44 = 34% 3434/44 = 77.3% /44 = 15.9% 4141/44 = 93.2% /44 = 6.8% 4444/44 = 100%

+ Describing Location in a Distribution Use the graph from page 88 to answer the following questions. Was Barack Obama, who was inaugurated at age 47,unusually young? Estimate and interpret the 65 th percentile of the distribution Interpreting Cumulative Relative Frequency Graphs

+ Example Example: State Median Household Incomes Here is a cumulative relative frequency graph showing thedistribution of median household incomes for the 50 statesand the District of Columbia. The point (50, 0.49) means that 49% of the states had a median household income of less than $50,000. a) California, with a median household income of $57,445, is at what percentile? Interpret this value. b) What is the 25 th percentile for this distribution? What is another name for this value?

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

+ Describing Location in a Distribution Using z -scores for Comparison We can use z-scores to compare the position of individuals in different distributions. The single-season home run record for major league baseball has been set just three times since Babe Ruth hit 60 home runs in Roger Maris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry Bonds hit 73 in In an absolute sense, Barry Bonds had the best performance of these four players, since he hit the most home runs in a single season. However, in a relative sense this may not be true. 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.

+ Example The single-season home run record for major league baseball has beenset just three times since Babe Ruth hit 60 home runs in RogerMaris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry Bonds hit73 in In an absolute sense, Barry Bonds had the bestperformance of these four players, since he hit the most home runs in asingle season. However, in a relative sense this may not be true. Tomake a fair comparison, we should see how these performances raterelative to others hitters during the same year. Calculate thestandardized score for each player and compare. Ruth: 60 – 7.2 = 5.44 McGwire: 70 – 20.7 = Maris: 61 – 18.8 = 3.16 Bonds: 73 – 21.4 = Although all 4 performances were outstanding, Babe Ruth can still lay claim to being the single-season home run champ, relatively speaking.

+ Describing Location in a Distribution 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. Adding the same number a (either positive, zero, or negative) to each observation: adds a to 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 Adding (or Subracting) a Constant

+ Example Example: Here is a graph and table of summary statisticsfor a sample of 30 test scores. The maximum possiblescore on the test was a 50. Suppose that the teacher is nice and wants to add 5points to each test score. How would this affect theshape, center and spread of the data?

+ Example The measures of center and position all increased by 5.However, the shape and spread of the distribution did notchange.

+ Describing Location in a Distribution Transforming Data Multiplying (or dividing) each observation by the same number b (positive, negative, or zero): multiplies (divides) measures of center and location by b multiplies (divides) measures of spread by |b|, but does not change the shape of the distribution Effect of Multiplying (or Dividing) by a Constant Example: Suppose that the teacher wants to convert each test score to percents. Since the test was out of 50 points, he should just multiply each score by 2 to make them out of 100. How would this change the shape, center and spread of the data? The measures of center, location and spread will all double. However, even though the distribution is more spread out, the shape will not change!!!

+ Describing Location in a Distribution Density Curves In Chapter 1, we developed a kit of graphical and numericaltools for describing distributions. Now, we ’ ll add one more step to the strategy. So far our strategy for exploring data is : 1. Graph the data to get an idea of the overall pattern 2. Calculate an appropriate numerical summary todescribe the center and spread of the distribution. Sometimes the overall pattern of a large number of observations is so regular, that we can describe it by asmooth curve, called a density curve.

+ Describing Location in a Distribution Density Curve Definition: A density curve is a curve that is always on or above the horizontal axis, and has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval. The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

+ Here ’ s the idea behind density curves:

+ A density curve has the following properties: The area under the density curve and above any rangeof values is the proportion of all observations that fall inthat range.

+ Density Curves NOTE: No real set of data is described by a density curve. It is simply an approximation that is easy to use and accurateenough for practical use. Describing Density Curves The median of the density curve is the “ equal-areas ” point. The mean of the density curve is the “ balance ” point.

+ Section 2.1 Describing Location in a Distribution In this section, we learned that… There are two ways of describing an individual’s location within a distribution – the percentile and z-score. A cumulative relative frequency graph allows us to examine location within a distribution. It is common to transform data, especially when changing units of measurement. Transforming data can affect the shape, center, and spread of a distribution. We can sometimes describe the overall pattern of a distribution by a density curve (an idealized description of a distribution that smooths out the irregularities in the actual data). Summary

+ Looking Ahead… We’ll learn about one particularly important class of density curves – the Normal Distributions We’ll learn The Rule The Standard Normal Distribution Normal Distribution Calculations, and Assessing Normality In the next Section…