Looking at data

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  • That's really what distinguishes these from discrete numerical
  • What are some others?
  • Does everybody know what I mean when I say percentiles? What is the median? Anyone?
  • 1. Bin sizes may be altered. 2. How many people do you think are in bin 125-135? 3. Where do you think the center of the data are (what's your best guess at the average weight)? 4. On average, how far do you think a given woman is from 127 -- the center/mean?
  • Balance the Bell Curve on a point. Where is the point of balance, average mass on each side.
  • 1. Bin sizes may be altered. 2. How many people do you think are in bin 125-135? 3. Where do you think the center of the data are (what's your best guess at the average weight)? 4. On average, how far do you think a given woman is from 127 -- the center/mean?
  • SAY: within 1 standard deviation either way of the mean within 2 standard deviations of the mean within 3 standard deviations either way of the mean WORKS FOR ALL NORMAL CURVES NO MATTER HOW SKINNY OR FAT
  • Looking at data

    1. 1. Looking at Data
    2. 2. Clinical Data Example <ul><li>1. Kline et al. (2002) </li></ul><ul><ul><li>The researchers analyzed data from 934 emergency room patients with suspected pulmonary embolism (PE). Only about 1 in 5 actually had PE. The researchers wanted to know what clinical factors predicted PE. </li></ul></ul><ul><ul><li>I will use four variables from their dataset today: </li></ul></ul><ul><ul><ul><li>Pulmonary embolism (yes/no) </li></ul></ul></ul><ul><ul><ul><li>Age (years) </li></ul></ul></ul><ul><ul><ul><li>Shock index = heart rate/systolic BP </li></ul></ul></ul><ul><ul><ul><li>Shock index categories = take shock index and divide it into 10 groups (lowest to highest shock index) </li></ul></ul></ul>
    3. 3. Descriptive Statistics
    4. 4. Types of Variables: Overview Categorical Quantitative continuous discrete ordinal nominal binary 2 categories + more categories + order matters + numerical + uninterrupted
    5. 5. Categorical Variables <ul><li>Also known as “qualitative.” </li></ul><ul><li>Categories. </li></ul><ul><ul><ul><li>treatment groups </li></ul></ul></ul><ul><ul><ul><li>exposure groups </li></ul></ul></ul><ul><ul><ul><li>disease status </li></ul></ul></ul>
    6. 6. Categorical Variables <ul><li>Dichotomous (binary) – two levels </li></ul><ul><ul><ul><li>Dead/alive </li></ul></ul></ul><ul><ul><ul><li>Treatment/placebo </li></ul></ul></ul><ul><ul><ul><li>Disease/no disease </li></ul></ul></ul><ul><ul><ul><li>Exposed/Unexposed </li></ul></ul></ul><ul><ul><ul><li>Heads/Tails </li></ul></ul></ul><ul><ul><ul><li>Pulmonary Embolism (yes/no) </li></ul></ul></ul><ul><ul><ul><li>Male/female </li></ul></ul></ul>
    7. 7. Categorical Variables <ul><li>Nominal variables – Named categories Order doesn’t matter! </li></ul><ul><ul><ul><li>The blood type of a patient (O, A, B, AB) </li></ul></ul></ul><ul><ul><ul><li>Marital status </li></ul></ul></ul><ul><ul><ul><li>Occupation </li></ul></ul></ul>
