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
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.
I will use four variables from their dataset today:
Pulmonary embolism (yes/no)
Age (years)
Shock index = heart rate/systolic BP
Shock index categories = take shock index and divide it into 10 groups (lowest to highest shock index)
4.
Types of Variables: Overview Categorical Quantitative continuous discrete ordinal nominal binary 2 categories + more categories + order matters + numerical + uninterrupted
5.
Categorical Variables
Also known as “qualitative.”
Categories.
treatment groups
exposure groups
disease status
6.
Categorical Variables
Dichotomous (binary) – two levels
Dead/alive
Treatment/placebo
Disease/no disease
Exposed/Unexposed
Heads/Tails
Pulmonary Embolism (yes/no)
Male/female
7.
Categorical Variables
Nominal variables – Named categories Order doesn’t matter!
The blood type of a patient (O, A, B, AB)
Marital status
Occupation
8.
Categorical Variables
Ordinal variable – Ordered categories. Order matters!
Staging in breast cancer as I, II, III, or IV
Birth order—1st, 2nd, 3rd, etc.
Letter grades (A, B, C, D, F)
Ratings on a scale from 1-5
Ratings on: always; usually; many times; once in a while; almost never; never
Age in categories (10-20, 20-30, etc.)
Shock index categories (Kline et al.)
9.
Quantitative Variables
Numerical variables; may be arithmetically manipulated.
Counts
Time
Age
Height
10.
Quantitative Variables
Discrete Numbers – a limited set of distinct values, such as whole numbers.
Number of new AIDS cases in CA in a year (counts)
Years of school completed
The number of children in the family (cannot have a half a child!)
The number of deaths in a defined time period (cannot have a partial death!)
Roll of a die
11.
Quantitative Variables
Continuous Variables - Can take on any number within a defined range.
Time-to-event (survival time)
Age
Blood pressure
Serum insulin
Speed of a car
Income
Shock index (Kline et al.)
12.
Looking at Data
How are the data distributed?
Where is the center?
What is the range?
What’s the shape of the distribution (e.g., Gaussian, binomial, exponential, skewed)?
Are there “outliers”?
Are there data points that don’t make sense?
13.
The first rule of statistics: USE COMMON SENSE! 90% of the information is contained in the graph.
14.
Frequency Plots (univariate)
Categorical variables
Bar Chart
Continuous variables
Box Plot
Histogram
15.
Bar Chart
Used for categorical variables to show frequency or proportion in each category.
Translate the data from frequency tables into a pictorial representation…
16.
Bar Chart: categorical variables no yes
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.
Box plot and histograms: for continuous variables
To show the distribution (shape, center, range, variation) of continuous variables.
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.
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.
100 bins (too much detail)
22.
2 bins (too little detail)
23.
Also shows the “right skew” 0.0 0.7 1.3 2.0 SI Box Plot: Shock Index Shock Index Units
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.
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.
Some histograms from last year’s class (n=18) Starting with politics…
27.
28.
29.
Feelings about math and writing…
30.
Optimism…
31.
Measures of central tendency
Mean
Median
Mode
32.
Central Tendency
Mean – the average; the balancing point
calculation: the sum of values divided by the sample size
In math shorthand:
33.
Mean: example
Some data :
Age of participants: 17 19 21 22 23 23 23 38
34.
Mean of age in Kline’s data
Means Section of AGE
Geometric Harmonic
Parameter Mean Median Mean Mean Sum Mode
Value 50.19334 49 46.66865 43.00606 46730 49
556.9546
0.0 4.7 9.3 14.0 0.0 33.3 66.7 100.0 Percent
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
36.
Mean of Pulmonary Embolism? (Binary variable?) 19.44% (181) 80.56% (750)
SSlide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
44.
Central Tendency
Mode – the value that occurs most frequently
45.
Mode: example
Some data :
Age of participants: 17 19 21 22 23 23 23 38
Mode = 23 (occurs 3 times)
46.
Mode of age in Kline’s data
Means Section of AGE
Geometric Harmonic
Parameter Mean Median Mean Mean Sum Mode
Value 50.19334 49 46.66865 43.00606 46730 49
47.
Mode of PE?
0 appears more than 1, so 0 is the mode.
48.
Measures of Variation/Dispersion
Range
Percentiles/quartiles
Interquartile range
Standard deviation/Variance
49.
Range
Difference between the largest and the smallest observations.
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
51.
Range of PE?
1-0 = 1
52.
Quartiles 25% 25% 25% 25%
The first quartile, Q 1 , is the value for which 25% of the observations are smaller and 75% are larger
Q 2 is the same as the median (50% are smaller, 50% are larger)
Only 25% of the observations are greater than the third quartile
Q1 Q2 Q3
53.
Interquartile Range
Interquartile range = 3 rd quartile – 1 st quartile = Q 3 – Q 1
54.
Interquartile Range: age Median (Q2) maximum minimum Q1 Q3 25% 25% 25% 25% 15 35 49 65 94 Interquartile range = 65 – 35 = 30
55.
Average (roughly) of squared deviations of values from the mean
Variance
56.
Why squared deviations?
Adding deviations will yield a sum of 0.
Absolute values are tricky!
Squares eliminate the negatives.
Result:
Increasing contribution to the variance as you go farther from the mean.
57.
Standard Deviation
Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data
58.
