Stat 226 - Lecture 5

01/28/14

Announcements

  • The project is now available on Blackboard. There will be 4 steps. The first step is due Feburary 10th.
  • Keep in mind you will need to collect data associated with the variables in the first step. Come up with a realistic example!
  • Reading quiz on Chapter 12 due this Sunday.
  • Go to office hours if you have problems with JMP!
  • Any questions?

plot of chunk head


          [,1]
  [88,89)    2
  [89,90)   23
  [90,91)   57
  [91,92)  100
  [92,93)  100
  [93,94)   32
  [94,95)    1
  • Histograms are essential to this class and statistics in general.
  • A histogram displays the distribution of a quantitative variable.
  • In simpler terms, it tells us where values of a variable are likely to occur.
  • These are absolute frequencies (counts).
  • In order to make inferential statements about a population, we use relative frequencies!

plot of chunk rel


          [,1] 
  [88,89) 0.006
  [89,90) 0.073
  [90,91) 0.181
  [91,92) 0.317
  [92,93) 0.317
  [93,94) 0.102
  [94,95) 0.003
  • To find relative frequencies, divide each count by the total number of observations.
  • Notice how the shape of histogram is preserved. Only the y-axis is changes!
  • Before, we plotted the number of pitches in each bin. Now we have the percent of pitches within each bin.
  • For example, only 0.6% of all these pitches are between 88 and 89 MPH!

Percentiles

plot of chunk percents


          [,1] 
  [88,89) 0.006
  [89,90) 0.073
  [90,91) 0.181
  [91,92) 0.317
  [92,93) 0.317
  [93,94) 0.102
  [94,95) 0.003
  • What percentage of pitches thrown by Rivera are less than 90 MPH?
  • Add up the relative frequencies…0.006 + 0.073 = 0.079 (about 8%)
  • We say that 90 MPH is the 8th percentile.

Percentiles continued

plot of chunk percents2


          [,1] 
  [88,89) 0.006
  [89,90) 0.073
  [90,91) 0.181
  [91,92) 0.317
  [92,93) 0.317
  [93,94) 0.102
  [94,95) 0.003
  • What percentage of pitches thrown by Rivera are less than 91 MPH?
  • Add up the relative frequencies…0.006 + 0.073 + .181 = 0.26
  • We say that 91 MPH is the 26th percentile.

Foreshadowing

plot of chunk dens

  • In practice, we find a “smooth curve” based on sample data in order to make inferential statements about parameters.
  • For instance, based on this data, I am about 99% confident the true average pitch speed is between 89 and 94 MPH.

A Mental Exercise

plot of chunk x

  • What's the difference between these two histograms?

    • There is “more variation” or “more spread” in the bottom histogram!

    Measures of spread (or variability)

    • Range: Maximum - Minimum
    • IQR: Q3 - Q1
    • Variance and Standard Deviation: see pages 75-79 in the lecture notes.

    Summarizing Measurements

    nominal ordinal quantitative
    mode Yes Yes Yes
    mean No No Yes
    variance No No Yes
    percentiles No Yes Yes
    • If you can calculate a percentile, you can calculate the 5 number summary.
    • If you can calculate variance, you can calculate standard deviation.

    Quantitative Measurements

    center spread
    skewed median IQR, Range
    symmetric mean Standard Deviation
    • Both the mean and standard deviation are influenced by outliers. That is why we don't use them for skewed distributions.
    • Still have to determine whether the distribution is skewed or not. How should we do this?

    Your Turn

    • Finish the heights worksheet, then consider the following:
    • For each of the following data scenarios, determine the appropriate measure of center and measure of spread. To answer the question, think about what values are plausible and whether a histogram would yield a more symmetric or a more skewed distribution.
      • Household income in the US.
      • In-state tuition at US public colleges.
      • Runs per game scored by the Chicago Cubs (or any baseball team)
      • Points per game scored by the Chicago Bulls (or any basketball team)