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+ Chapter 2: Modeling Distributions of Data Section 2.1 Describing Location in a Distribution The Practice of Statistics, 4th edition - For AP* STARNES, YATES, MOORE + Chapter 2 Modeling Distributions of Data 2.1 Describing Location in a Distribution 2.2 Normal Distributions + Section 2.1 Describing Location in a Distribution Learning Objectives 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 One way to describe the location of a value in a distribution is to tell what percent of observations are less than it. Definition: The pth percentile of a distribution is the value with p percent of the observations less than it. Example, p. 85 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 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. Describing Location in a Distribution Position: Percentiles + Measuring A cumulative relative frequency graph (or ogive) displays the cumulative relative frequency of each class of a frequency distribution. Age of First 44 Presidents When They Were Inaugurated Age Frequency Relative frequency Cumulative frequency Cumulative relative frequency 4044 2 2/44 = 4.5% 2 2/44 = 4.5% 4549 7 7/44 = 15.9% 9 9/44 = 20.5% 5054 13 13/44 = 29.5% 22 22/44 = 50.0% 5559 12 12/44 = 34% 34 34/44 = 77.3% 6064 7 7/44 = 15.9% 41 41/44 = 93.2% 6569 3 3/44 = 6.8% 44 44/44 = 100% + Relative Frequency Graphs Describing Location in a Distribution Cumulative Interpreting Cumulative Relative Frequency Graphs Was Barack Obama, who was inaugurated at age 47, unusually young? Estimate and interpret the 65th percentile of the distribution 65 11 47 58 Describing Location in a Distribution Use the graph from page 88 to answer the following questions. + Constructing a Histogram + Comparing data sets • How do we compare results when they are measured on two completely different scales? • One solution might be to look at percentiles • What might you say about a woman that is in the 50th percentile and a man in the 15th percentile? AP Statistics, Section 2.2, Part 1 8 Another way of comparing • Another way of comparing: Look at whether the data point is above or below the mean, and by how much. • Example: Compare an SAT score of 1080 to an ACT score of 28 given the mean SAT is 896 with standard deviation of 174 & the mean ACT is 20.6 with a standard deviation of 5.2 AP Statistics, Section 2.2, Part 1 9 A z-score tells us how 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: x mean z standard deviation 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 6.07. What is her standardized score? x mean 86 80 z 0.99 standard deviation 6.07 Describing Location in a Distribution Position: z-Scores + Measuring + z-scores for Comparison We can use z-scores to compare the position of individuals in different distributions. Example, p. 91 Jenny earned a score of 86 on her statistics test. The class mean was 80 and the standard deviation was 6.07. She earned a score of 82 on her chemistry test. The chemistry scores had a fairly symmetric distribution with a mean 76 and standard deviation of 4. On which test did Jenny perform better relative to the rest of her class? 82 76 4 86 80 zstats 6.07 zchem zstats 0.99 zchem 1.5 Describing Location in a Distribution Using Standardizing with z-scores (cont.) Standardized values have no units. z-scores measure the distance of each data value from the mean in standard deviations. A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean. Depending on the situation, it may be desirable to either have a positive or negative z-score: Grades on a test or Finishing times in a race Copyright © 2010, 2007, 2004 Pearson Education, Inc. Benefits of Standardizing Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean. Thus, we can compare values that are measured on different scales, with different units, or from different populations. Copyright © 2010, 2007, 2004 Pearson Education, Inc. When Is a z-score BIG? A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean. A data value that sits right at the mean, has a zscore equal to 0. A z-score of 1 means the data value is 1 standard deviation above the mean. A z-score of –1 means the data value is 1 standard deviation below the mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. When Is a z-score BIG? How far from 0 does a z-score have to be to be interesting or unusual? There is no universal standard, but the larger a zscore is (negative or positive), the more unusual it is. Remember that a negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. CHEBYSHEV’S INEQUALITY There is also an interesting result that describes the percent of observations in any distribution that fall within a specified number of standard deviations of the mean. It is known as Copyright © 2010, 2007, 2004 Pearson Education, Inc. CHEBYSHEV'S THEOREM for k = 2 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 2) standard deviations of the mean? 1 3 At least 1 2 2 4 75 % of the data falls within 2 standard deviations of the mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. CHEBYSHEV'S THEOREM for k = 3 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 3) standard deviations of the mean? 1 8 At least 1 3 2 9 88 . 