    8. 8. Categorical Variables <ul><li>Ordinal variable – Ordered categories. Order matters! </li></ul><ul><ul><ul><li>Staging in breast cancer as I, II, III, or IV </li></ul></ul></ul><ul><ul><ul><li>Birth order—1st, 2nd, 3rd, etc. </li></ul></ul></ul><ul><ul><ul><li>Letter grades (A, B, C, D, F) </li></ul></ul></ul><ul><ul><ul><li>Ratings on a scale from 1-5 </li></ul></ul></ul><ul><ul><ul><li>Ratings on: always; usually; many times; once in a while; almost never; never </li></ul></ul></ul><ul><ul><ul><li>Age in categories (10-20, 20-30, etc.) </li></ul></ul></ul><ul><ul><ul><li>Shock index categories (Kline et al.) </li></ul></ul></ul>
    9. 9. Quantitative Variables <ul><li>Numerical variables; may be arithmetically manipulated. </li></ul><ul><ul><li>Counts </li></ul></ul><ul><ul><li>Time </li></ul></ul><ul><ul><li>Age </li></ul></ul><ul><ul><li>Height </li></ul></ul>
    10. 10. Quantitative Variables <ul><li>Discrete Numbers – a limited set of distinct values, such as whole numbers. </li></ul><ul><ul><ul><li>Number of new AIDS cases in CA in a year (counts) </li></ul></ul></ul><ul><ul><ul><li>Years of school completed </li></ul></ul></ul><ul><ul><ul><li>The number of children in the family (cannot have a half a child!) </li></ul></ul></ul><ul><ul><ul><li>The number of deaths in a defined time period (cannot have a partial death!) </li></ul></ul></ul><ul><ul><ul><li>Roll of a die </li></ul></ul></ul>
    11. 11. Quantitative Variables <ul><li>Continuous Variables - Can take on any number within a defined range. </li></ul><ul><ul><ul><li>Time-to-event (survival time) </li></ul></ul></ul><ul><ul><ul><li>Age </li></ul></ul></ul><ul><ul><ul><li>Blood pressure </li></ul></ul></ul><ul><ul><ul><li>Serum insulin </li></ul></ul></ul><ul><ul><ul><li>Speed of a car </li></ul></ul></ul><ul><ul><ul><li>Income </li></ul></ul></ul><ul><ul><ul><li>Shock index (Kline et al.) </li></ul></ul></ul>
    12. 12. Looking at Data <ul><li> How are the data distributed? </li></ul><ul><ul><li>Where is the center? </li></ul></ul><ul><ul><li>What is the range? </li></ul></ul><ul><ul><li>What’s the shape of the distribution (e.g., Gaussian, binomial, exponential, skewed)? </li></ul></ul><ul><li> Are there “outliers”? </li></ul><ul><li> Are there data points that don’t make sense? </li></ul>
    13. 13. The first rule of statistics: USE COMMON SENSE! 90% of the information is contained in the graph.
    14. 14. Frequency Plots (univariate) <ul><ul><li>Categorical variables </li></ul></ul><ul><ul><li>Bar Chart </li></ul></ul><ul><ul><li>Continuous variables </li></ul></ul><ul><ul><li>Box Plot </li></ul></ul><ul><ul><li>Histogram </li></ul></ul>
    15. 15. Bar Chart <ul><li>Used for categorical variables to show frequency or proportion in each category. </li></ul><ul><li>Translate the data from frequency tables into a pictorial representation… </li></ul>
    16. 16. Bar Chart: categorical variables no yes
    17. 17. Much easier to extract information from a bar chart than from a table! Bar Chart for SI categories Number of Patients Shock Index Category 0.0 16.7 33.3 50.0 66.7 83.3 100.0 116.7 133.3 150.0 166.7 183.3 200.0 1 2 3 4 5 6 7 8 9 10
    18. 18. Box plot and histograms: for continuous variables <ul><li>To show the distribution (shape, center, range, variation) of continuous variables. </li></ul>
    19. 19. 0.0 0.7 1.3 2.0 SI Box Plot: Shock Index Shock Index Units “ whisker” Q3 + 1.5IQR = .8+1.5(.25)=1.175 75th percentile (0.8) 25th percentile (0.55) maximum (1.7) interquartile range (IQR) = .8-.55 = .25 minimum (or Q1-1.5IQR) Outliers median (.66)
