1. The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers CHAPTER 1 Exploring Data Introduction Data Analysis: Making Sense of Data

2. Learning Objectives After this section, you should be able to: The Practice of Statistics, 5 th Edition2 IDENTIFY the individuals and variables in a set of data CLASSIFY variables as categorical or quantitative Data Analysis: Making Sense of Data

3. The Practice of Statistics, 5 th Edition3 Data Analysis Statistics is the science of data. Data Analysis is the process of organizing, displaying, summarizing, and asking questions about data. Individuals objects described by a set of data Variable any characteristic of an individual Individuals objects described by a set of data Variable any characteristic of an individual Categorical Variable places an individual into one of several groups or categories. Categorical Variable places an individual into one of several groups or categories. Quantitative Variable takes numerical values for which it makes sense to find an average. Quantitative Variable takes numerical values for which it makes sense to find an average.

4. The Practice of Statistics, 5 th Edition4 A variable generally takes on many different values. We are interested in how often a variable takes on each value. Distribution tells us what values a variable takes and how often it takes those values. Distribution tells us what values a variable takes and how often it takes those values. Variable of Interest: MPG Variable of Interest: MPG Dotplot of MPG Distribution Data Analysis

5. The Practice of Statistics, 5 th Edition5 Examine each variable by itself. Then study relationships among the variables. Examine each variable by itself. Then study relationships among the variables. Start with a graph or graphs Start with a graph or graphs How to Explore Data Add numerical summaries

6. The Practice of Statistics, 5 th Edition6 Population Sample Collect data from a representative Sample... Perform Data Analysis, keeping probability in mind… Make an Inference about the Population. From Data Analysis to Inference

7. Section Summary In this section, we learned that… The Practice of Statistics, 5 th Edition7 A dataset contains information on individuals. For each individual, data give values for one or more variables. Variables can be categorical or quantitative. The distribution of a variable describes what values it takes and how often it takes them. Inference is the process of making a conclusion about a population based on a sample set of data. Data Analysis: Making Sense of Data

8. Learning Objectives After this section, you should be able to: The Practice of Statistics, 5 th Edition8 DISPLAY categorical data with a bar graph IDENTIFY what makes some graphs of categorical data deceptive CALCULATE and DISPLAY the marginal distribution of a categorical variable from a two-way table CALCULATE and DISPLAY the conditional distribution of a categorical variable for a particular value of the other categorical variable in a two-way table DESCRIBE the association between two categorical variables Analyzing Categorical Data

9. The Practice of Statistics, 5 th Edition9 Categorical Variables Categorical variables place individuals into one of several groups or categories. Frequency Table FormatCount of Stations Adult Contemporary1556 Adult Standards1196 Contemporary Hit569 Country2066 News/Talk2179 Oldies1060 Religious2014 Rock869 Spanish Language750 Other Formats1579 Total13838 Relative Frequency Table FormatPercent of Stations Adult Contemporary11.2 Adult Standards8.6 Contemporary Hit4.1 Country14.9 News/Talk15.7 Oldies7.7 Religious14.6 Rock6.3 Spanish Language5.4 Other Formats11.4 Total99.9 Count Percent Variable Values

10. The Practice of Statistics, 5 th Edition10 Frequency tables can be difficult to read. Sometimes is is easier to analyze a distribution by displaying it with a bar graph or pie chart. Displaying Categorical Data Frequency Table FormatCount of Stations Adult Contemporary1556 Adult Standards1196 Contemporary Hit569 Country2066 News/Talk2179 Oldies1060 Religious2014 Rock869 Spanish Language750 Other Formats1579 Total13838 Relative Frequency Table FormatPercent of Stations Adult Contemporary11.2 Adult Standards8.6 Contemporary Hit4.1 Country14.9 News/Talk15.7 Oldies7.7 Religious14.6 Rock6.3 Spanish Language5.4 Other Formats11.4 Total99.9

11. The Practice of Statistics, 5 th Edition11 Graphs: Good and Bad Bar graphs compare several quantities by comparing the heights of bars that represent those quantities. Our eyes, however, react to the area of the bars as well as to their height. When you draw a bar graph, make the bars equally wide. It is tempting to replace the bars with pictures for greater eye appeal. Don’t do it! There are two important lessons to keep in mind: (1)beware the pictograph, and (2)watch those scales. There are two important lessons to keep in mind: (1)beware the pictograph, and (2)watch those scales.

