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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. Turning Data Into Information Chapter 2

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Questionnaire What is your gender (1=female, 0=male)? How many hours did you sleep last night? Randomly pick a letter – S or Q. Randomly pick a number between 1 and 10. What is your height in inches? What is your right handspan in centimeters? What is your left handspan in centimeters? What’s the fastest you’ve ever driven a car?

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Raw Data Raw data are for numbers and category labels that have been collected but have not yet been processed in any way. When measurements are taken from a subset of a population, they represent sample data. When all individuals in a population are measured, the measurements represent population data. Descriptive statistics: summary numbers for either population or a sample.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Types of Data Raw data from categorical variables consist of group or category names that don’t necessarily have a logical ordering. Examples: eye color, country of residence. Categorical variables for which the categories have a logical ordering are called ordinal variables. Examples: highest educational degree earned, tee shirt size (S, M, L, XL). Raw data from quantitative variables consist of numerical values taken on each individual. Examples: height, number of siblings.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 5 Asking the Right Questions One Categorical Variable Question 1a: How many and what percentage of individuals fall into each category? Example: What percentage of college students favor the legalization of marijuana, and what percentage of college students oppose legalization of marijuana? Question 1b: Are individuals equally divided across categories, or do the percentages across categories follow some other interesting pattern? Example: When individuals are asked to choose a number from 1 to 10, are all numbers equally likely to be chosen?

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 6 Asking the Right Questions Two Categorical Variables Question 2a: Is there a relationship between the two variables, so that the category into which individuals fall for one variable seems to depend on which category they are in for the other variable? Example: In Case Study 1.6, we asked if the risk of having a heart attack was different for the physicians who took aspirin than for those who took a placebo. Question 2b: Do some combinations of categories stand out because they provide information that is not found by examining the categories separately? Example: The relationship between smoking and lung cancer was detected, in part, because someone noticed that the combination of being a nonsmoker and having lung cancer is unusual.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 7 Asking the Right Questions One Quantitative Variable Question 3a: What are the interesting summary measures, like the average or the range of values, that help us understand the collection of individuals who were measured? Example: What is the average handspan measurement, and how much variability is there in handspan measurements? Question 3b: Are there individual data values that provide interesting information because they are unique or stand out in some way? Example: What is the oldest recorded age of death for a human? Are there many people who have lived nearly that long, or is the oldest recorded age a unique case?

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 8 Asking the Right Questions One Categorical and One Quantitative Variable Question 4a: Are the measurements similar across categories? Example: Do men and women drive at the same “fastest speeds” on average? Question 4b: When the categories have a natural ordering (an ordinal variable), does the measurement variable increase or decrease, on average, in that same order? Example: Do high school dropouts, high school graduates, college dropouts, and college graduates have increasingly higher average incomes?

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 9 Asking the Right Questions Two Quantitative Variables Question 5a: If the measurement on one variable is high (or low), does the other one also tend to be high (or low)? Example: Do taller people also tend to have larger handspans? Question 5b: Are there individuals whose combination of data values provides interesting information because that combination is unusual? Example: An individual who has a very low IQ score but can perform complicated arithmetic operations very quickly may shed light on how the brain works. Neither the IQ nor the arithmetic ability may stand out as uniquely low or high, but it is the combination that is interesting.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 10 Explanatory and Response Variables Many questions are about the relationship between two variables. It is useful to identify one variable as the explanatory variable and the other variable as the response variable. In general, the value of the explanatory variable for an individual is thought to partially explain the value of the response variable for that individual.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Summarizing One or Two Categorical Variables Count how many fall into each category. Calculate the percent in each category. If two variables, have the categories of the explanatory variable define the rows and compute row percentages. Numerical Summaries

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 12 Example 2.1 Importance of Order Survey of n = 190 college students. About half (92) given the question: “Randomly pick a letter --- S or Q.” Note: 66% picked the first choice of S. Other half (98) given the question: “Randomly pick a letter --- Q or S.” Note: 54% picked the first choice of Q.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 13 Example 2.2 Lighting the Way to Nearsightedness Survey of n = 479 children. Those who slept with nightlight or in fully lit room before age 2 had higher incidence of nearsightedness (myopia) later in childhood. Note: Study does not prove sleeping with light actually caused myopia in more children.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 14 Pie Charts: useful for summarizing a single categorical variable if not too many categories. Bar Graphs: useful for summarizing one or two categorical variables and particularly useful for making comparisons when there are two categorical variables. Visual Summaries for Categorical Variables

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 15 Example 2.3 Humans Are Not Good Randomizers Survey of n = 190 college students. “Randomly pick a number between 1 and 10.” Results: Most chose 7, very few chose 1 or 10.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 16 Example 2.4 Revisiting Nightlights and Nearsightedness Survey of n = 479 children. Response: Degree of Myopia Explanatory: Amount of Sleeptime Lighting

