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Chapter 2 Turning Data Into I f Information i Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 2.1 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 p population p p data. • Descriptive statistics: summary numbers for either population or a sample. sample Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 2 2.2 Types of Data • Categorical variables consist of group or g y names that don’t necessarilyy have a category 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 variables. earned, tee shirt size (S, M, L, XL). • Quantitative variables consist of numerical values taken on each individual. Examples: g , number of siblings. g height, Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 3 Asking the Right Questions One Categorical Variable Question: How many and what percentage of individuals fall into each category? Example: p What ppercentage g of college g students favor the legalization of marijuana, and what percentage of college students oppose legalization of marijuana? Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 4 Asking the Right Questions Two Categorical Variables Question: 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: l In Case C S d 11.6, Study 6 we asked k d if the h risk i k off having h i a heart attack was different for the physicians who took aspirin than for those who took a placebo. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 5 Asking the Right Questions One Quantitative Variable Question: What are the interestingg summaryy measures,, like Q 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? Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 6 Asking the Right Questions One Categorical and One Quantitative Variable Question: Are the measurements similar across categories? Example: Do men and women drive at the same “f “fastest speeds” d ” on average?? Question: 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, d college ll dropouts, d andd college ll graduates d have increasingly higher average incomes? Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 7 Asking the Right Questions Two Quantitative Variables Question: If the measurement on one variable is high Q g (or low), does the other one also tend to be high (or low)? Example: Do taller people also tend to have larger handspans? Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 8 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. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 9 2.3 Summarizing One or Two Categorical Variables Numerical Summaries • Count how many fall into each category. • Calculate the percent in each category. • If two variables, have the categories of the explanatory e planator variable ariable define the rows ro s and compute row percentages. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 10 Example 2.1 Importance of Order Survey of n = 190 college students. About half (92) given the question: “Randomly d l pick i k a letter l --- S or Q.” Note: 66% picked the first choice of S. Oth half Other h lf (98) given i th question: the ti “Randomly pick a letter --- Q or S.” Note: 54% picked the first choice of Q. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 11 Example 2.2 Lighting the Way to Nearsightedness N i h d Survey of n = 479 children. Th Those who h slept l t with ith nightlight i htli ht or in i fully f ll 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. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 12 Visual V sua Su Summaries a es • 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 ti l l useful f l for f making ki comparisons i when there are two categorical variables. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 13 Example 2.3 Humans Are Not G dR Good Randomizers d i Survey of n = 190 college students. “Randomly d l pick i k a number b between b 1 andd 10.” R lt Most Results: M t chose h 7, 7 very few f chose h 1 or 10. 10 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 14 Example 2.4 Revisiting Nightlights and d Nearsightedness N i h d Survey of n = 479 children. hild Response: Degree of Myopia Explanatory: Amount of Sl ti Sleeptime Lighting Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 15 2.4 Finding Information in Quantitative Data Long list of numbers – needs to be organized to obtain answers to questions of interest. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 16 Five-Number Summaries • Find extremes (high, low), the median, and the quartiles (medians of lower and upper halves of the values). • Quick Q i k overview i off the h data d values. l • Information about the center, spread, and shape of data. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 17 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 23.25 cm. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 18 Interesting Features of Quantitative Variables • Location: center or average. e g median e.g. • Spread: variability e.g. difference between two extremes or two quartiles. q • Shape: (later in Section 2.5) Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 19 Outliers and How to Handle Them Outlier: a data point that is not consistent with the bulk of the data. • L Lookk for f them h via i graphs. h • Can have bigg influence on conclusions. • Can cause complications in some statistical anal analyses. ses • Cannot discard without justification. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 20 Example 2.6 Ages of Death off U.S. U S Fi First L Ladies di Partial Data Listingg and five-number summary: y Extremes are more interesting here: Who died at 34? Martha Jefferson Who lived to be 97? Bess Truman Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 21 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. spread Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 22 2.5 Pictures for Quantitative Data • Histograms: Hi t similar i il to t bar b graphs, h usedd for any number of data values. • Stem-and-leaf plots and dotplots: ppresent 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. groups Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 23 Interpreting Histograms, Stemplots, and dD Dotplots t l t • Values are centered around 20 cm. • Two possible low outliers. • Apart from outliers, spans range from about 16 to 23 cm. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 24 Describing g Shape p • S Symmetric, t i bell-shaped b ll h d • Symmetric, y , not bell-shaped p • Skewed Right: values trail off to the right • Skewed Left: values trail off to the left Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 25 Example 2.8 Big Music Collection About how many CDs do you own? Stem is ‘100s’ and leaf unit is ‘10s’. Final digit is truncated. N b ranged Numbers d ffrom 0 tto about b t 450, 450 with 450 being a clear outlier and most values ranging from 0 to 99. 99 The shape is skewed right. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 26 2.6 Numerical Summaries of Quantitative Data Notation N t ti ffor R Raw D Data: t n = number of individuals in a data set x1, x2 , x3,…, xn representt individual i di id l raw data d t values l 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 x1= 21, x2 = 19, x3 = 20, x4 = 20, x5 = 22, and x6 = 19 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 27 Describing the Location of a Data Set • M Mean: the th numerical i l 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 g mean > median Skewed Right: Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 28 Determining the Mean and Median x ∑ x= i The Mean where ∑x i n means “add together all the values” The Median If n is i odd: dd M = middle iddl off ordered d d values. l Count (n + 1)/2 down from top of ordered list. If n is even: M = average of middle two ordered values. values Average values that are (n/2) and (n/2) + 1 down from top of ordered list. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 29 The Influence of Outliers on the Mean and Median Larger influence on mean than median. High g outliers will increase the mean. Low outliers will decrease the mean. If ages att ddeath th are: 70 70, 72 72, 74, 74 76, 76 andd 78 then mean = median = 74 years. If ages at death are: 35, 72, 74, 76, and 78 then median = 74 but mean = 67 years. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 30 Describing Spread: Range and Interquartile Range • Range = high value – low value • Interquartile I t til R Range (IQR) = upper quartile – lower quartile • Standard Deviation ((covered later in Section 2.7)) Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 31 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 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 32 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. Q1 = lower l quartile il = median of data values that are below the median Q3 = upper pp q quartile = median of data values that are above the median Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 33 Example 2.10 Fastest Speeds (cont) Ordered Data ((in rows of 10 values) for the 87 males: 55 60 80 80 80 80 85 85 85 85 90 90 90 90 90 92 94 95 95 95 95 95 95 100 100 100 100 100 100 100 100 100 101 102 105 105 105 105 105 105 105 105 109 110 110 110 110 110 110 110 110 110 110 110 110 112 115 115 115 115 115 115 120 120 120 120 120 120 120 120 120 120 124 125 125 125 125 125 125 130 130 140 140 140 140 145 150 • Median = (87+1)/2 = 44th value in the list = 110 mph • Q1 = median of the 43 values below the median = (43+1)/2 = 22nd value from the start of the list = 95 mph • Q3 = median of the 43 values above the median = (43+1)/2 = 22nd value from the end of the list = 120 mph Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 34 Percentiles The kth 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 qquartile = 25th ppercentile • Median = 50th percentile • Upper quartile = 75th percentile Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 35 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 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 36 How to Draw a Boxplot off a Q Quantitative i i Variable V i bl 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. St 33: Draw Step D a li line through th h the th box b att the th median di M. M Step 4: Draw a line from Q1 end of box to smallest data value that is not further than 1.5 1 5 × IQR from Q1. 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. outliers Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 37 2.7 Bell-Shaped Distributions of Numbers Many measurements follow a predictable pattern: • Most individuals are clumped p 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. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 38 Example 2.11 Bell-Shaped B i i h Women’s British W ’ Heights H i h Data: representative p sample p of 199 married British couples. p Below shows a histogram of the wives’ heights with a normal curve superimposed. The mean height = 1602 millimeters. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 39 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. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 40 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. 10 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 41 Calculating the Standard Deviation Formula for the (sample) standard deviation: ∑ (x − x ) 2 s= i n −1 The value of s2 is called the (sample) variance. An equivalent formula, easier to compute, is: s= ∑x 2 i − nx 2 n −1 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 42 Calculating the Standard Deviation Consider four pulse rates: 62, 68, 74, 76 Step 1: 62 + 68 + 74 + 76 280 x= = = 70 4 4 Steps 2 and 3: 120 Step p 4: s = = 40 4 −1 2 Step 5: s = 40 = 6.3 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 43 Population Standard Deviation Data sets usually represent a sample from a larger population If the data set includes measurements for population. 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 μ (“mu”), and the population standard deviation is ∑ (x − μ ) 2 σ= i n Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 44 Interpreting the Standard Deviation for Bell-Shaped Curves: The Empirical Rule For any bell-shaped curve, approximately • 68% off the h values l fall f ll within i hi 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 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 45 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. Range R s≈ 6 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 46 Example 2.11 Women’s Heights (cont) Mean height for the 199 British women is 1602 mm andd standard d d ddeviation i i is i 62.4 62 4 mm. • 68% of the 199 heights would fall in the range 1602 ± 62.4, or 1539.6 to 1664.4 mm • 95% of the heights would fall in the interval 1602 ± 2(62.4), or 1477.2 to 1726.8 mm • 99.7% 99 7% of the heights would fall in the interval 1602 ± 3(62.4), or 1414.8 to 1789.2 mm Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 47 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: Range 350 s≈ = = 58.3 mm 6 6 Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 48 Standardized z-Scores Standardized score or z-score: Observed value − Mean z= Standard deviation E Example: l Mean M resting ti pulse l rate t for f adult d lt men is i 70 beats per minute (bpm), standard deviation is 8 bpm. The standardized score for a resting pulse rate of 80: 80 − 70 z= = 1.25 8 A pulse rate of 80 is 1.25 standard deviations above the mean pulse rate for adult men men. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 49 The Empirical Rule Restated For bell-shaped data, • About 68% of the values have zz-scores scores between –11 and +1. +1 • About 95% of the values have z-scores between b t –22 and d +2. +2 • About 99.7% of the values have z-scores between –3 and +3. Copyright ©2004 Brooks/Cole, a division of Thomson Learning, Inc. 50