What is Statistics

1
What is Statistics?

• A branch of science concerned with selecting a
  subset of a large collection of data with
  inference as the objective. Wackerly, et
  al.
• Population the large collection of data in
  which we are interested
• Populations can be actual or conceptual
• Sample the subset which is selected from the
  population
• Inference drawing conclusions about the
  population based on the sample

? John J Currano, 07/02/2007
2

• Samples and populations always consist of
  numbers.
• Sometimes this is clear and straightforward
• What is the mean (average) U. S. household
  income ?
• What is the mean weight of all 15 oz. boxes of
  Cheerios produced by General Mills in the last
  month?
• Sometimes it is not so clear what the numbers
  are or should be, or even if it is a statistical
  question
• If the election were today, would Candidate X
  win?
• What is the acceleration due to gravity?

3

• When we make an inference about a population
  (e.g., get an estimate of the mean family income
  in the U.S.), we would also like to have some
  measure of how good the estimate is
• How close to the true value is it?
• How certain can we be that it is this close?
• To answer questions like these we need to study
  probability and probability distributions.

4
Characterizing a Set of Measurements - Graphical

• Data (200 observations)
• Histogram
• Relative Frequency Histogram

5
Characterizing a Set of Measurements - Graphical

• In both histograms, the area of a rectangle over
  a subinterval is proportional to the fraction of
  the observations in the subinterval.
• A useful variant of the relative frequency
  histogram is obtained by scaling the y-axis so as
  to make the total area 1 ? multiply all heights
  by 1/(total area).

6
Characterizing a Set of Measurements - Graphical

• A relative frequency distribution is a relative
  frequency histogram of the population in which
  the total area is 1.
• Depending on the population, it may look like a
  relative frequency histogram, or it may be
  smoothed out.

7
Numeric Characterizations of a Sample (p. 8)

• Sample Mean (measure of central tendency)
• Sample Variance (measure of dispersion, spread,
  variation)
• Sample Standard Deviation, s
• Corresponding Population Parameters
• Mean µ, variance ? 2, standard deviation ?
• will be defined later

8
Empirical Rule (p. 10)

• For a set of measurements (data) coming from a
  population whose distribution is approximately
  normal (bell- or mound-shaped) with mean ? and
  standard deviation ?, the interval
• (??, ??) contains approx. 68 of the
  measurements
• (?2?, ?2?) contains approx. 95 of the
  measurements
• (?3?, ?3?) contains almost all of the
  measurements