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Finding Information in Quantitative Data Long list of numbers – needs to be organized to obtain answers to questions of interest.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 18 Find extremes (high, low), the median, and the quartiles (medians of lower and upper halves of the values). Quick overview of the data values. Information about the center, spread, and shape of data. Five-Number Summaries

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 19 Example 2.5 Right Handspans About 25% of handspans of females are between 12.5 and 19.0 centimeters, about 25% are between 19 and 20 cm, about 25% are between 20 and 21 cm, and about 25% are between 21 and cm.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 20 Location: center or average. e.g. median Spread: variability e.g. difference between two extremes or two quartiles. Shape: later… Interesting Features of Quantitative Variables

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 21 Outlier: a data point that is not consistent with the bulk of the data. Outliers and How to Handle Them Look for them via graphs. Can have big influence on conclusions. Can cause complications in some statistical analyses. Cannot discard without justification.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 22 Example 2.6 Ages of Death of U.S. First Ladies Partial Data Listing and five-number summary: Extremes are more interesting here: Who died at 34? Martha Jefferson Who lived to be 97? Bess Truman

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 23 Possible Reasons for Outliers and Reasonable Actions Mistake made while taking measurement or entering it into computer. If verified, should be discarded/corrected. Individual in question belongs to a different group than bulk of individuals measured. Values may be discarded if summary is desired and reported for the majority group only. Outlier is legitimate data value and represents natural variability for the group and variable(s) measured. Values may not be discarded — they provide important information about location and spread.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 24 Example 2.7 Tiny Boatsmen Weights (in pounds) of 18 men on crew team: Note: last weight in each list is unusually small. They are the coxswains for their teams, while others are rowers. Cambridge:188.5, 183.0, 194.5, 185.0, 214.0, 203.5, 186.0, 178.5, Oxford: 186.0, 184.5, 204.0, 184.5, 195.5, 202.5, 174.0, 183.0, 109.5

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Pictures for Quantitative Data Histograms: similar to bar graphs, used for any number of data values. Stem-and-leaf plots and dotplots: present all individual values, useful for small to moderate sized data sets. Boxplot or box-and-whisker plot: useful summary for comparing two or more groups.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 26 Values are centered around 20 cm. Two possible low outliers. Apart from outliers, spans range from about 16 to 23 cm. Interpreting Histograms, Stemplots, and Dotplots

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 27 Describing Shape Symmetric, bell-shaped Symmetric, not bell-shaped Skewed Right: values trail off to the right Skewed Left: values trail off to the left

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 28 Creating a Histogram Step 1: Decide how many equally spaced (same width) intervals to use for the horizontal axis. Between 6 and 15 intervals is a good number. Step 2: Decide to use frequencies (count) or relative frequencies (proportion) on the vertical axis. Step 3: Draw equally spaced intervals on horizontal axis covering entire range of the data. Determine frequency or relative frequency of data values in each interval and draw a bar with corresponding height. Decide rule to use for values that fall on the border between two intervals.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Numerical Summaries of Quantitative Data Notation for Raw Data: n = number of individuals in a data set x 1, x 2, x 3,…, x n represent individual raw data values Example: A data set consists of handspan values in centimeters for six females; the values are 21, 19, 20, 20, 22, and 19. Then, n = 6 x 1 = 21, x 2 = 19, x 3 = 20, x 4 = 20, x 5 = 22, and x 6 = 19

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 30 Describing the Location of a Data Set Mean: the numerical average Median: the middle value (if n odd) or the average of the middle two values (n even) Symmetric: mean = median Skewed Left: mean < median Skewed Right: mean > median

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 31 Determining the Mean and Median The Mean where means “add together all the values” The Median If n is odd: M = middle of ordered values. Count (n + 1)/2 down from top of ordered list. If n is even: M = average of middle two ordered values. Average values that are (n/2) and (n/2) + 1 down from top of ordered list.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 32 Example 2.9 Will “Normal” Rainfall Get Rid of Those Odors? Mean = inches Median = inches Data: Average rainfall (inches) for Davis, California for 47 years In , a company with odor problem blamed it on excessive rain. That year rainfall was inches. More rain occurred in 4 other years.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 33 The Influence of Outliers on the Mean and Median Larger influence on mean than median. High outliers will increase the mean. Low outliers will decrease the mean. If ages at death are: 70, 72, 74, 76, and 78 then mean = median = 74 years. If ages at death are: 35, 72, 74, 76, and 78 then median = 74 but mean = 67 years.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 34 Describing Spread: Range and Interquartile Range Range = high value – low value Interquartile Range (IQR) = upper quartile – lower quartile Standard Deviation (later …)

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 35 Example 2.10 Fastest Speeds Ever Driven Five-Number Summary for 87 males Median = 110 mph measures the center of the data Two extremes describe spread over 100% of data Range = 150 – 55 = 95 mph Two quartiles describe spread over middle 50% of data Interquartile Range = 120 – 95 = 25 mph

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 36 Notation and Finding the Quartiles Split the ordered values into the half that is below the median and the half that is above the median. Q 1 = lower quartile = median of data values that are below the median Q 3 = upper quartile = median of data values that are above the median

