1. Descriptive Statistics
Tabular and Graphical Displays
Frequency Distribution - List of intervals of values for a variable, and the number of occurrences per interval
Relative Frequency - Proportion (often reported as a percentage) of observations falling in the interval
Histogram/Bar Chart - Graphical representation of a Relative Frequency distribution
Stem and Leaf Plot - Horizontal tabular display of data, based on 2 digits (stem/leaf)

2. Constructing Pie Charts
Select a small number of categories (say 5 or 6 at most) to avoid many narrow “slivers”
If possible, arrange categories in ascending or descending order for categorical variables

3. Monthly Philly Rainfall 1825-1869 (1/100 in)

4. Constructing Bar Charts
Put frequencies on one axis (typically vertical, unless many categories) and categories on other
Draw rectangles over categories with height=frequency
Leave spaces between categories

5. Constructing Histograms
Used for numeric variables, so need Class Intervals
Let Range = Largest - Smallest Measurement
Break range into (say) 5-20 intervals depending on sample size
Make the width of the subintervals a convenient unit, and make “break points” so that no observations fall on them
Obtain Class Frequencies, the number in each subinterval
Obtain Relative Frequencies, proportion in each subinterval
Construct Histogram
Draw bars over each subinterval with height representing class frequency or relative frequency (shape will be the same)
Leave no space between bars to imply adjacency of class intervals

7. Interpreting Histograms
Probability: Heights of bars over the class intervals are proportional to the “chances” an individual chosen at random would fall in the interval
Unimodal: A histogram with a single major peak
Bimodal: Histogram with two distinct peaks (often evidence of two distinct groups of units)
Uniform: Interval heights are approximately equal
Symmetric: Right and Left portions are same shape
Right-Skewed: Right-hand side extends further
Left-Skewed: Left-hand side extends further

8. Stem-and-Leaf Plots
Simple, crude approach to obtaining shape of distribution without losing individual measurements to class intervals. Procedure:
Split each measurement into 2 sets of digits (stem and leaf)
List stems from smallest to largest
Line corresponding leaves aside stems from smallest to largest
If too cramped/narrow, break stems into two groups: low with leaves 0-4 and high with leaves 5-9
When numbers have many digits, trim off right-most (less significant) digits. Leaves should always be a single digit.

9. Comparing Groups Side-by-side bar charts 3 dimensional histograms
Back-to-back stem and leaf plots
Goal: Compare 2 (or more) groups wrt variable(s) being measured
Do measurements tend to differ among groups?

10. Summarizing Data of More than One Variable
Contingency Table: Cross-tabulation of units based on measurements of two qualitative variables simultaneously
Stacked Bar Graph: Bar chart with one variable represented on the horizontal axis, second variable as subcategories within bars
Cluster Bar Graph: Bar chart with one variable forming “major groupings” on horizontal axis, second variable used to make side-by-side comparisons within major groupings (displays all combinations in factorial expt)
Scatterplot: Plot with quantitaive variables y and x plotted against each other for each unit
Side-by-Side Boxplot: Compares distributions by groups

11. Example - Ginkgo and Acetazolamide for Acute Mountain Syndrome Among Himalayan Trekkers
Contingency Table (Counts)
Percent Outcome by Treatment

15. Sample & Population Distributions
Distributions of Samples and Populations- As samples get larger, the sample distribution gets smoother and looks more like the population distribution
U-shaped - Measurements tend to be large or small, fewer in middle range of values
Bell-shaped - Measurements tend to cluster around the middle with few extremes (symmetric)
Skewed Right - Few extreme large values
Skewed Left - Few extreme small values

16. Measures of Central Tendency
Mean - Sum of all measurements divided by the number of observations (even distribution of outcomes among cases). Can be highly influenced by extreme values.
Notation: Sample Measurements labeled Y1,...,Yn

17. Median, Percentiles, Mode
Median - Middle measurement after data have been ordered from smallest to largest. Appropriate for interval and ordinal scales
Pth percentile - Value where P% of measurements fall below and (100-P)% lie above. Lower quartile(25th), Median(50th), Upper quartile(75th) often reported
Mode - Most frequently occurring outcome. Typically reported for ordinal and nominal data.

18. Measures of Variation
Measures of how similar or different individual’s measurements are
Range -- Largest-Smallest observation
Deviation -- Difference between ith individual’s outcome and the sample mean:
Variance of n observations Y1,...,Yn is the “average” squared deviation:

19. Measures of Variation
Standard Deviation - Positive square root of the variance (measure in original units):
Properties of the standard deviation:
s  0, and only equals 0 if all observations are equal
s increases with the amount of variation around the mean
Division by n-1 (not n) is due to technical reasons (later)
s depends on the units of the data (e.g. $1000s vs $)

20. Empirical Rule
If the histogram of the data is approximately bell-shaped, then:
Approximately 68% of measurements lie within 1 standard deviation of the mean.
Approximately 95% of measurements lie within 2 standard deviations of the mean.
Virtually all of the measurements lie within 3 standard deviations of the mean.

21. Other Measures and Plots
Interquartile Range (IQR)-- 75th%ile - 25th%ile (measures the spread in the middle 50% of data)
Box Plots - Display a box containing middle 50% of measurements with line at median and lines extending from box. Breaks data into four quartiles
Outliers - Observations falling more than 1.5IQR above (below) upper (lower) quartile

22. Dependent and Independent Variables
Dependent variables are outcomes of interest to investigators. Also referred to as Responses or Endpoints
Independent variables are Factors that are often hypothesized to effect the outcomes (levels of dependent variables). Also referred to as Predictor or Explanatory Variables
Research ??? Does I.V.  D.V.