    20. 20. Note the “right skew” Bins of size 0.1 0.0 8.3 16.7 25.0 0.0 0.7 1.3 2.0 Histogram of SI SI Percent
    21. 21. 100 bins (too much detail)
    22. 22. 2 bins (too little detail)
    23. 23. Also shows the “right skew” 0.0 0.7 1.3 2.0 SI Box Plot: Shock Index Shock Index Units
    24. 24. 0.0 33.3 66.7 100.0 AGE Box Plot: Age Variables Years More symmetric 75th percentile 25th percentile maximum interquartile range minimum median
    25. 25. Histogram: Age Not skewed, but not bell-shaped either… 0.0 4.7 9.3 14.0 0.0 33.3 66.7 100.0 AGE (Years) Percent
    26. 26. Some histograms from last year’s class (n=18) Starting with politics…
    27. 29. Feelings about math and writing…
    28. 30. Optimism…
    29. 31. Measures of central tendency <ul><li>Mean </li></ul><ul><li>Median </li></ul><ul><li>Mode </li></ul>
    30. 32. Central Tendency <ul><li>Mean – the average; the balancing point </li></ul><ul><ul><li>calculation: the sum of values divided by the sample size </li></ul></ul>In math shorthand:
    31. 33. Mean: example <ul><ul><li>Some data : </li></ul></ul><ul><ul><li>Age of participants: 17 19 21 22 23 23 23 38 </li></ul></ul>
    32. 34. Mean of age in Kline’s data <ul><ul><ul><li>Means Section of AGE </li></ul></ul></ul><ul><ul><ul><li>Geometric Harmonic </li></ul></ul></ul><ul><ul><ul><li>Parameter Mean Median Mean Mean Sum Mode </li></ul></ul></ul><ul><ul><ul><li>Value 50.19334 49 46.66865 43.00606 46730 49 </li></ul></ul></ul><ul><ul><ul><li>556.9546 </li></ul></ul></ul>0.0 4.7 9.3 14.0 0.0 33.3 66.7 100.0 Percent
    33. 35. Mean of age in Kline’s data The balancing point 0.0 4.7 9.3 14.0 0.0 33.3 66.7 100.0 Percent
    34. 36. Mean of Pulmonary Embolism? (Binary variable?) 19.44% (181) 80.56% (750)
    35. 37. Mean <ul><li>The mean is affected by extreme values (outliers) </li></ul>0 1 2 3 4 5 6 7 8 9 10 Mean = 3 0 1 2 3 4 5 6 7 8 9 10 Mean = 4 <ul><li>Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall </li></ul>
    36. 38. Central Tendency <ul><li>Median – the exact middle value </li></ul><ul><li>Calculation: </li></ul><ul><li>If there are an odd number of observations, find the middle value </li></ul><ul><li>If there are an even number of observations, find the middle two values and average them. </li></ul>
    37. 39. Median: example <ul><ul><li>Some data : </li></ul></ul><ul><ul><li>Age of participants: 17 19 21 22 23 23 23 38 </li></ul></ul>Median = (22+23)/2 = 22.5
    38. 40. Median of age in Kline’s data <ul><ul><ul><li>Means Section of AGE </li></ul></ul></ul><ul><ul><ul><li>Geometric Harmonic </li></ul></ul></ul><ul><ul><ul><li>Parameter Mean Median Mean Mean Sum Mode </li></ul></ul></ul><ul><ul><ul><li>Value 50.19334 49 46.66865 43.00606 46730 49 </li></ul></ul></ul>0.0 4.7 9.3 14.0 0.0 33.3 66.7 100.0 AGE (Years) Percent
    39. 41. Median of age in Kline’s data 0.0 4.7 9.3 14.0 0.0 33.3 66.7 100.0 Percent 50% of mass 50% of mass
    40. 42. Does PE have a median? <ul><li>Yes, if you line up the 0’s and 1’s, the middle number is 0. </li></ul>
    41. 43. Median <ul><li>The median is not affected by extreme values (outliers). </li></ul>0 1 2 3 4 5 6 7 8 9 10 Median = 3 0 1 2 3 4 5 6 7 8 9 10 Median = 3 <ul><ul><ul><ul><ul><li>SSlide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall </li></ul></ul></ul></ul></ul>
