+ Chapter 1 Exploring Data

Transcription

1 Chapter 1 Exploring Data Introduction: Data Analysis: Making Sense of Data 1.1 Analyzing Categorical Data 1.2 Displaying Quantitative Data with Graphs 1.3 Describing Quantitative Data with Numbers

2 Introduction Data Analysis: Making Sense of Data Learning Objectives After this section, you should be able to DEFINE Individuals and Variables DISTINGUISH between Categorical and Quantitative variables DEFINE Distribution DESCRIBE the idea behind Inference

3 Statistics is the science of data. Data Analysis is the process of organizing, displaying, summarizing, and asking questions about data. Definitions: Individuals objects (people, animals, things) described by a set of data Variable - any characteristic of an individual Data Analysis 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.

4 Read Example : Census at school ( P-3) Now do Exercise 3 on P- 7 Answers: a) Individuals: AP Stat students who completed a questionnaire on the first day of class. b) Categorical: gender, handedness, favorite type of music Quantitative: height( inches), amount of time the students expects to spend on HW (mins), the total value of coin(cents) c) Female, right-handed, 58 inches tall, spends 60 mins on HW, prefers alternative music, had 76 cents in her pocket.

5 Do Activity : Hiring Discrimination It just won t fly Follow the directions on Page 5 Perform 5 repetitions of your simulation. Turn in your results to your teacher. (Make Groups of 4)

6 A variable generally takes on many different values. In data analysis, we are interested in how often a variable takes on each value. Definition: Distribution tells us what values a variable takes and how often it takes those values 2009 Fuel Economy Guide Example MODEL MPG Acura RL 922 Audi A6 Quattro 1023 Bentley Arnage 1114 BMW Buick Lacrosse 1328 Cadillac CTS 1425 Chevrolet Malibu 1533 Chrysler Sebring 1630 Dodge Avenger 1730 Variable of Interest: MPG 2009 Fuel Economy Guide MODEL MPG <new> MODEL MPG <new> Dodge Avenger 1630 Hyundai Elantra 1733 Jaguar XF 1825 Kia Optima 1932 Lexus GS Lincolon MKZ 2128 Mazda Mercedes-Benz E Mercury Milan Fuel Economy Guide Mercedes-Benz E Mercury Milan 29 Mitsubishi Galant 27 Nissan Maxima 26 Rolls Royce Phantom 18 Saturn Aura 33 Toyota Camry 31 Volkswagen Passat 29 Volvo S80 25 Dotplot of MPG Distribution

7 Examine each variable by itself. Then study relationships among the variables. How to Explore Data 2009 Fuel Economy Guide MODEL 2009 Fuel Economy Guide MPG 2009 Fuel Economy Guide MODEL MPG <new> MODEL MPG <new> Acura RL 922 Dodge Avenger 1630 Mercedes-Benz E Audi A6 Quattro 1023 Hyundai Elantra 1733 Mercury Milan 29 Bentley Arnage 1114 Jaguar XF 1825 Mitsubishi Galant 27 BMW Kia Optima 1932 Nissan Maxima 26 Buick Lacrosse 1328 Lexus GS Rolls Royce Phantom 18 Cadillac CTS 1425 Lincolon MKZ 2128 Saturn Aura 33 Chevrolet Malibu 1533 Mazda Toyota Camry 31 Chrysler Sebring 1630 Mercedes-Benz E Volkswagen Passat 29 Dodge Avenger 1730 Mercury Milan 2429 Volvo S80 25 Start with a graph or graphs Data Analysis Add numerical summaries

8 From Data Analysis to Inference Population Sample Collect data from a representative Sample... Data Analysis Make an Inference about the Population. Perform Data Analysis, keeping probability in mind

9 Inference: Inference is the process of making a conclusion about a population based on a sample set of data.

10 Introduction Data Analysis: Making Sense of Data Summary In this section, we learned that 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.

11 Do Notes worksheet : First page

12 Looking Ahead In the next Section We ll learn how to analyze categorical data. Bar Graphs Pie Charts Two-Way Tables Conditional Distributions We ll also learn how to organize a statistical problem.

