• What are the observational units, variable(s), and variable type(s) in each plot?
• What did the code I used to make the plots look like?
• What statistics should we use for the center and spread?

• The Nearly Normal condition:
  • Distribution is unimodal
  • Distribution is (approximately) symmetric

• For any variable with a nearly normal distribution, we can use the same rules to calculate:
  1. Percentiles/quantiles
  2. The proportion of the data that are less than a given value.
• Lots of variables have a nearly normal distribution!

• \(N(\mu, \sigma)\)
• Read: "normal distribution with mean \(\mu\) and standard deviation \(\sigma\)"
• \(\mu\) and \(\sigma\) are parameters

• To use the model with real data, we estimate \(\mu\) and \(\sigma\) with the sample mean \(\bar{y}\) and standard deviation \(s\)

summarize(car_speeds, mean_speed = mean(speed), sd_speed = sd(speed))

## # A tibble: 1 x 2
##   mean_speed sd_speed
##        <dbl>    <dbl>
## 1    23.8439 3.563338

• Example: red curve is a \(N(23.8, 3.6)\) distribution

• If driver speeds in a 20 MPH speed zone can be represented by a \(N(23.8, 3.6)\) model, find the following:
  • The proportion of drivers who drive between 20.2 and 27.6 MPH.
  • The proportion of drivers who drive less than 20.2 MPH
  • The 2.5th percentile of driver speeds
  • The 50th percentile of driver speeds

summarize(pizza, mean_price = mean(Price), sd_price = sd(Price))

## # A tibble: 1 x 2
##   mean_price  sd_price
##        <dbl>     <dbl>
## 1   2.619038 0.1559685

• If the cost of a slice of pizza can be represented by a \(N(2.62, 0.16)\) model, find the following:
  • The proportion of pizza shops where a slice of pizza costs less than $2.30.
  • The 84th percentile of pizza slice costs
  • A lower and upper bound on the 99th percentile of pizza slice costs

• To calculate percentiles, we only need to know the number of standard devations above or below the mean a particular value is.
• This is the \(z\)-score:

\[z = \frac{y - \mu}{\sigma}\]

• Ex: Suppose a police officer pulls over someone who was going 31MPH in a 20MPH zone. Assume a \(N(23.8, 3.6)\) model applies.
  • How many standard deviations above the mean was that driver going?
  • What percentile of driving speeds were they at?
• Ex: Suppose a slice of pizza costs $2.94. Assume a \(N(2.62, 0.16)\) model applies.
  • How many standard deviations above the mean did that piece of pizza cost?
  • What percentile of costs was that slice at?

• Ex: Suppose a police officer pulls over someone who was going 31MPH in a 20MPH zone. Assume a \(N(23.8, 3.6)\) model applies.
  • How many standard deviations above the mean was that driver going?
  • What percentile of driving speeds were they at?
• Ex: Suppose a slice of pizza costs $2.94. Assume a \(N(2.62, 0.16)\) model applies.
  • How many standard deviations above the mean did that piece of pizza cost?
  • What percentile of costs was that slice at?
• Apparently, driving 31 MPH in a 20 MPH zone is as rare as getting a slice of pizza for $2.94

• Use qnorm to calculate quantiles (remember – essentially the same thing as percentiles)
  • What is the 90th percentile of speeds in a 20 MPH speed zone? Assume a \(N(23.8, 3.6)\) model applies.

qnorm(p = 0.90, mean = 23.8, sd = 3.6)

## [1] 28.41359

• Use pnorm to calculate proportion of data that are less than a particular value
  • What proportion of drivers travel less than 30 MPH?

pnorm(q = 30, mean = 23.8, sd = 3.6)

## [1] 0.9574854

• For you to do (draw a picture!):
  • What proportion of drivers travel more than 30 MPH?
  • What proportion of drivers travel between 20 and 25 MPH?

1. Everything in this chapter is for
   • one quantitative variable
   • that satisfies the nearly normal condition (unimodal, symmetric)
2. There are 2 basic types of calculations:
   • Find a percentile/quantile
   • Find the proportion of the data that are in a given range of values.
3. We can do calculations using either:
   • The 68-95-99.7 rule – often only approximate
   • R (qnorm for quantiles, pnorm for proportion of data less than a given number)