    42. 44. Central Tendency <ul><li>Mode – the value that occurs most frequently </li></ul>
    43. 45. Mode: example <ul><ul><li>Some data : </li></ul></ul><ul><ul><li>Age of participants: 17 19 21 22 23 23 23 38 </li></ul></ul>Mode = 23 (occurs 3 times)
    44. 46. Mode of age in Kline’s data <ul><ul><ul><li>Means Section of AGE </li></ul></ul></ul><ul><ul><ul><li>Geometric Harmonic </li></ul></ul></ul><ul><ul><ul><li>Parameter Mean Median Mean Mean Sum Mode </li></ul></ul></ul><ul><ul><ul><li>Value 50.19334 49 46.66865 43.00606 46730 49 </li></ul></ul></ul>
    45. 47. Mode of PE? <ul><li>0 appears more than 1, so 0 is the mode. </li></ul>
    46. 48. Measures of Variation/Dispersion <ul><li>Range </li></ul><ul><li>Percentiles/quartiles </li></ul><ul><li>Interquartile range </li></ul><ul><li>Standard deviation/Variance </li></ul>
    47. 49. Range <ul><li>Difference between the largest and the smallest observations. </li></ul>
    48. 50. 0.0 4.7 9.3 14.0 0.0 33.3 66.7 100.0 Range of age: 94 years-15 years = 79 years AGE (Years) Percent
    49. 51. Range of PE? <ul><li>1-0 = 1 </li></ul>
    50. 52. Quartiles 25% 25% 25% 25% <ul><li>The first quartile, Q 1 , is the value for which 25% of the observations are smaller and 75% are larger </li></ul><ul><li>Q 2 is the same as the median (50% are smaller, 50% are larger) </li></ul><ul><li>Only 25% of the observations are greater than the third quartile </li></ul>Q1 Q2 Q3
    51. 53. Interquartile Range <ul><li>Interquartile range = 3 rd quartile – 1 st quartile = Q 3 – Q 1 </li></ul>
    52. 54. Interquartile Range: age Median (Q2) maximum minimum Q1 Q3 25% 25% 25% 25% 15 35 49 65 94 Interquartile range = 65 – 35 = 30
    53. 55. <ul><li>Average (roughly) of squared deviations of values from the mean </li></ul>Variance
    54. 56. Why squared deviations? <ul><li>Adding deviations will yield a sum of 0. </li></ul><ul><li>Absolute values are tricky! </li></ul><ul><li>Squares eliminate the negatives. </li></ul><ul><li>Result: </li></ul><ul><ul><li>Increasing contribution to the variance as you go farther from the mean. </li></ul></ul>
    55. 57. Standard Deviation <ul><li>Most commonly used measure of variation </li></ul><ul><li>Shows variation about the mean </li></ul><ul><li>Has the same units as the original data </li></ul>
    56. 58. Calculation Example: Sample Standard Deviation Age data (n=8) : 17 19 21 22 23 23 23 38 n = 8 Mean = X = 23.25
    57. 59. 0.0 4.7 9.3 14.0 0.0 33.3 66.7 100.0 AGE (Years) Percent Std. dev is a measure of the “average” scatter around the mean. Estimation method: if the distribution is bell shaped, the range is around 6 SD, so here rough guess for SD is 79/6 = 13
    58. 60. Std. Deviation age <ul><ul><ul><li>Variation Section of AGE </li></ul></ul></ul><ul><ul><ul><li>Standard </li></ul></ul></ul><ul><ul><ul><li>Parameter Variance Deviation </li></ul></ul></ul><ul><ul><ul><li>Value 333.1884 18.25345 </li></ul></ul></ul>
    59. 61. 0.0 62.5 125.0 187.5 250.0 0.0 0.5 1.0 1.5 2.0 Std Dev of Shock Index SI Count Estimation method: if the distribution is bell shaped, the range is around 6 SD, so here rough guess for SD is 1.4/6 =.23 Std. dev is a measure of the “average” scatter around the mean.