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 37 Example 2.10 Fastest Speeds (cont) Ordered Data (in rows of 10 values) for the 87 males: Median = (87+1)/2 = 44 th value in the list = 110 mph Q 1 = median of the 43 values below the median = (43+1)/2 = 22 nd value from the start of the list = 95 mph Q 3 = median of the 43 values above the median = (43+1)/2 = 22 nd value from the end of the list = 120 mph

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 38 Percentiles The k th percentile is a number that has k% of the data values at or below it and (100 – k)% of the data values at or above it. Lower quartile = 25 th percentile Median = 50 th percentile Upper quartile = 75 th percentile

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 39 Picturing Location and Spread with Boxplots Boxplots for right handspans of males and females. Box covers the middle 50% of the data Line within box marks the median value Possible outliers are marked with asterisk Apart from outliers, lines extending from box reach to min and max values.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 40 How to Draw a Boxplot of a Quantitative Variable Step 1: Label either a vertical axis or a horizontal axis with numbers from min to max of the data. Step 2: Draw box with lower end at Q1 and upper end at Q3. Step 3: Draw a line through the box at the median M. Step 4: Draw a line from Q1 end of box to smallest data value that is not further than 1.5 IQR from Q1. Draw a line from Q3 end of box to largest data value that is not further than 1.5 IQR from Q3. Step 5: Mark data points further than 1.5 IQR from either edge of the box with an asterisk. Points represented with asterisks are considered to be outliers.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Bell-Shaped Distributions of Numbers Many measurements follow a predictable pattern: Most individuals are clumped around the center The greater the distance a value is from the center, the fewer individuals have that value. Variables that follow such a pattern are said to be “bell-shaped”. A special case is called a normal distribution or normal curve.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 42 Example 2.11 Bell-Shaped British Women’s Heights Data: representative sample of 199 married British couples. Below shows a histogram of the wives’ heights with a normal curve superimposed. The mean height = 1602 millimeters.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 43 Describing Spread with Standard Deviation Standard deviation measures variability by summarizing how far individual data values are from the mean. Think of the standard deviation as roughly the average distance values fall from the mean.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 44 Describing Spread with Standard Deviation Both sets have same mean of 100. Set 1: all values are equal to the mean so there is no variability at all. Set 2: one value equals the mean and other four values are 10 points away from the mean, so the average distance away from the mean is about 10.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 45 Formula for the (sample) standard deviation: The value of s 2 is called the (sample) variance. An equivalent formula, easier to compute, is: Calculating the Standard Deviation

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 46 Step 1: Calculate, the sample mean. Step 2: For each observation, calculate the difference between the data value and the mean. Step 3: Square each difference in step 2. Step 4: Sum the squared differences in step 3, and then divide this sum by n – 1. Step 5: Take the square root of the value in step 4. Calculating the Standard Deviation

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 47 Consider four pulse rates: 62, 68, 74, 76 Calculating the Standard Deviation Step 1: Steps 2 and 3: Step 4: Step 5:

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 48 Data sets usually represent a sample from a larger population. If the data set includes measurements for an entire population, the notations for the mean and standard deviation are different, and the formula for the standard deviation is also slightly different. A population mean is represented by the symbol (“mu”), and the population standard deviation is Population Standard Deviation

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 49 Interpreting the Standard Deviation for Bell-Shaped Curves: The Empirical Rule For any bell-shaped curve, approximately 68% of the values fall within 1 standard deviation of the mean in either direction 95% of the values fall within 2 standard deviations of the mean in either direction 99.7% of the values fall within 3 standard deviations of the mean in either direction

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 50 The Empirical Rule, the Standard Deviation, and the Range Empirical Rule => the range from the minimum to the maximum data values equals about 4 to 6 standard deviations for data with an approximate bell shape. You can get a rough idea of the value of the standard deviation by dividing the range by 6.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 51 Example 2.11 Women’s Heights (cont) Mean height for the 199 British women is 1602 mm and standard deviation is 62.4 mm. 68% of the 199 heights would fall in the range 1602 62.4, or to mm 95% of the heights would fall in the interval 1602 2(62.4), or to mm 99.7% of the heights would fall in the interval 1602 3(62.4), or to mm

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 52 Example 2.11 Women’s Heights (cont) Summary of the actual results: Note: The minimum height = 1410 mm and the maximum height = 1760 mm, for a range of 1760 – 1410 = 350 mm. So an estimate of the standard deviation is:

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 53 Standardized z-Scores Standardized score or z-score: Example: Mean resting pulse rate for adult men is 70 beats per minute (bpm), standard deviation is 8 bpm. The standardized score for a resting pulse rate of 80: A pulse rate of 80 is 1.25 standard deviations above the mean pulse rate for adult men.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 54 The Empirical Rule Restated For bell-shaped data, About 68% of the values have z-scores between –1 and +1. About 95% of the values have z-scores between –2 and +2. About 99.7% of the values have z-scores between –3 and +3.

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