12. The Practice of Statistics, 5 th Edition12 Two-Way Tables and Marginal Distributions When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables. A two-way table describes two categorical variables, organizing counts according to a row variable and a column variable. What are the variables described by this two-way table? How many young adults were surveyed?

13. The Practice of Statistics, 5 th Edition13 Two-Way Tables and Marginal Distributions The marginal distribution of one of the categorical variables in a two- way table of counts is the distribution of values of that variable among all individuals described by the table. Note: Percents are often more informative than counts, especially when comparing groups of different sizes. How to examine a marginal distribution: 1)Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. 2)Make a graph to display the marginal distribution. How to examine a marginal distribution: 1)Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. 2)Make a graph to display the marginal distribution.

14. The Practice of Statistics, 5 th Edition14 Two-Way Tables and Marginal Distributions ResponsePercent Almost no chance 194/4826 = 4.0% Some chance712/4826 = 14.8% A 50-50 chance1416/4826 = 29.3% A good chance1421/4826 = 29.4% Almost certain1083/4826 = 22.4% Examine the marginal distribution of chance of getting rich.

15. The Practice of Statistics, 5 th Edition15 Relationships Between Categorical Variables A conditional distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. How to examine or compare conditional distributions: 1) Select the row(s) or column(s) of interest. 2) Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). 3) Make a graph to display the conditional distribution. Use a side-by-side bar graph or segmented bar graph to compare distributions. How to examine or compare conditional distributions: 1) Select the row(s) or column(s) of interest. 2) Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). 3) Make a graph to display the conditional distribution. Use a side-by-side bar graph or segmented bar graph to compare distributions.

16. The Practice of Statistics, 5 th Edition16 Relationships Between Categorical Variables ResponseMale Almost no chance 98/2459 = 4.0% Some chance 286/2459 = 11.6% A 50-50 chance 720/2459 = 29.3% A good chance 758/2459 = 30.8% Almost certain 597/2459 = 24.3% Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion. Female 96/2367 = 4.1% 426/2367 = 18.0% 696/2367 = 29.4% 663/2367 = 28.0% 486/2367 = 20.5%

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18. The Practice of Statistics, 5 th Edition18 Relationships Between Categorical Variables Caution! Even a strong association between two categorical variables can be influenced by other variables lurking in the background. Caution! Even a strong association between two categorical variables can be influenced by other variables lurking in the background. Can we say there is an association between gender and opinion in the population of young adults? Making this determination requires formal inference, which will have to wait a few chapters.

19. Section Summary In this section, we learned that… The Practice of Statistics, 5 th Edition19 DISPLAY categorical data with a bar graph IDENTIFY what makes some graphs of categorical data deceptive CALCULATE and DISPLAY the marginal distribution of a categorical variable from a two-way table CALCULATE and DISPLAY the conditional distribution of a categorical variable for a particular value of the other categorical variable in a two-way table DESCRIBE the association between two categorical variables Data Analysis: Making Sense of Data

20. Learning Objectives After this section, you should be able to: The Practice of Statistics, 5 th Edition20 MAKE and INTERPRET dotplots and stemplots of quantitative data DESCRIBE the overall pattern of a distribution and IDENTIFY any outliers IDENTIFY the shape of a distribution MAKE and INTERPRET histograms of quantitative data COMPARE distributions of quantitative data Displaying Quantitative Data with Graphs

21. The Practice of Statistics, 5 th Edition21 Dotplots One of the simplest graphs to construct and interpret is a dotplot. Each data value is shown as a dot above its location on a number line. How to make a dotplot: 1)Draw a horizontal axis (a number line) and label it with the variable name. 2)Scale the axis from the minimum to the maximum value. 3)Mark a dot above the location on the horizontal axis corresponding to each data value. How to make a dotplot: 1)Draw a horizontal axis (a number line) and label it with the variable name. 2)Scale the axis from the minimum to the maximum value. 3)Mark a dot above the location on the horizontal axis corresponding to each data value.

