Day 04: STA205 - Introduction to Statistical Methods

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Today:

We'll decide which write-up you'll hand in for you first "quiz grade"
- #8, 9, 7
At this point, quizzes become a distinct possibility. Each day it's possible that you will have a quiz over the material studied one week prior. On Tuesday we finished up chapter 2: today you should ask questions over chapter 2, and it's all fair game for a quiz next Tuesday....
Today we'll finish up section 3.1, and perhaps 3.2: therefore 3.1 is a distinct possibility for a quiz next Thursday.
Questions over problems from homework?
Section 3.1: Describing the center of a distribution of data
Measures of central tendency: measures typical (central) values
- mean
  - Also known as the arithmetic average
  - center of mass of histogram
  - Notation (see p. 64):
    
    population ("mu") versus sample means ("xbar")
    sigma notation
  - Sensitive to unusual, extreme values
  - We computed some means of M&M colors
- median
  - the middle value: insensitive to unusual, extreme values; but not as good as the mean for purposes of statistical inference.
  - In the event that there are an even number of data values, the median is the taken to be mean of the two values "in the middle".
- mode
  - the most "popular" value (or midpoint of the class with largest frequency)
  - You may have more than one mode
- Relationship between measures of central tendency with skewness of a histogram
  - symmetrically uni-modal (a centered hump):
    mean and median tend to be the same (example)
  - skewed right (more extremely large values):
    as the mean is sensitive to extremes, it chases the tail to the right. (example)
  - skewed left (more extremely small values):
    as the mean is sensitive to extremes, it chases the tail to the left. (example)
- Example: Computing mean and median for heights of NKU students, using StatCrunch
Section 3.2: Describing the Variability of a Distribution of Data
- Measures of Variability
  - A measure of variability is a statistic that describes the spread, or variation, or dispersion in a distribution of data.
  - range
    is equal to the largest value in the data minus the smallest value
  - sample variance
    the mean of the squared deviations (differences of individual data value from the mean)
    Notation - population variance (sigma squared) versus sample variance (s²)
    Notation, formulas, examples -- sample versus population
  - sample standard deviation
    is calculated as the square root of the sample variance, and represented by s.
    It is always a non-negative number.
    It has the same units as the data, and as the mean.
    Notation:
- Example for StatCrunch: shapes of distributions and variation
- When is
  - Variability good?
  - Variability bad?
- The empirical rule - the 68/95/99 rule.
  This is a rule based on the standard deviation that you'll need to know.
  - Based on the standard deviation, which serves as a "typical deviation"
  - 68% of the data is within one standard deviation of the mean
  - 95% of the data is within two standard deviation of the mean
  - 99% of the data is within three standard deviation of the mean
  - Works well when the distribution is bell-shaped
  - May work okay in any event.
  - Example
- Replacing a histogram by a smooth curve....
  - exam03-08
  - exam03-11

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