08: Variance

Overview

$$s^2={\displaystyle \frac{1}{n-1}{\displaystyle \sum _{i=1}^n(x_i-\overline{x}{)}^2}}$$

Today’s main questions are “What is variance and what is a standard deviation?” We will go through

• the formulas and calculations
• demostrations
• applications (word problems)

Notation

We tend to study a relatively small sample to understand the behavior of a much larger population.

Example: Nathan’s Hot Dog Eating Contest

Each year on July 4, the Nathan’s Hot Dog Eating Contest takes place on Coney Island in New York. The rule is simple: eat as many hot dogs (and buns) as you can in 10 minutes. The past 5 winning amounts were: 63, 70, 72, 74, 71. Compute the variance.

$$s^2={\displaystyle \frac{1}{n-1}{\displaystyle \sum _{i=1}^n(x_i-\overline{x}{)}^2}}$$

Units?

square hot dogs?

Demostrations

Setup

For each of the following sets

• $A=\{1,2,3,4,5,6,7\}$
• $B=\{3,4,5,6,7,8,9\}$
• $C=\{-3,-2,-1,0,1,2,3\}$
• $D=\{-9,-6,-3,0,3,6,9\}$

we will compute the sample mean, sample median, and sample standard deviation.

R

A <- seq(1, 7, 1)
B <- A + 2
C <- seq(-3, 3, 3)
D <- 3*C

A to B

mean(A)

[1] 4

mean(B)

[1] 6

median(A)

[1] 4

median(B)

[1] 6

sd(A)

[1] 2.160247

sd(B)

[1] 2.160247

C to D

mean(C)

[1] 0

mean(D)

[1] 0

median(C)

[1] 0

median(D)

[1] 0

sd(C)

[1] 3

sd(D)

[1] 9

Standardization

$$z={\displaystyle \frac{x-\mu }{\sigma }}$$

To standardize data, compute a z-score by

• subtracting by the mean
• then dividing by the standard deviation

This calculation is considered to be “unitless”, and the units are usually said as “[number of] standard deviations above/below the mean”

Most data falls within two standard deviations of the mean,

$$\text{usually }z\in (-2,2)$$

but $z\in (-\mathrm{\infty },\mathrm{\infty })$

Example: Dating Website Data

Scenario 1

According to OkCupid data, if men rate women on a scale from 1 = “least attractive” to 7 = “most attractive”, the average score is 3.99 with a sample standard deviation of 1.6401.

• What is the $z$-score of a woman rated a “6”?
• What is the attractiveness score of a woman at a $z$-score of 1.5?

Scenario 2

According to OkCupid data, if women rate men on a scale from 1 = “least attractive” to 7 = “most attractive”, the average score is 2.43 with a sample standard deviation of 1.2510.

• What is the $z$-score of a man rated a “6”?
• What is the attractiveness score of a man at a $z$-score of 1.5?

Looking Ahead

• due Fri., Feb. 3:
  • WHW3
  • JHW1
  • Identity Statement (short essay)
• and the before-lecture quizzes

Exam 1 will be on Wed., Mar. 1

• more information will be in the weekly announcements

source