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

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

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

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

• 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

• 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:

• 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

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.

BoxplotA 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 Boxplots1) 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.