# The Normal Distribution

Math 145

### Distributions

• What is a distribution?
• What does it tell you?

### Density Curves

The histogram shows the actual vocabulary scores of a group of 7-th grade children.

What do you notice?

• The smooth curve is the idealized curve what we imagine we would get if we took the population of all such children and made the bin widths very small.
• The smooth curve is called a density curve.

### Features

• Symmetric
• Bunched around a central value
• Mean is around 7
• Standard deviation? Maybe 2?

### Differences between continuous and discrete distributions

• Discrete/categorical distribtions convey information by the heights of the bars
• Continuous distributions convey information using the areas under the curve

### The Normal Distribution

Mean: 68.2 inches

SD: 2.7 inches

How would the shape of this distribution change if we changed the mean or the standard deviation?

### What might be normal?

• Income in a city?
• Scores on a standardized test?
• Age of IC students?

### Rules of thumb

IF you have a normal distribution:

• ~68% of the data lies within 1 sd of the mean
• ~95% of the data lies within 2 std of the mean
• ~99.7% of the data lies within 3 std of the mean

### Z-Scores

https://fivethirtyeight.com/features/the-20-most-extreme-cases-of-the-book-was-better-than-the-movie/

$z = \dfrac{Value - Mean}{Standard \ deviation} = \dfrac{\bar{x} - \mu}{\sigma}$

• Mean of z-scores is 0
• SD of z-scores is 1

### Example

Women’s heights are normally distributed with mean 65 inches (165 cm) and standard deviation 2.5 inches (6.4 cm)

• What proportion of women are less than 60 inches? (5ft)
• What proportion of women are less than 70 inches?
• Approiximately what proportion is more than 72 inches? (6ft)
• What if we want to know a different number? More than 63 inches?

We can use a table or we can use R, r-fiddle.org

Notation: $$N(\mu, \sigma)$$

### Using z-scores

On the 2008 SAT, which of the following scores represents the best performance: 580 on reading, 595 on math, or 575 on writing?

Subject Reading Math Writing
Mean 501 515 493
SD 112 116 111

Reading z = 0.71

Math z = 0.69

Writing z = 0.74

### Finding proportions from z-scores

Many methods, we'll use tables first:

• Find the proportion of data that has z-score less than 0.7
• Find the proportion with z-score above 1.2
• Find the proportion with z-scores between -0.20 and 1.4
• What z-score is at the 70th percentile?
• What z-score has 75% of the data above it?
• What percentile is a SAT reading score of 700? (Mean reading score is 503; standard deviation is 113)
• What SAT math score is at the 90th percentile? (Mean math score is 518; standard deviation is 115)

### Using R

Subject Reading Math Writing
Mean 501 515 493
SD 112 116 111
• What percent of reading scores are less than 450?
pnorm(450,501,112)

[1] 0.3244262

• What percent of z-scores are above 2?
pnorm(2,0,1, lower.tail=FALSE)

[1] 0.02275013


or use:

pnorm(2,0,1)

[1] 0.9772499


What z-score is at the 70th percentile?

qnorm(.70,0,1)

[1] 0.5244005


What SAT math score would be at the 81st percentile?

Subject Reading Math Writing
Mean 501 515 493
SD 112 116 111
qnorm(.81,515,116)

[1] 616.836


### Analyzing normality

x <- rnorm(500)
histogram(~x)


qqnorm(x)


### Analyzing normality

x <- rnorm(500)
x2 <- x^2
histogram(~x2)


qqnorm(x2)