    60. 62. Std. Deviation SI <ul><ul><ul><li>Variation Section of SI </li></ul></ul></ul><ul><ul><ul><li>Standard Std Error Interquartile </li></ul></ul></ul><ul><ul><ul><li>Parameter Variance Deviation of Mean Range Range </li></ul></ul></ul><ul><ul><ul><li>Value 4.155749E-02 0.2038566 6.681129E-03 0.2460432 1.430856 </li></ul></ul></ul>
    61. 63. Std. Dev of binary variable, PE Std. dev is a measure of the “average” scatter around the mean. 19.44% 80.56%
    62. 64. Std. Deviation PE <ul><ul><ul><li>Variation Section of PE </li></ul></ul></ul><ul><ul><ul><li>Standard </li></ul></ul></ul><ul><ul><ul><li>Parameter Variance Deviation </li></ul></ul></ul><ul><ul><ul><li>Value 0.156786 0.3959621 </li></ul></ul></ul>
    63. 65. Comparing Standard Deviations Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 S = 0.926 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 S = 4.570 Data C <ul><li>SSlide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall </li></ul>
    64. 66. <ul><li>Regardless of how the data are distributed, a certain percentage of values must fall within K standard deviations from the mean: </li></ul>Bienaym é- Chebyshev Rule within At least (1 - 1/1 2 ) = 0% …….….. k=1 ( μ ± 1 σ ) (1 - 1/2 2 ) = 75% …........ k=2 ( μ ± 2 σ ) (1 - 1/3 2 ) = 89% ……….... k=3 ( μ ± 3 σ ) Note use of  (sigma) to represent “standard deviation.” Note use of  (mu) to represent “mean”.
    65. 67. Symbol Clarification <ul><li>S = Sample standard deviation (example of a “sample statistic”) </li></ul><ul><li> = Standard deviation of the entire population (example of a “population parameter”) or from a theoretical probability distribution </li></ul><ul><li>X = Sample mean </li></ul><ul><li>µ = Population or theoretical mean </li></ul>
    66. 68. **The beauty of the normal curve: No matter what  and  are, the area between  -  and  +  is about 68%; the area between  -2  and  +2  is about 95%; and the area between  -3  and  +3  is about 99.7%. Almost all values fall within 3 standard deviations.
    67. 69. 68-95-99.7 Rule 68% of the data 95% of the data 99.7% of the data
    68. 70. Summary of Symbols <ul><li>S 2 = Sample variance </li></ul><ul><li>S = Sample standard dev </li></ul><ul><li> 2 = Population (true or theoretical) variance </li></ul><ul><li> = Population standard dev. </li></ul><ul><li>X = Sample mean </li></ul><ul><li>µ = Population mean </li></ul><ul><li>IQR = interquartile range (middle 50%) </li></ul>
    69. 71. Examples of bad graphics
    70. 72. What’s wrong with this graph? from : ER Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983, p.69
    71. 73. From: Visual Revelations: Graphical Tales of Fate and Deception from Napoleon Bonaparte to Ross Perot Wainer, H. 1997, p.29. Notice the X-axis
    72. 74. Correctly scaled X-axis…
    73. 75. Report of the Presidential Commission on the Space Shuttle Challenger Accident , 1986 (vol 1, p. 145) The graph excludes the observations where no O-rings failed.
