Descriptive Statistics

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
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?

3
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

4
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

5
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.

6
Measures of Variation

• Measures of how similar or different individuals
  measurements are
• Range -- Largest-Smallest observation
• Deviation -- Difference between ith individuals
  outcome and the sample mean

• Variance of n observations Y1,...,Yn is the
  average squared deviation

7
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 )

8
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.

9
Other Measures and Plots

• Interquartile Range (IQR)-- 75thile - 25thile
  (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

10
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.

11
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 partners 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).
12
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?

13
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.