13 Section 1.1 Analyzing Categorical Data Learning Objectives After this section, you should be able to CONSTRUCT and INTERPRET bar graphs and pie charts RECOGNIZE good and bad graphs CONSTRUCT and INTERPRET two-way tables DESCRIBE relationships between two categorical variables ORGANIZE statistical problems

14 Categorical Variables place individuals into one of several groups or categories The values of a categorical variable are labels for the different categories The distribution of a categorical variable lists the count or percent of individuals who fall into each category. Example, page 8 Variable Format Frequency Table Count of Stations Adult Contemporary 1556 Adult Standards 1196 Contemporary Hit 569 Country 2066 News/Talk 2179 Format Relative Frequency Table Percent of Stations Adult Contemporary 11.2 Adult Standards 8.6 Contemporary Hit 4.1 Country 14.9 News/Talk 15.7 Analyzing Categorical Data Values Oldies 1060 Religious 2014 Rock 869 Spanish Language 750 Other Formats 1579 Total Oldies 7.7 Religious 14.6 Rock 6.3 Count Spanish Language 5.4 Percent Other Formats 11.4 Total 99.9

15 Displaying categorical data 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. Format Frequency Table Count of Stations Adult Contemporary 1556 Adult Standards 1196 Contemporary Hit 569 Country 2066 News/Talk 2179 Oldies 1060 Format Relative Frequency Table Percent of Stations Adult Contemporary 11.2 Adult Standards 8.6 Contemporary Hit 4.1 Country 14.9 News/Talk 15.7 Oldies 7.7 Analyzing Categorical Data Religious 2014 Rock 869 Spanish Language 750 Other Formats 1579 Total Religious 14.6 Rock 6.3 Spanish Language 5.4 Other Formats 11.4 Total 99.9

16 Graphs: Good and Bad Bar graphs compare several quantities by comparing the heights of bars that represent those quantities. Our eyes react to the area of the bars as well as height. Be sure to make your bars equally wide. Avoid the temptation to replace the bars with pictures for greater appeal this can be misleading! Analyzing Categorical Data Alternate Example This ad for DIRECTV has multiple problems. How many can you point out?

17 Example: Bar graph What personal media do you own? Here are the percents of year olds who own the following personal media devices. Device % who won Cell Phone 85 MP3 player 83 Handheld Video game player 41 Laptop 38 Portable CD/tape player 20 a) Make a well-labeled bar graph. What do you see? b) Would it be appropriate to make a pie chart for these data? Why or why not?

18 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. Definition: Two-way Table describes two categorical variables, organizing counts according to a row variable and a column variable. Example, p. 12 Young adults by gender and chance of getting rich Female Male Total Almost no chance Some chance, but probably not A chance A good chance Almost certain Total What are the variables described by this twoway table? How many young adults were surveyed? Analyzing Categorical Data

19 Two-Way Tables and Marginal Distributions Definition: 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. Analyzing Categorical Data 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.

20 Percent + Two-Way Tables and Marginal Distributions Example, p. 13 Young adults by gender and chance of getting rich Female Male Total Almost no chance Some chance, but probably not A chance A good chance Almost certain Total Response Percent Almost no chance 194/4826 = 4.0% Some chance 712/4826 = 14.8% A chance 1416/4826 = 29.3% A good chance 1421/4826 = 29.4% Almost certain 1083/4826 = 22.4% Examine the marginal distribution of chance of getting rich. Chance of being wealthy by age 30 Almost none Some chance chance Good chance Survey Response Almost certain Analyzing Categorical Data

21 Relationships Between Categorical Variables Marginal distributions tell us nothing about the relationship between two variables. Definition: A Conditional Distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. 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.

22 Percent Percent + Two-Way Tables and Conditional Distributions Example, p. 15 Young adults by gender and chance of getting rich Female Male Total 100% Almost no chance % Some chance, but probably not % A chance % % A good chance % Almost certain % Total % Response Male Almost no chance 98/2459 = 4.0% Some chance 286/2459 = 11.6% A chance 720/2459 = 29.3% A good chance 758/2459 = 30.8% Almost certain 597/2459 = 24.3% Female 96/2367 = 4.1% 426/2367 = 18.0% 696/2367 = 29.4% 663/2367 = 28.0% 486/2367 = 20.5% % 10% 0% Chance of being wealthy by age 30 Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion. Chance of of being wealthy by by age age Males Opinion Females Almost no Some Some chance chancegood Good chance chance chance chance chance Almost certain Almost certain Good chance chance Some chance Almost no chance Males Males Females

23 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? Analyzing Categorical Data Do: Make graphs and carry out needed calculations. Conclude: Give your practical conclusion in the setting of the real-world problem.

