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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)

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.

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

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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