Summarizing Numerical Data

STAT 20: Introduction to Probability and Statistics

Concept Questions

Describing Shape

Which of these variables do you expect to be uniformly distributed?

1. bill length of Gentoo penguins
2. salaries of a random sample of people from California
3. house sale prices in San Francisco
4. birthdays of classmates (day of the month)

Please vote at pollev.com.

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Correct answer: 4.T

This lends itself very well to boardwork: you can ask a student to tell you how to draw the distribution birthdays. Topics to bring up:

1. Barchart shows more information than a histgram (which would aggregate)
2. For discrete numerical, barchart is fine as long as the number of categories isn’t too great.
3. Birthdays isn’t quite uniform: you would expect 29, 30, 31 to have fewer since not all months have those dayes.

You can continue the same approach to drawing the remaining three distribution: 1) bimodal (separated out by species) and symmetric (as demonstrated on the next slide) 2) unimodal and heavily right skewed 3) unimodal and right skewed.

Concept Activity - Measures of Center

Mean, median, mode: which is best?

It depends on your desiderata: the nature of your data and what you seek to capture in your summary.

Get out a piece of paper. You’ll be watching a 3 minute video that discusses characteristics of a typical human. Note which numerical summaries are used and what for.

After the video, you can write on the board each of mean, median, mode and have students list measures that were reported. There are many modes because lots of the data that they collect is categorical.

General Advice

1. Means are often a good default for symmetric data.

2. Means are sensitive to very large and small values, so can be deceptive on skewed data. > Use a median

3. Modes are often the only option for categorical data.

But there are other notions of typical… what about a maximum?

This is an interesting example of when we don’t use a measure of central tendency to characterize the distribution. The answer to, “What will the temperature be tomorrow?” is generally answered not with an average but with a max. This is a good question to put to students: why do we characterize daily temperatures by their max?

This is a nice but not necessary example of an excellent visualization of summary statistics. Every dot is a max temp on a day of the year over a 40 year span. The annotations of the PWN heat dome are very effective in showing by how much they are the max of a distribution of maxes.

Concept Question 3 - Measures of Spread

• Why are measures of spread so important? Consider the following question.

There are two new food delivery services that open in Berkeley: Oski Eats and Cal Cravings. A friend of yours that took Stat 20 collected data on each and noted that Oski Eats has a mean delivery time of 29 minutes and Cal Cravings a mean delivery time of 27 minutes. Which would would you rather order from?

This starts off as a softball, so you probably don’t want to give too much time or bother with a poll. Simply ask students why they voted for Cal Cravings and then give them a bit more information…

One possible reality

Which would would you rather order from?

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Now you can ask students to make a good argument for both options. Cal Cravings is shorter on average but is very variable; its good for the gamblers. For students who value predictability, Oski Eats is better because of its smaller spread.

Worksheet: Summarizing Numerical Data

25:00

Break

05:00

Lab: Computing on the Data

25:00