24 Do : P- 25 # 26. State: Do the data support the idea that people who get angry easily tend to have more heart disease? Plan: We suspect that people with different anger levels will have different rates of CHD, so we will compare the conditional distributions of CHD for each anger level. Do: We will display the conditional distributions in a table to compare the rate of CHD occurrence for each of the 3 anger levels Conclude: The conditional distributions show that while CHD occurrence is quite small overall, the percent of the population with CHD does increase as eth anger level increases.

25 Section 1.1 Analyzing Categorical Data Summary In this section, we learned that The distribution of a categorical variable lists the categories and gives the count or percent of individuals that fall into each category. Pie charts and bar graphs display the distribution of a categorical variable. A two-way table of counts organizes data about two categorical variables. The row-totals and column-totals in a two-way table give the marginal distributions of the two individual variables. There are two sets of conditional distributions for a two-way table.

26 Section 1.1 Analyzing Categorical Data Summary, continued In this section, we learned that We can use a side-by-side bar graph or a segmented bar graph to display conditional distributions. To describe the association between the row and column variables, compare an appropriate set of conditional distributions. Even a strong association between two categorical variables can be influenced by other variables lurking in the background. You can organize many problems using the four steps state, plan, do, and conclude.

27 Looking Ahead In the next Section We ll learn how to display quantitative data. Dotplots Stemplots Histograms We ll also learn how to describe and compare distributions of quantitative data.

28 Section 1.2 Displaying Quantitative Data with Graphs Learning Objectives After this section, you should be able to CONSTRUCT and INTERPRET dotplots, stemplots, and histograms DESCRIBE the shape of a distribution COMPARE distributions USE histograms wisely

29 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. Number of Goals Scored Per Game by the 2004 US Women s Soccer Team Displaying Quantitative Data

30 Examining the Distribution of a Quantitative Variable The purpose of a graph is to help us understand the data. After you make a graph, always ask, What do I see? How to Examine the Distribution of a Quantitative Variable In any graph, look for the overall pattern and for striking departures from that pattern. Describe the overall pattern of a distribution by its: Shape Center Spread Outliers Don t forget your SOCS! Note individual values that fall outside the overall pattern. These departures are called outliers. Displaying Quantitative Data

31 Examine this data The table and dotplot below displays the Environmental Protection Agency s estimates of highway gas mileage in miles per gallon (MPG) for a sample of 24 model year 2009 midsize cars Fuel Economy Guide Example, page 28 MODEL MPG Acura RL 922 Audi A6 Quattro 1023 Bentley Arnage 1114 BMW Buick Lacrosse 1328 Cadillac CTS 1425 Chevrolet Malibu 1533 Chrysler Sebring 1630 Dodge Avenger Fuel Economy Guide MODEL MPG <new> MODEL MPG <new> Dodge Avenger 1630 Hyundai Elantra 1733 Jaguar XF 1825 Kia Optima 1932 Lexus GS Lincolon MKZ 2128 Mazda Mercedes-Benz E Mercury Milan Fuel Economy Guide Mercedes-Benz E Mercury Milan 29 Mitsubishi Galant 27 Nissan Maxima 26 Rolls Royce Phantom 18 Saturn Aura 33 Toyota Camry 31 Volkswagen Passat 29 Volvo S80 25 Displaying Quantitative Data Describe the shape, center, and spread of the distribution. Are there any outliers?

32 Describing Shape When you describe a distribution s shape, concentrate on the main features. Look for rough symmetry or clear skewness. Definitions: 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 Displaying Quantitative Data

33 You Do: Smart Phone Battery Life Here is the estimated battery life for each of 9 different smart phones (in minutes) according to Smart Phone Battery Life (minutes) Make a dot plot. Apple iphone 300 * Then describe the shape, center, and spread. Motorola Droid 385 Palm Pre 300 Blackberry Bold Blackberry Storm Motorola Cliq 360 Samsung Moment Blackberry Tour HTC Droid 460 of the distribution. Are there any outliers?

