Chapter 2: Modeling Distributions of Data Section 2.1 The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE
Chapter 2 Modeling Distributions of Data 2.1 2.2 Normal Distributions
Section 2.1 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
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. 6 7 7 2334 7 5777899 8 00123334 8 569 9 03 Definition: The p th 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? 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.
Measuring Position: Percentiles Alternate Example The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2009. 5 9 6 2455 7 00455589 8 0345667778 9 123557 10 3 a) 80 th Percentile b) 97 th Percentile c) 10 th Percentile Find the percentiles for the following teams: (a) The Colorado Rockies, who won 92 games. (b) The New York Yankees, who won 103 games. (c) The Kansas City Royals and Cleveland Indians, who both won 65 games.
Cumulative relative frequency (%) + Cumulative Relative Frequency Graphs A cumulative relative frequency graph 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 40-44 45-49 50-54 55-59 60-64 65-69 2 2/44 = 4.5% 7 7/44 = 15.9% 13 13/44 = 29.5% 12 12/44 = 34% 7 7/44 = 15.9% 3 3/44 = 6.8% Cumulative frequency Cumulative relative frequency 2 2/44 = 4.5% 9 9/44 = 20.5% 22 22/44 = 50.0% 34 34/44 = 77.3% 41 41/44 = 93.2% 44 44/44 = 100% 100 80 60 40 20 0 40 45 50 55 60 65 70 Age at inauguration
Interpreting Cumulative Relative Frequency Graphs 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 65 11 47 58
Cumulative relative frequency (%) + Median Income $1000s 35 to <40 40 to <45 45 to <50 50 to <55 55 to <60 Alternate Example Here is a table showing the distribution of median household incomes for the 50 states and the District of Columbia. Freq Relative frequency 1 1/51=0.020 or 2.0% 10 10/51=0.196 or 19.6% 14 14/51=0.275 or 27.5% 12 12/51=0.236 or 23.6% 5 5/51=0.098 or 9.8% Cumulative frequency Cumulative relative frequency 1 1/51=0.020 or 2.0% 11 11/51=0.216 or 21.6% 25 25/51=0.490 or 49.0% 37 37/51=0.725 or 72.5% 42 42/51=0.824 or 82.4% 100 80 60 40 20 60 to <65 65 to <70 6 6/51=0.118 or 11.8% 3 3/51=0.059 or 5.9% 48 48/51=0.941 or 94.1% 51 51/51=1.000 or 100% 0 35 40 45 50 55 60 65 70 Median Income ($1000s)
Cumulative relative frequency (%) + Interpreting Cumulative Relative Frequency Graphs Use the graph from the alternate example to answer the following questions. At what percentile is California, with a median income of $57,445? Estimate and interpret the first quartile of the distribution. 100 80 60 40 20 0 35 40 45 50 55 60 65 70 Median Income ($1000s)
Measuring Position: z-scores 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: z x mean 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? z x mean standard deviation 86 80 6.07 0.99
Using 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? z stats 86 80 6.07 z stats 0.99 z chem z chem 1.5 82 76 4
Using z-scores for Comparison Alternate Example The single-season home run record for MLB has been set just three times since Babe Ruth hit 60 home runs in 1927. In an absolute sense, Barry Bonds had the best performance of the four players, since 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. To make a fair comparison, we should see how these performances rate relative to those of other hitters during the same year. Year Player HR Mean SD 1927 Babe Ruth 1961 Roger Maris 1998 Mark McGwire 2001 Barry Bonds 60 7.2 9.7 61 18.8 13.4 70 20.7 12.7 73 21.4 13.2 Compute the standardized scores for each performance. Which player had the most outstanding performance relative to his peers? Babe Ruth 5.44 Roger Maris 3.16 Mark McGwire 3.87 Barry Bonds 3.91
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 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). Example, p. 93 n Mean s x Min Q 1 M Q 3 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
Transforming Data Effect of Multiplying (or Dividing) by a Constant Multiplying (or dividing) each observation by the same number b (positive, negative, or zero): Example, p. 95 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 Mean s x Min Q 1 M Q 3 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
Transforming Data Example, p. 96 During the winter months, the temperatures at the Starnes Colorado cabin can stay well below freezing for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets thermostat at 50F and records the temperature each night at midnight, but in degrees Celsius. a) Find the mean temp in degrees Fahrenheit. Does the thermostat seem accurate? b) Calculate the standard deviation of the readings in degrees Fahrenheit. Interpret this value in context. c) The 90 th percentile of the readings was 11C. What is the 90 th percentile temp in degrees Fahrenheit?
Density Curves In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we ll add one more step to the strategy. Exploring Quantitative Data 1. Always plot your data: make a graph. 2. Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers. 3. Calculate a numerical summary to briefly describe center and spread. 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.
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.
Density Curve 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 mean are the same for a symmetric density curve. The both lie in 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 Density Curves 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.
Section 2.1 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).
Looking Ahead In the next Section We ll learn about one particularly important class of density curves the Normal Distributions We ll learn The 68-95-99.7 Rule The Standard Normal Distribution Normal Distribution Calculations, and Assessing Normality