    74. 76. <ul><li>http:// www.math.yorku.ca /SCS/Gallery/ </li></ul>Smooth curve at least shows the trend toward failure at high and low temperatures…
    75. 77. Even better: graph all the data (including non-failures) using a logistic regression model Tappin, L. (1994). &quot;Analyzing data relating to the Challenger disaster&quot;. Mathematics Teacher , 87, 423-426
    76. 78. What’s wrong with this graph? from : ER Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983, p.74
    77. 80. What’s the message here? Diagraphics II , 1994
    78. 81. Diagraphics II , 1994
    79. 82. For more examples… <ul><li>http:// www.math.yorku.ca /SCS/Gallery/ </li></ul>
    80. 83. “Lying” with statistics <ul><li>More accurately, misleading with statistics… </li></ul>
    81. 84. Example 1: projected statistics <ul><li>Lifetime risk of melanoma: </li></ul><ul><li>1935: 1/1500 </li></ul><ul><li>1960: 1/600 </li></ul><ul><li>1985: 1/150 </li></ul><ul><li>2000: 1/74 </li></ul><ul><li>2006: 1/60 </li></ul><ul><li>http://www.melanoma.org/mrf_facts.pdf </li></ul>
    82. 85. Example 1: projected statistics <ul><li>How do you think these statistics are calculated? </li></ul><ul><li>How do we know what the lifetime risk of a person born in 2006 will be? </li></ul>
    83. 86. Example 1: projected statistics <ul><li>Interestingly, a clever clinical researcher recently went back and calculated (using SEER data) the actual lifetime risk (or risk up to 70 years) of melanoma for a person born in 1935. </li></ul><ul><li>The answer? </li></ul><ul><li>Closer to 1/150 (one order of magnitude off) </li></ul><ul><li>(Martin Weinstock of Brown University, AAD conference 2006) </li></ul>
    84. 87. Example 2: propagation of statistics <ul><li>In many papers and reviews of eating disorders in women athletes, authors cite the statistic that 15 to 62% of female athletes have disordered eating. </li></ul><ul><li>I’ve found that this statistic is attributed to about 50 different sources in the literature and cited all over the place with or without citations... </li></ul>
    85. 88. For example… <ul><li>In a recent review (Hobart and Smucker, The Female Athlete Triad, American Family Physician, 2000): </li></ul><ul><li>“ Although the exact prevalence of the female athlete triad is unknown, studies have reported disordered eating behavior in 15 to 62 percent of female college athletes.” </li></ul><ul><li>No citations given. </li></ul>
    86. 89. And… <ul><li>Fact Sheet on eating disorders: </li></ul><ul><li>“ Among female athletes, the prevalence of eating disorders is reported to be between 15% and 62%.” Citation given: Costin, Carolyn. (1999) The Eating Disorder Source Book: A comprehensive guide to the causes, treatment, and prevention of eating disorders. 2 nd edition. Lowell House: Los Angeles. </li></ul>
    87. 90. And… <ul><li>From a Fact Sheet on disordered eating from a college website: </li></ul><ul><li>“ Eating disorders are significantly higher (15 to 62 percent) in the athletic population than the general population.” </li></ul><ul><li>No citation given. </li></ul>
    88. 91. And… <ul><li>“ Studies report between 15% and 62% of college women engage in problematic weight control behaviors (Berry & Howe, 2000).” (in The Sport Journal , 2004) </li></ul><ul><li>Citation: Berry, T.R. & Howe, B.L. (2000, Sept). Risk factors for disordered eating in female university athletes. Journal of Sport Behavior, 23(3), 207-219. </li></ul>
    89. 92. And… <ul><li>1999 NY Times article </li></ul><ul><li>“But informal surveys suggest that 15 percent to 62 percent of female athletes are affected by disordered behavior that ranges from a preoccupation with losing weight to anorexia or bulimia.” </li></ul>
    90. 93. And <ul><li>“ It has been estimated that the prevalence of disordered eating in female athletes ranges from 15% to 62%.” ( in Journal of General Internal Medicine   15  (8), 577-590.) Citations: </li></ul><ul><li>Steen SN. The competitive athlete. In: Rickert VI, ed. Adolescent Nutrition: Assessment and Management . New York, NY: Chapman and Hall; 1996:223 47. </li></ul><ul><li>Tofler IR, Stryer BK, Micheli LJ. Physical and emotional problems of elite female gymnasts. N Engl J Med . 1996; 335 :281 3. </li></ul>
    91. 94. Where did the statistics come from? The 15%: Dummer GM, Rosen LW, Heusner WW, Roberts PJ, and Counsilman JE. Pathogenic weight-control behaviors of young competitive swimmers. Physician Sportsmed 1987; 15: 75-84. The “to”: Rosen LW, McKeag DB, O’Hough D, Curley VC. Pathogenic weight-control behaviors in female athletes. Physician Sportsmed . 1986; 14: 79-86. The 62%:Rosen LW, Hough DO. Pathogenic weight-control behaviors of female college gymnasts. Physician Sportsmed 1988; 16:140-146.