Calculation Example: Sample Standard Deviation Age data (n=8) : 17 19 21 22 23 23 23 38 n = 8 Mean = X = 23.25
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
60.
Std. Deviation age
Variation Section of AGE
Standard
Parameter Variance Deviation
Value 333.1884 18.25345
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.
62.
Std. Deviation SI
Variation Section of SI
Standard Std Error Interquartile
Parameter Variance Deviation of Mean Range Range
Value 4.155749E-02 0.2038566 6.681129E-03 0.2460432 1.430856
63.
Std. Dev of binary variable, PE Std. dev is a measure of the “average” scatter around the mean. 19.44% 80.56%
64.
Std. Deviation PE
Variation Section of PE
Standard
Parameter Variance Deviation
Value 0.156786 0.3959621
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
SSlide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
66.
Regardless of how the data are distributed, a certain percentage of values must fall within K standard deviations from the mean:
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”.
67.
Symbol Clarification
S = Sample standard deviation (example of a “sample statistic”)
= Standard deviation of the entire population (example of a “population parameter”) or from a theoretical probability distribution
X = Sample mean
µ = Population or theoretical mean
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.
69.
68-95-99.7 Rule 68% of the data 95% of the data 99.7% of the data
70.
Summary of Symbols
S 2 = Sample variance
S = Sample standard dev
2 = Population (true or theoretical) variance
= Population standard dev.
X = Sample mean
µ = Population mean
IQR = interquartile range (middle 50%)
71.
Examples of bad graphics
72.
What’s wrong with this graph? from : ER Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983, p.69
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
74.
Correctly scaled X-axis…
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.
76.
http:// www.math.yorku.ca /SCS/Gallery/
Smooth curve at least shows the trend toward failure at high and low temperatures…
77.
Even better: graph all the data (including non-failures) using a logistic regression model Tappin, L. (1994). "Analyzing data relating to the Challenger disaster". Mathematics Teacher , 87, 423-426
78.
What’s wrong with this graph? from : ER Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983, p.74
79.
80.
What’s the message here? Diagraphics II , 1994
81.
Diagraphics II , 1994
82.
For more examples…
http:// www.math.yorku.ca /SCS/Gallery/
83.
“Lying” with statistics
More accurately, misleading with statistics…
84.
Example 1: projected statistics
Lifetime risk of melanoma:
1935: 1/1500
1960: 1/600
1985: 1/150
2000: 1/74
2006: 1/60
http://www.melanoma.org/mrf_facts.pdf
85.
Example 1: projected statistics
How do you think these statistics are calculated?
How do we know what the lifetime risk of a person born in 2006 will be?
86.
Example 1: projected statistics
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.
The answer?
Closer to 1/150 (one order of magnitude off)
(Martin Weinstock of Brown University, AAD conference 2006)
87.
Example 2: propagation of statistics
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.
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...
88.
For example…
In a recent review (Hobart and Smucker, The Female Athlete Triad, American Family Physician, 2000):
“ 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.”
No citations given.
89.
And…
Fact Sheet on eating disorders:
“ 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.
90.
And…
From a Fact Sheet on disordered eating from a college website:
“ Eating disorders are significantly higher (15 to 62 percent) in the athletic population than the general population.”
No citation given.
91.
And…
“ Studies report between 15% and 62% of college women engage in problematic weight control behaviors (Berry & Howe, 2000).” (in The Sport Journal , 2004)
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.
92.
And…
1999 NY Times article
“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.”
93.
And
“ 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:
Steen SN. The competitive athlete. In: Rickert VI, ed. Adolescent Nutrition: Assessment and Management . New York, NY: Chapman and Hall; 1996:223 47.
Tofler IR, Stryer BK, Micheli LJ. Physical and emotional problems of elite female gymnasts. N Engl J Med . 1996; 335 :281 3.
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.
95.
Where did the statistics come from?
Study design? Control group?
Cross-sectional survey (all)
No non-athlete control groups
Population/sample size?
Convenience samples
Rosen et al. 1986: 182 varsity athletes from two midwestern universities (basketball, field hockey, golf, running, swimming, gymnastics, volleyball, etc.)
Dummer et al. 1987: 486 9-18 year old swimmers at a swim camp
Rosen et al. 1988: 42 college gymnasts from 5 teams at an athletic conference
96.
Where did the statistics come from?
Measurement?
Instrument: Michigan State University Weight Control Survey
Disordered eating = at least one pathogenic weight control behavior:
Self-induced vomiting
fasting
Laxatives
Diet pills
Diuretics
In the 1986 survey, they required use 1/month; in the 1988 survey, they required use twice-weekly
In the 1988 survey, they added fluid restriction
97.
Where did the statistics come from?
Findings?
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)
Dummer et al. 1987: 15.4% of swimmers used at least one of these behaviors
Rosen et al. 1988: 62% of gymnasts used at least one of these behaviors
98.
References
http://www.math.yorku.ca/SCS/Gallery/
Kline et al. Annals of Emergency Medicine 2002; 39: 144-152.
Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
Tappin, L. (1994). "Analyzing data relating to the Challenger disaster". Mathematics Teacher , 87, 423-426
Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983.
Visual Revelations: Graphical Tales of Fate and Deception from Napoleon Bonaparte to Ross Perot Wainer, H. 1997.