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Chapter 12: Describing Distributions with Numbers. We create graphs to give us a picture of the data. We also need numbers to summarize the center and spread of a distribution. Two types of descriptive statistics for categorical variables: 1) Counts (Frequencies)

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chapter 12 describing distributions with numbers
Chapter 12: Describing Distributions with Numbers
  • We create graphs to give us a picture of the data.
  • We also need numbers to summarize the center and spread of a distribution.
  • Two types of descriptive statistics for categorical variables:

1) Counts (Frequencies)

2) Rates or Proportions (Relative Frequencies)

  • Many statistics available to summarize quantitative variables.
homeruns in baseball
Homeruns in Baseball

Question: Who is the best home run hitter ever in major league baseball?

Players with high numbers of homeruns in seasons:

  • Babe Ruth
  • Roger Maris
  • Mark McGwire
  • Sammy Sosa
  • Barry Bonds
median and quartiles
Median and Quartiles

The median (M) is the midpoint of a distribution when the observations are arranged in increasing order. Number such that half the observations are smaller and the other half are larger. (p. 219)

  • List the data in order from smallest to largest
  • If n is odd, the median is the middle value.
  • If n is even, the median is the mean of the middle two values.
m for sosa and maris
M for Sosa and Maris

Calculate M for Sosa’s homeruns in a season (8 seasons, to 1999).

  • Data: 15, 10, 33, 25, 36, 40, 36, 66

Calculate M for Maris’s homeruns in a season (11 seasons).

  • Data: 14, 28, 16, 39, 61, 33, 23, 26, 13, 9, 5
percentiles
Percentiles
  • p×100% percentile – the value of a variable such that p×100% of the values are below it and (1-p)×100%of the values are above it where 0 < p < 1.
  • For the 35th percentile, p=0.35.
  • Where have you seen percentiles before?
quartiles
Quartiles
  • First Quartile (Q1): The value such that 25% of the data values lie below Q1 and 75% of the data values lie above Q1. (25th percentile)
  • Third Quartile (Q3): The value such that 75% of the data values lie below Q3 and 25% of the data values lie above Q3. (75th percentile)
  • The median is the second quartile (Q2) . (50th percentile)
calculating percentiles
Calculating percentiles:
  • Let n be the number of data values.
  • Order the n values from largest to smallest.
  • Calculate the product, n×p.
    • If the product is not an integer (0,1,2,3,…), then round it up to the next integer and take the corresponding ordered value.
    • If the product is an integer, say k, then average the kth and (k+1)-st ordered values.
5 number summary
5-Number Summary

The 5-number summary of a data set consists of the following descriptive statistics (p. 221):

Minimum, First Quartile (Q1), Median, Third Quartile (Q3), Maximum

Give the 5-number summaries for Sosa and Maris’s homeruns.

boxplot
Boxplot

A boxplot is a graphical representation of the 5-number summary. (p. 221)

  • A central box spans the quartiles (Q1 to Q3)

Inter-quartile Range = IQR = Q3 - Q1

  • A line in the box marks the median
  • Lines (whiskers) extend from box to the minimum and maximum observations.
constructing boxplots
Constructing Boxplots

1) Compute the 5-number summary.

2) Draw a vertical line at the Q1 and Q3.

3) Draw two horizontal lines to complete the box.

4) Draw a vertical line at the median.

5) Draw “whiskers” to the extremes (Min and Max).

Draw boxplots for Sosa and Maris’s homeruns.

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