    92. 95. Where did the statistics come from? <ul><li>Study design? Control group? </li></ul><ul><ul><li>Cross-sectional survey (all) </li></ul></ul><ul><ul><li>No non-athlete control groups </li></ul></ul><ul><li>Population/sample size? </li></ul><ul><ul><li>Convenience samples </li></ul></ul><ul><ul><li>Rosen et al. 1986: 182 varsity athletes from two midwestern universities (basketball, field hockey, golf, running, swimming, gymnastics, volleyball, etc.) </li></ul></ul><ul><ul><li>Dummer et al. 1987: 486 9-18 year old swimmers at a swim camp </li></ul></ul><ul><ul><li>Rosen et al. 1988: 42 college gymnasts from 5 teams at an athletic conference </li></ul></ul>
    93. 96. Where did the statistics come from? <ul><li>Measurement? </li></ul><ul><ul><li>Instrument: Michigan State University Weight Control Survey </li></ul></ul><ul><ul><li>Disordered eating = at least one pathogenic weight control behavior: </li></ul></ul><ul><ul><ul><li>Self-induced vomiting </li></ul></ul></ul><ul><ul><ul><li>fasting </li></ul></ul></ul><ul><ul><ul><li>Laxatives </li></ul></ul></ul><ul><ul><ul><li>Diet pills </li></ul></ul></ul><ul><ul><ul><li>Diuretics </li></ul></ul></ul><ul><ul><ul><li>In the 1986 survey, they required use 1/month; in the 1988 survey, they required use twice-weekly </li></ul></ul></ul><ul><ul><ul><li>In the 1988 survey, they added fluid restriction </li></ul></ul></ul>
    94. 97. Where did the statistics come from? <ul><li>Findings? </li></ul><ul><ul><li>Rosen et al. 1986: 32% used at least one “pathogenic weight-control behavior” (ranges: 8% of 13 basketball players to 73.7% of 19 gymnasts) </li></ul></ul><ul><ul><li>Dummer et al. 1987: 15.4% of swimmers used at least one of these behaviors </li></ul></ul><ul><ul><li>Rosen et al. 1988: 62% of gymnasts used at least one of these behaviors </li></ul></ul>
    95. 98. References <ul><li>http://www.math.yorku.ca/SCS/Gallery/ </li></ul><ul><li>Kline et al. Annals of Emergency Medicine 2002; 39: 144-152. </li></ul><ul><li>Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall </li></ul><ul><li>Tappin, L. (1994). &quot;Analyzing data relating to the Challenger disaster&quot;. Mathematics Teacher , 87, 423-426 </li></ul><ul><li>Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983. </li></ul><ul><li>Visual Revelations: Graphical Tales of Fate and Deception from Napoleon Bonaparte to Ross Perot Wainer, H. 1997. </li></ul>

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