34 Solution: Collection 1 Dot Plot BatteryLife (minutes) Shape: There is a peak at 300 and the distribution has a long tail to the right (skewed to the right). Center: The middle value is 330 minutes. Spread: The range is = 160 minutes. Outliers: There is one phone with an unusually long battery life, the HTC Droid at 460 minutes.

35 Place U.K South Africa + 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. Example, page 32 Compare the distributions of household size for these two countries. Don t forget your SOCS! ( Compare each characteristic)

36 Stemplots (Stem-and-Leaf Plots) Another simple graphical display for small data sets is a stemplot. 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. Displaying Quantitative Data 5)Provide a key that explains in context what the stems and leaves represent.

37 Stemplots (Stem-and-Leaf Plots) 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 Key: 4 9 represents a female student who reported having 49 pairs of shoes. Displaying Quantitative Data Stems Add leaves Order leaves Add a key

38 Splitting Stems and Back-to-Back 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. Females Males split stems Females Males Key: 4 9 represents a student who reported having 49 pairs of shoes.

39 From both sides, compare and write about the center, spread, shape and possible outliers. Do CYU : P- 34. Check Your Understanding, page 34: 1. In general, it appears that females have more pairs of shoes than males. The median report for the males was 9 pairs while the female median was 26 pairs. The females also have a larger range of 57-13=44 in comparison to the range of 38-4=34 for the males. Finally, both males and females have distributions that are skewed to the right, though the distribution for the males is more heavily skewed as evidenced by the three likely outliers at 22, 35 and 38. The females do not have any likely outliers. 2. b 3. b 4. b

40 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.

41 Number of States + Example, page 35 Making a Histogram The table on page 35 presents data on the percent of residents from each state who were born outside of the U.S. Frequency Table Class 0 to < to < to < to < to < to <30 1 Total 50 Count Percent of foreign-born residents Displaying Quantitative Data

42 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)Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. 3)Just because a graph looks nice, it s not necessarily a meaningful display of data. ( P- 40: Read the example)

43 Try Check your understanding: P 39 Many people.. (Also put these #s in the calculator.) Solution: 2. The distribution is roughly symmetric and bell-shaped. The median IQ appears to be between 110 and 120 and the IQ s vary from 80 to 150. There do not appear to be any outliers.

44 Section 1.2 Displaying Quantitative Data with Graphs Summary In this section, we learned that You can use a dotplot, stemplot, or histogram to show the distribution of a quantitative variable. When examining any graph, look for an overall pattern and for notable departures from that pattern. Describe the shape, center, spread, and any outliers. Don t forget your SOCS! Some distributions have simple shapes, such as symmetric or skewed. The number of modes (major peaks) is another aspect of overall shape. When comparing distributions, be sure to discuss shape, center, spread, and possible outliers. Histograms are for quantitative data, bar graphs are for categorical data. Use relative frequency histograms when comparing data sets of different sizes.

45 Do # 56 P- 46

46 Looking Ahead In the next Section We ll learn how to describe quantitative data with numbers. Mean and Standard Deviation Median and Interquartile Range Five-number Summary and Boxplots Identifying Outliers We ll also learn how to calculate numerical summaries with technology and how to choose appropriate measures of center and spread.

47 Section 1.3 Describing Quantitative Data with Numbers Learning Objectives After this section, you should be able to MEASURE center with the mean and median MEASURE spread with standard deviation and interquartile range IDENTIFY outliers CONSTRUCT a boxplot using the five-number summary CALCULATE numerical summaries with technology

48 Measuring Center: The Mean The most common measure of center is the ordinary arithmetic average, or mean. Definition: 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: x x sum of observations x 1 x 2... x n n n In mathematics, the capital Greek letter Σis short for add them all up. Therefore, the formula for the mean can be written in more compact notation: Describing Quantitative Data x x i n

49 Measuring Center: The Median Another common measure of center is the median. In section 1.2, we learned that the median describes the midpoint of a distribution. Definition: The median M 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 M is the center observation in the ordered list. 3)If the number of observations n is even, the median M is the average of the two center observations in the ordered list. Describing Quantitative Data

50 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. Example, page x Key: 4 5 represents a New York worker who reported a 45- minute travel time to work. M minutes 22.5 minutes Describing Quantitative Data

