Review

1
Review

• Statistics the science of collecting,
  organizing, and interpreting data.
• Data collection.
• Data analysis - organize summarize data to
  bring out main features and clarify their
  underlying structure.
• Inference and decision theory extract the info
  provided by the data and may be used as a guide
  for further action.

2
Fundamental concepts

• Population the entire group of individuals that
  we want information about.
• Sample a part of the population that we actually
  examine in order to gather information.
• Sample size number of observations/individuals
  in a sample.
• Statistical inference to make an inference about
  a population based on the information contained
  in a sample.

3

• Data contains
• Individuals the objects described by the data
• Variable any characteristic of an individual. A
  variable can take different values for different
  individuals
• A categorical variable places an individual into
  one of several groups or categories.
• A quantitative variable takes numerical values
  for which arithmetic operations such as adding
  and averaging make sense.

4

• Distributions of Variables
• The distribution of a variable indicates what
  values a variable takes and how often it takes
  these values.
• For a categorical variable, distribution
  categories count/percent for each category
• For a quantitative variable, distribution
  pattern of variation of its values

5
Examining distribution

• Overall Pattern
• Shape
• Graphical techniques to display distributions
• Bar graph
• Pie chart
• Stemplot
• Histogram
• Modes peaks of a distribution.
• Unimodal or Bimodal
• Symmetric or skewed (left/right)?

6
Examining distribution

• Center
• Mean
• easily calculated
• easy to work with algebraically
• highly affected by outliers
• Not a resistant measure
• Median
• can be time consuming to calculate
• more resistant to a few extreme observations
  (sometimes outliers)
• robust
• Mode, Mean and Median
• Relative locations for skewed/symmetric dist.
• Which one to use

7
Examining distribution

• Spread
• Standard deviation and variance
• Definition and calculation
• Sum always 0
• Why (n-1)?
• Quartiles
• Definition and calculation
• IQR
• Rule to identify outliers
• The five-number summary
• Boxplots
• Comparison with histograms and stemplots
• Range, IQR, S.D.

8
Examining distribution

• Deviations
• Outliers some values that fall outside the
  overall pattern.
• IQR can help to identify outliers
• Modified boxplots
• Strategies
• Detect them, investigate their causes, correct
  them, or delete them, or give them individual
  attention.
• Use resistant methods such as median to reduce
  the influence of the outliers.

9
Linear transformation

• form XabX
• its effects on shape, center and spread

10
Density curves

• pdf
• properties of a pdf
• comparison with histograms
• mode, median, mean, quartiles and s.d. of density
  curves

11
The normal distributions

• normal distributions
• shape
• Symmetric around mean
• Single-peaked (Unimodal)
• Bell-shaped.
• center and spread

12
The normal distributions

• Standardizing and z-Scores
• Effects of Standardizing
• Standardizing is a linear transformation.
• The standardized values for any distribution
  always have mean 0 and standard deviation 1.
• Effects on shape, center and spread.
• Linear transformation normal into normal.

13
The normal distributions

• Standard normal distribution
• Standard normal table
• Normal probability calculation
• Normal quantile plot

• Standard normal distribution
• Standard normal table
• Normal probability calculation
• Normal quantile plot
• Relationships between variables
• Scatterplot
• Response and explanatory variables
• Form, direction, strength
• Adding categorical variable
• Correlation r
• -1ltrlt1
• The sign indicates positive/negative association
• What it means when r0, -1, 1
• Invariant to scaling
• Sensitive to outliers

14
Relationships between variables

• Scatterplot
• Response and explanatory variables
• Form, direction, strength
• Adding categorical variable
• Correlation r
• -1ltrlt1
• The sign indicates positive/negative association
• What it means when r0, -1, 1
• Invariant to scaling
• Sensitive to outliers