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CHAPTER 10 Data Analysis and Probability Chapter Focus In this chapter you will learn how to organize and analyze categorical data by using relative frequencies. You will analyze numerical data by calculating statistics that locate the center and measure the spread of the data. Then you will see how displaying numerical data in various ways helps you make sense of it, especially when the amount of data is substantial. You will also learn about probability of events, and distinguish between dependent and independent events in order to calculate their probabilities. Chapter at a Glance Standards for Mathematical Content 10-1 Organizing and Displaying Data CC.9-12.S.ID.5 10-2 Frequency and Histograms CC.9-12.S.ID.1, CC.9-12.S.ID.2 10-3 Data Distributions CC.9-12.S.ID.1, CC.9-12.S.ID.2 10-4 Misleading Graphs and Statistics CC.9-12.S.ID.1, CC.9-12.S.ID.2, CC.9-12.S.ID.3 10-5 Experimental Probability CC.9-12.S.CP.1 10-6 Theoretical Probability CC.9-12.S.CP.1 10-7 Independent and Dependent Events CC.9-12.S.CP.2, CC.9-12.S.CP.3, CC.9-12.S.CP.8(+) Performance Tasks Assessment Readiness Chapter 10 545 Data Analysis and Probability chapter 10 © Houghton Mifflin Harcourt Publishing Company Lesson Unpacking the Standards Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this chapter. Represent data with plots on the real number line (dot plots, histograms, and box plots). Key Vocabulary Lessons 10-2, 10-3, 10-4 Displaying numerical data on the real number line gives you an instant visual image of how the data are distributed, and helps you draw conclusions about the center and spread of the data. Histogram EXAMPLE Golf Tournament Scores A histogram gives you an overall picture of how data are distributed, but does not indicate any particular values or statistics. 6 5 4 3 2 1 0 chapter 10 EXAMPLE 0 –7 68 3 –7 71 6 –7 74 Scores 77 –7 9 80 –8 2 Box-and-whisker plot A box-and-whisker plot includes five statistical values. First quartile Minimum 0 CC.9-12.S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Key Vocabulary 2 6 What It Means For You 8 Maximum 10 12 14 Lesson 10-4 Always examine the displays and statistics for a data set in its own particular context so that you can draw valid conclusions. EXAMPLE Most of data Mean outlier (valor extremo) A data value that is far removed from the rest of the data. Median Chapter 10 4 Third quartile Median 546 Outliers Much different value The outlier pulls the mean to the right. The median better represents a “typical” value. Data Analysis and Probability © Houghton Mifflin Harcourt Publishing Company; Photo credit: © Corbis histogram (histograma) A bar graph used to display data grouped in intervals. box-and-whisker plot (gráfica de mediana y rango) A method of showing how data are distributed by using the median, quartiles, and minimum and maximum values; also called a box plot. What It Means For You Frequency CC.9-12.S.ID.1 What It Means For You CC.9-12.S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. frequency table (tabla de frequencia) A table that lists the number of times, or frequency, that each data value occurs. joint relative frequency (frecuencia relativa conjunta) The ratio of the frequency in a particular category divided by the total number of data values. marginal relative frequency (frecuencia relativa marginal) The sum of the joint relative frequencies in a row or column of a two-way table. conditional relative frequency (frecuencia relativa condicional) The ratio of a joint relative frequency to a related marginal relative frequency in a two-way table. CC.9-12.S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Key Vocabulary independent events (sucesos independientes)) Events for which the occurrence or non-occurrence of one event does not affect the probability of the other event. probability (probabilidad) A number from 0 to 1 (or 0% to 100%) that is the measure of how likely an event is to occur. Two-way frequency tables give you a visual way to organize data categorized by two different variables so that you can more easily identify relationships. EXAMPLE A two-way relative frequency table The table shows the portions of households in a study that own a dog, a cat, both, or neither. Joint relative frequencies Owns a dog Owns a cat Yes No Total Yes 0.15 0.24 0.39 No 0.18 0.43 0.61 Total 0.33 0.67 1 Marginal relative frequencies Here are a few conclusions you can draw from the table: • 39% own a dog, and 33% own a cat. • 15% own a dog and a cat, and 43% own neither. 15 % ≈ 38% also own a cat. • Of dog owners, __ 39 15 • Of cat owners, __ 33 % ≈ 45% also own a dog. What It Means For You Lessons 10-6, 10-7 Two events are independent if the occurrence of one event does not affect the occurrence of the other. When two events A and B are independent, the probability of both events occurring together is P(A and B) = P(A) · P(B). EXAMPLE Independent Events You choose a marble without looking and put it back. Then you choose a second marble.. 1 P(blue and then black) = _ 1 · _ 1 = _ 2 2 4 NON-EXAMPLE Dependent Events You choose a marble without looking and don’t put it back. Then you choose a second marble. P(blue and then black) = _ 1 · _ 2 = _ 1 2 3 3 Chapter 10 547 Data Analysis and Probability chapter 10 © Houghton Mifflin Harcourt Publishing Company; Photo credit: © Corbis Key Vocabulary Lesson 10-1 Key Vocabulary box-and-whisker plot (gráfica de mediana y rango) A method of showing how data are distributed by using the median, quartiles, and minimum and maximum values; also called a box plot. complement of an event __ (complemento de un suceso) All outcomes in the sample space that are not in an event E, denoted E . conditional probability (probabilidad condicional) The probability of event B, given that event A has already occurred or is certain to occur, denoted P(B | A); used to find probability of dependent events. dependent events (sucesos dependientes) Events for which the occurrence or nonoccurrence of one event affects the probability of the other event. first quartile (primer cuartil) The median of the lower half of a data set, denoted Q1. Also called lower quartile. histogram (histograma) A bar graph used to display data grouped in intervals independent events (sucesos independientes) Events for which the occurrence or non-occurrence of one event does not affect the probability of the other event. chapter 10 interquartile range (IQR) (rango entre cuartiles) The difference of the third (upper) and first (lower) quartiles in a data set, representing the middle half of the data. intersection (intersección de conjuntos) The intersection of two sets is the set of all elements that are common to both sets, denoted by ⋂. mean (media) The sum of all the values in a data set divided by the number of data values. Also called the average. median (mediana) For an ordered data set with an odd number of values, the median is the middle value. For an ordered data set with an even number of values, the median is the average of the two middle values. outlier (valor extremo) A data value that is far removed from the rest of the data. probability (probabilidad) A number from 0 to 1 (or 0% to 100%) that is the measure of how likely an event is to occur. range of a data set (rango de un conjunto de datos) The difference of the greatest and least values in the data set. standard deviation (desviación estándar) A measure of dispersion of a data set. The standard deviation σ is the square root of the variance. third quartile (tercer cuartil) The median of the upper half of a data set. Also called upper quartile. union (unión) The union of two sets is the set of all elements that are in either set, denoted by ⋃. Chapter 10 548 Data Analysis and Probability