22. The Practice of Statistics, 5 th Edition22 The purpose of a graph is to help us understand the data. After you make a graph, always ask, “What do I see?” Examining the Distribution of a Quantitative Variable How to Examine the Distribution of a Quantitative Variable 1)In any graph, look for the overall pattern and for striking departures from that pattern. 2)Describe the overall pattern of a distribution by its: Shape Center Spread 3)Note individual values that fall outside the overall pattern. These departures are called outliers. How to Examine the Distribution of a Quantitative Variable 1)In any graph, look for the overall pattern and for striking departures from that pattern. 2)Describe the overall pattern of a distribution by its: Shape Center Spread 3)Note individual values that fall outside the overall pattern. These departures are called outliers. Don’t forget your SOCS!

23. The Practice of Statistics, 5 th Edition23 Describing Shape When you describe a distribution’s shape, concentrate on the main features. Look for rough symmetry or clear skewness. A distribution is roughly symmetric if the right and left sides of the graph are approximately mirror images of each other. A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. A distribution is roughly symmetric if the right and left sides of the graph are approximately mirror images of each other. A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. Symmetric Skewed-left Skewed-right

24. The Practice of Statistics, 5 th Edition24 Comparing Distributions Some of the most interesting statistics questions involve comparing two or more groups. Always discuss shape, center, spread, and possible outliers whenever you compare distributions of a quantitative variable. Compare the distributions of household size for these two countries. Don’t forget your SOCS!

25. The Practice of Statistics, 5 th Edition25 Stemplots Another simple graphical display for small data sets is a stemplot. (Also called a stem-and-leaf plot.) Stemplots give us a quick picture of the distribution while including the actual numerical values. How to make a stemplot: 1)Separate each observation into a stem (all but the final digit) and a leaf (the final digit). 2)Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column. 3)Write each leaf in the row to the right of its stem. 4)Arrange the leaves in increasing order out from the stem. 5)Provide a key that explains in context what the stems and leaves represent. How to make a stemplot: 1)Separate each observation into a stem (all but the final digit) and a leaf (the final digit). 2)Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column. 3)Write each leaf in the row to the right of its stem. 4)Arrange the leaves in increasing order out from the stem. 5)Provide a key that explains in context what the stems and leaves represent.

26. The Practice of Statistics, 5 th Edition26 Stemplots These data represent the responses of 20 female AP Statistics students to the question, “How many pairs of shoes do you have?” Construct a stemplot. 5026 31571924222338 13501334233049131551 Stems 1234512345 Add leaves 1 93335 2 664233 3 1840 4 9 5 0701 Order leaves 1 33359 2 233466 3 0148 4 9 5 0017 Add a key Key: 4|9 represents a female student who reported having 49 pairs of shoes.

27. The Practice of Statistics, 5 th Edition27 Stemplots When data values are “bunched up”, we can get a better picture of the distribution by splitting stems. Two distributions of the same quantitative variable can be compared using a back-to-back stemplot with common stems. 5026 31571924222338 13501334233049131551 001122334455001122334455 Key: 4|9 represents a student who reported having 49 pairs of shoes. Females 1476512388710 1145227510357 Males 0 4 0 555677778 1 0000124 1 2 3 3 58 4 5 Females 333 95 4332 66 410 8 9 100 7 Males “split stems”

28. The Practice of Statistics, 5 th Edition28 Histograms Quantitative variables often take many values. A graph of the distribution may be clearer if nearby values are grouped together. The most common graph of the distribution of one quantitative variable is a histogram. How to make a histogram: 1)Divide the range of data into classes of equal width. 2)Find the count (frequency) or percent (relative frequency) of individuals in each class. 3)Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals. How to make a histogram: 1)Divide the range of data into classes of equal width. 2)Find the count (frequency) or percent (relative frequency) of individuals in each class. 3)Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals.

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31. The Practice of Statistics, 5 th Edition31 Histograms This table presents data on the percent of residents from each state who were born outside of the U.S. Frequency Table ClassCount 0 to <520 5 to <1013 10 to <159 15 to <205 20 to <252 25 to <301 Total50 Percent of foreign-born residents Number of States

32. The Practice of Statistics, 5 th Edition32 Using Histograms Wisely Here are several cautions based on common mistakes students make when using histograms. Cautions! 1)Don’t confuse histograms and bar graphs. 2)Don’t use counts (in a frequency table) or percents (in a relative frequency table) as data. 3)Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. 4)Just because a graph looks nice, it’s not necessarily a meaningful display of data. Cautions! 1)Don’t confuse histograms and bar graphs. 2)Don’t use counts (in a frequency table) or percents (in a relative frequency table) as data. 3)Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. 4)Just because a graph looks nice, it’s not necessarily a meaningful display of data.