9 % of the data falls within 3 standard deviations of the mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. CHEBYSHEV'S THEOREM for k =4 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 4) standard deviations of the mean? 1 15 At least 1 4 2 16 93 . 8 % of the data falls within 4 standard deviations of the mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Using Chebyshev’s Theorem A mathematics class completes an examination and it is found that the class mean is 77 and the standard deviation is 6. According to Chebyshev's Theorem, between what two values would at least 75% of the grades be? Copyright © 2010, 2007, 2004 Pearson Education, Inc. In Summary For specific values of k Chebyshev’s Rule reads At least 75% of the observations are within 2 standard deviations of the mean. At least 89% of the observations are within 3 standard deviations of the mean. At least 90% of the observations are within 3.16 standard deviations of the mean. At least 94% of the observations are within 4 standard deviations of the mean. At least 96% of the observations are within 5 standard deviations of the mean. At least 99% of the observations are with 10 standard deviations of the mean. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Transforming Data Effect of Adding (or Subracting) a Constant 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). n Example, p. 93 Mean sx Min Q1 M Q3 Max IQR Range Guess(m) 44 16.02 7.14 8 11 15 17 40 6 32 Error (m) 44 3.02 7.14 -5 -2 2 4 27 6 32 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Describing Location in a Distribution 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. Transforming Data •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 n Example, p. 95 Mean sx Min Q1 M Q3 Max IQR Range Error(ft) 44 9.91 23.43 -16.4 -6.56 6.56 13.12 88.56 19.68 104.96 Error (m) 44 3.02 7.14 -5 -2 2 4 27 6 32 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Describing Location in a Distribution Effect of Multiplying (or Dividing) by a Constant Multiplying (or dividing) each observation by the same number b (positive, negative, or zero): Copyright © 2010, 2007, 2004 Pearson Education, Inc. Density Curve So far our strategy for exploring data is Graph the data to get an idea of the overall pattern Calculate an appropriate numerical summary to describe 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 a smooth curve, called a density curve AP Statistics, Section 2.1, Copyright © 2010, 2007, 2004 Pearson Education, Inc.1 Part 25 A Density Curve is an idealized description of a distribution of data Characteristics Always above the x- axis Area always equal to 1 The area under the curve and above any range of values is the proportion of all observations that fall in that range. AP Statistics, Section 2.1, Part 1 26 Our measures of center and spread apply to density curves as well as to actual sets of observations. Distinguishing the Median and Mean of a Density Curve The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail. Describing Location in a Distribution Density Curves + Describing TYPES OF DENSITY CURVES Since a Density Curve is an idealized description of the data, we classify the types of density curves by the shapes of the distributions they represent. Finding the area under the density curve is like finding the proportion of data values on a particular interval The time it takes for students to drive to school is evenly distributed with a minimum of 5 minutes and a range of 35 minutes. What is the height of the rectangle? a)Draw the distribution Where should the rectangle end? 1/35 5 40 b) What is the probability that it takes less than 20 minutes to drive to school? P(X < 20) = (15)(1/35) = .4286 1/35 5 40 Uniform Distribution Is a continuous distribution that is evenly (or uniformly) distributed Has a density curve in the shape of a rectangle EX: The Citrus Sugar Company packs sugar in bags labeled 5 pounds. However, the packaging isn’t perfect and the actual weights are uniformly distributed with a mean of 4.98 pounds and a range of .12 pounds. a)Constructing the the uniform distribution we draw a rectangle centered at the mean of 4.98 extending .06 in either direction. What is the height of this rectangle? What shape does a uniform distribution have? How long is this rectangle? 1/.12 4.92 4.98 5.04 What is the probability that a randomly selected bag will weigh more than 4.97 pounds? P(X > 4.97) = .07(1/.12) .5833 What is =the length of the shaded region? 1/.12 4.92 4.98 5.04 Find the probability that a randomly selected bag weighs between 4.93 and 5.03 pounds. P(4.93<X<5.03) = What is = the length of the shaded region? .1(1/.12) .8333 1/.12 4.92 4.98 5.04 + Section 2.1 Describing Location in a Distribution Summary 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). Our measures of center and spread apply to density curves as well as to actual sets of observations. Distinguishing the Median and Mean of a Density Curve The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail. Describing Location in a Distribution Density Curves + Describing