51 Comparing the Mean and the Median The mean and median measure center in different ways, and both are useful. Don t confuse the average value of a variable (the mean) with its typical value, which we might describe by the median. Comparing the Mean and the Median The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median. Describing Quantitative Data

52 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 Interquartile Range To calculate the quartiles: 1)Arrange the observations in increasing order and locate the median M. 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: Describing Quantitative Data IQR = Q 3 Q 1

53 Do CYU P- 55

54 Find and Interpret the IQR Example, page 57 Travel times to work for 20 randomly selected New Yorkers Q 1 = 15 M = 22.5 Q 3 = 42.5 IQR = Q 3 Q 1 = = 27.5 minutes Describing Quantitative Data Interpretation: The range of the middle half of travel times for the New Yorkers in the sample is 27.5 minutes.

55 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. Definition: 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. Example, page 57 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) = Q x IQR = = Q x IQR = = Any travel time shorter than minutes or longer than minutes is considered an outlier Describing Quantitative Data

56 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. Definition: 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 M Q 3 Maximum Describing Quantitative Data

57 Calculator: Command to get 5 number summary: Put all #s in the list. Stat -> Edit-> List L1-> Put the numbers Now, for the command for 5 number summary, Stat -> Calc -> choose 1:1-VarStats L1 Enter

58 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 Draw and label a number line that includes the range of the distribution. Draw a central box from Q 1 to Q 3. Note the median M inside the box. Describing Quantitative Data Extend lines (whiskers) from the box out to the minimum and maximum values that are not outliers.

59 Construct a Boxplot Example Consider our NY travel times data. Construct a boxplot Min=5 Q 1 = 15 M = 22.5 Q 3 = 42.5 Max=85 Recall, this is an outlier by the 1.5 x IQR rule Describing Quantitative Data

60 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. Let s explore it! 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 deviation: 1-5 = -4 deviation: 8-5 = 3 Describing Quantitative Data x = 5

61 Measuring Spread: The Standard Deviation 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 x i (x i -mean) (x i -mean) = -4 (-4) 2 = = -2 (-2) 2 = = -1 (-1) 2 = = -1 (-1) 2 = = -1 (-1) 2 = = 0 (0) 2 = = 2 (2) 2 = = 3 (3) 2 = = 4 (4) 2 = 16 Sum=? Sum=? This is the variance. Describing Quantitative Data Standard deviation = square root of variance =

62 Measuring Spread: The Standard Deviation Definition: 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. This average squared distance is called the variance. variance = s x 2 (x 1 x )2 (x 2 x ) 2... (x n x ) 2 n 1 standard deviation = s x 1 (x i x ) 2 n 1 1 (x i x ) 2 n 1 Describing Quantitative Data

63 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. Describing Quantitative Data NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA!

64 Section 1.3 Describing Quantitative Data with Numbers Summary In this section, we learned that A numerical summary of a distribution should report at least its center and spread. The mean and median describe the center of a distribution in different ways. The mean is the average and the median is the midpoint of the values. When you use the median to indicate the center of a distribution, describe its spread using the quartiles. The interquartile range (IQR) is the range of the middle 50% of the observations: IQR = Q 3 Q 1.

65 Section 1.3 Describing Quantitative Data with Numbers Summary In this section, we learned that An extreme observation is an outlier if it is smaller than Q 1 (1.5xIQR) or larger than Q 3 +(1.5xIQR). The five-number summary (min, Q 1, M, Q 3, max) provides a quick overall description of distribution and can be pictured using a boxplot. The variance and its square root, the standard deviation are common measures of spread about the mean as center. The mean and standard deviation are good descriptions for symmetric distributions without outliers. The median and IQR are a better description for skewed distributions.

66 Looking Ahead In the next Chapter We ll learn how to model distributions of data Describing Location in a Distribution Normal Distributions

67 92 (a) The median is the average of the ranked scores in the middle two positions (the 15th and 16th ranked scores). The median is Half of the students scored less than and half scored more than Q1 is the score one-quarter up the list of ordered scores, 82. Q3 is the score three-quarters up the ordered list of scores, 93. IQR = =11 The middle 50% of the scores have a range of 11 points. Any observation above Q IQR = (11) = or below Q1-1.5 IQR = (11) = 65.5 is considered an outlier. Thus, the scores 43 and 45 are outliers.

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