23. Example - Clinical Trials of Cialis
Clinical trials conducted worldwide to study efficacy and safety of Cialis (Tadalafil) for ED
Patients randomized to Placebo, 10mg, and 20mg
Co-Primary outcomes:
Change from baseline in erectile dysfunction domain if the International Index of Erectile Dysfunction (Numeric)
Response to: “Were you able to insert your P… into your partner’s V…?” (Nominal: Yes/No)
Response to: “Did your erection last long enough for you to have succesful intercourse?” (Nominal: Yes/No)
Source: Carson, et al. (2004).

24. Example - Clinical Trials of Cialis
Population: All adult males suffering from erectile dysfunction
Sample: 2102 men with mild-to-severe ED in 11 randomized clinical trials
Dependent Variable(s): Co-primary outcomes listed on previous slide
Independent Variable: Cialis Dose: (0, 10, 20 mg)
Research Questions: Does use of Cialis improve erectile function?

25. Contingency Tables
Tables representing all combinations of levels of explanatory and response variables
Numbers in table represent Counts of the number of cases in each cell
Row and column totals are called Marginal counts

26. 2x2 Tables - Notation n1+n2 (n1+n2)-(X1+X2) X1+X2 Outcome Total n2
Group 2 n1
n1-X1 X1
Group 1
Group
Absent
Present

27. Example - Firm Type/Product Quality
172
134 38
Outcome
Total 84 79 5
Vertically
Integrated 88 55 33
Not
Group
Low
Quality
High
Groups: Not Integrated (Weave only) vs Vertically integrated (Spin and Weave) Cotton Textile Producers
Outcomes: High Quality (High Count) vs Low Quality (Count)
Source: Temin (1988)

28. Scatterplots
Identify the explanatory and response variables of interest, and label them as x and y
Obtain a set of individuals and observe the pairs (xi , yi) for each pair. There will be n pairs.
Statistical convention has the response variable (y) placed on the vertical (up/down) axis and the explanatory variable (x) placed on the horizontal (left/right) axis. (Note: economists reverse axes in price/quantity demand plots)
Plot the n pairs of points (x,y) on the graph

29. France August,2003 Heat Wave Deaths
Individuals: 13 cities in France
Response: Excess Deaths(%) Aug1/19,2003 vs
Explanatory Variable: Change in Mean Temp in period (C)
Data:

30. France August,2003 Heat Wave Deaths

31. Sample Statistics/Population Parameters
Sample Mean and Standard Deviations are most commonly reported summaries of sample data. They are random variables since they will change from one sample to another.
Population Mean (m) and Standard Deviation (s) computed from a population of measurements are fixed (unknown in practice) values called parameters.

32. Example 1.3 - Grapefruit Juice Study
To import an EXCEL file, click on:
FILE  OPEN  DATA then change FILES OF TYPE to EXCEL (.xls)
To import a TEXT or DATA file, click on:
FILE  OPEN  DATA then change FILES OF TYPE to TEXT (.txt) or
DATA (.dat)
You will be prompted through a series of dialog boxes to import dataset

33. Descriptive Statistics-Numeric Data
After Importing your dataset, and providing names to variables, click on:
ANALYZE  DESCRIPTIVE STATISTICS DESCRIPTIVES
Choose any variables to be analyzed and place them in box on right
Options include:

34. Example 1.3 - Grapefruit Juice Study

35. Descriptive Statistics-General Data
After Importing your dataset, and providing names to variables, click on:
ANALYZE  DESCRIPTIVE STATISTICS FREQUENCIES
Choose any variables to be analyzed and place them in box on right
Options include (For Categorical Variables):
Frequency Tables
Pie Charts, Bar Charts
Options include (For Numeric Variables)
Frequency Tables (Useful for discrete data)
Measures of Central Tendency, Dispersion, Percentiles
Pie Charts, Histograms

36. Example 1.4 - Smoking Status

37. Vertical Bar Charts and Pie Charts
After Importing your dataset, and providing names to variables, click on:
GRAPHS  BAR…  SIMPLE (Summaries for Groups of Cases)  DEFINE
Bars Represent N of Cases (or % of Cases)
Put the variable of interest as the CATEGORY AXIS
GRAPHS  PIE… (Summaries for Groups of Cases)  DEFINE
Slices Represent N of Cases (or % of Cases)
Put the variable of interest as the DEFINE SLICES BY

38. Example 1.5 - Antibiotic Study

39. Histograms
After Importing your dataset, and providing names to variables, click on:
GRAPHS  HISTOGRAM
Select Variable to be plotted
Click on DISPLAY NORMAL CURVE if you want a normal curve superimposed (see Chapter 4).

40. Example 1.6 - Drug Approval Times

41. Side-by-Side Bar Charts
After Importing your dataset, and providing names to variables, click on:
GRAPHS  BAR…  Clustered (Summaries for Groups of Cases)  DEFINE
Bars Represent N of Cases (or % of Cases)
CATEGORY AXIS: Variable that represents groups to be compared (independent variable)
DEFINE CLUSTERS BY: Variable that represents outcomes of interest (dependent variable)

42. Example 1.7 - Streptomycin Study