33. Section Summary In this section, we learned that… The Practice of Statistics, 5 th Edition33 MAKE and INTERPRET dotplots and stemplots of quantitative data DESCRIBE the overall pattern of a distribution IDENTIFY the shape of a distribution MAKE and INTERPRET histograms of quantitative data COMPARE distributions of quantitative data Data Analysis: Making Sense of Data

34. Learning Objectives After this section, you should be able to: The Practice of Statistics, 5 th Edition34 CALCULATE measures of center (mean, median). CALCULATE and INTERPRET measures of spread (range, IQR, standard deviation). CHOOSE the most appropriate measure of center and spread in a given setting. IDENTIFY outliers using the 1.5 × IQR rule. MAKE and INTERPRET boxplots of quantitative data. USE appropriate graphs and numerical summaries to compare distributions of quantitative variables. Describing Quantitative Data with Numbers

35. The Practice of Statistics, 5 th Edition35 Measuring Center: The Mean The most common measure of center is the ordinary arithmetic average, or mean. To find the mean (pronounced “x-bar”) of a set of observations, add their values and divide by the number of observations. If the n observations are x 1, x 2, x 3, …, x n, their mean is: In mathematics, the capital Greek letter Σ is short for “add them all up.” Therefore, the formula for the mean can be written in more compact notation:

36. The Practice of Statistics, 5 th Edition36 Measuring Center: The Median Another common measure of center is the median. The median describes the midpoint of a distribution. The median is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1. Arrange all observations from smallest to largest. 2. If the number of observations n is odd, the median is the center observation in the ordered list. 3. If the number of observations n is even, the median is the average of the two center observations in the ordered list.

37. The Practice of Statistics, 5 th Edition37 Measuring Center Use the data below to calculate the mean and median of the commuting times (in minutes) of 20 randomly selected New York workers. 103052540201015302015208515651560 4045 0 5 1 005555 2 0005 3 00 4 005 5 6 005 7 8 5 Key: 4|5 represents a New York worker who reported a 45- minute travel time to work.

38. The Practice of Statistics, 5 th Edition38 Measuring Spread: The Interquartile Range (IQR) A measure of center alone can be misleading. A useful numerical description of a distribution requires both a measure of center and a measure of spread. How To Calculate The Quartiles And The IQR: To calculate the quartiles: 1.Arrange the observations in increasing order and locate the median. 2.The first quartile Q 1 is the median of the observations located to the left of the median in the ordered list. 3.The third quartile Q 3 is the median of the observations located to the right of the median in the ordered list. The interquartile range (IQR) is defined as: IQR = Q 3 – Q 1 How To Calculate The Quartiles And The IQR: To calculate the quartiles: 1.Arrange the observations in increasing order and locate the median. 2.The first quartile Q 1 is the median of the observations located to the left of the median in the ordered list. 3.The third quartile Q 3 is the median of the observations located to the right of the median in the ordered list. The interquartile range (IQR) is defined as: IQR = Q 3 – Q 1

39. The Practice of Statistics, 5 th Edition39 Find and Interpret the IQR 510 15 20 2530 40 4560 6585 103052540201015302015208515651560 4045 510 15 20 2530 40 4560 6585 Median = 22.5 Q 3 = 42.5 Q 1 = 15 IQR= Q 3 – Q 1 = 42.5 – 15 = 27.5 minutes Interpretation: The range of the middle half of travel times for the New Yorkers in the sample is 27.5 minutes. Travel times for 20 New Yorkers:

40. The Practice of Statistics, 5 th Edition40 Identifying Outliers In addition to serving as a measure of spread, the interquartile range (IQR) is used as part of a rule of thumb for identifying outliers. The 1.5 x IQR Rule for Outliers Call an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile. In the New York travel time data, we found Q 1 =15 minutes, Q 3 =42.5 minutes, and IQR=27.5 minutes. For these data, 1.5 x IQR = 1.5(27.5) = 41.25 Q 1 - 1.5 x IQR = 15 – 41.25 = -26.25 Q 3 + 1.5 x IQR = 42.5 + 41.25 = 83.75 Any travel time shorter than -26.25 minutes or longer than 83.75 minutes is considered an outlier. 0 5 1 005555 2 0005 3 00 4 005 5 6 005 7 8 5

41. The Practice of Statistics, 5 th Edition41 The Five-Number Summary The minimum and maximum values alone tell us little about the distribution as a whole. Likewise, the median and quartiles tell us little about the tails of a distribution. To get a quick summary of both center and spread, combine all five numbers. The five-number summary of a distribution consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest. Minimum Q 1 Median Q 3 Maximum

42. The Practice of Statistics, 5 th Edition42 Boxplots (Box-and-Whisker Plots) The five-number summary divides the distribution roughly into quarters. This leads to a new way to display quantitative data, the boxplot. How To Make A Boxplot: A central box is drawn from the first quartile (Q 1 ) to the third quartile (Q 3 ). A line in the box marks the median. Lines (called whiskers) extend from the box out to the smallest and largest observations that are not outliers. Outliers are marked with a special symbol such as an asterisk (*). How To Make A Boxplot: A central box is drawn from the first quartile (Q 1 ) to the third quartile (Q 3 ). A line in the box marks the median. Lines (called whiskers) extend from the box out to the smallest and largest observations that are not outliers. Outliers are marked with a special symbol such as an asterisk (*).

43. The Practice of Statistics, 5 th Edition43 Construct a Boxplot Consider our New York travel time data: Median = 22.5 Q 3 = 42.5 Q 1 = 15 Min=5 103052540201015302015208515651560 4045 510 15 20 2530 40 4560 6585 Max=85 Recall, this is an outlier by the 1.5 x IQR rule Max=85 Recall, this is an outlier by the 1.5 x IQR rule

44. The Practice of Statistics, 5 th Edition44 Measuring Spread: The Standard Deviation The most common measure of spread looks at how far each observation is from the mean. This measure is called the standard deviation. Consider the following data on the number of pets owned by a group of 9 children. 1)Calculate the mean. 2)Calculate each deviation. deviation = observation – mean = 5 deviation: 1 - 5 = - 4 deviation: 8 - 5 = 3

45. The Practice of Statistics, 5 th Edition45 Measuring Spread: The Standard Deviation xixi (x i -mean)(x i -mean) 2 11 - 5 = -4(-4) 2 = 16 33 - 5 = -2(-2) 2 = 4 44 - 5 = -1(-1) 2 = 1 44 - 5 = -1(-1) 2 = 1 44 - 5 = -1(-1) 2 = 1 55 - 5 = 0(0) 2 = 0 77 - 5 = 2(2) 2 = 4 88 - 5 = 3(3) 2 = 9 99 - 5 = 4(4) 2 = 16 Sum=? 3) Square each deviation. 4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance. 5) Calculate the square root of the variance…this is the standard deviation. “average” squared deviation = 52/(9-1) = 6.5 This is the variance. Standard deviation = square root of variance =

46. The Practice of Statistics, 5 th Edition46 Measuring Spread: The Standard Deviation The standard deviation s x measures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root. The average squared distance is called the variance.

47. The Practice of Statistics, 5 th Edition47 Choosing Measures of Center and Spread We now have a choice between two descriptions for center and spread Mean and Standard Deviation Median and Interquartile Range Choosing Measures of Center and Spread The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers. NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA! Choosing Measures of Center and Spread The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers. NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA!

48. The Practice of Statistics, 5 th Edition48 Organizing a Statistical Problem As you learn more about statistics, you will be asked to solve more complex problems. Here is a four-step process you can follow. How to Organize a Statistical Problem: A Four-Step Process State: What’s the question that you’re trying to answer? Plan: How will you go about answering the question? What statistical techniques does this problem call for? Do: Make graphs and carry out needed calculations. Conclude: Give your conclusion in the setting of the real-world problem. How to Organize a Statistical Problem: A Four-Step Process State: What’s the question that you’re trying to answer? Plan: How will you go about answering the question? What statistical techniques does this problem call for? Do: Make graphs and carry out needed calculations. Conclude: Give your conclusion in the setting of the real-world problem.

49. Section Summary In this section, we learned that… The Practice of Statistics, 5 th Edition49 CALCULATE measures of center (mean, median). CALCULATE and INTERPRET measures of spread (range, IQR, standard deviation). CHOOSE the most appropriate measure of center and spread in a given setting. IDENTIFY outliers using the 1.5 × IQR rule. MAKE and INTERPRET boxplots of quantitative data. USE appropriate graphs and numerical summaries to compare distributions of quantitative variables. Data Analysis: Making Sense of Data