Math 145

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

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*.

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

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

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?

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

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

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

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

Womenâ€™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: \( N(\mu, \sigma) \)

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

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)

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
```

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

```
qqnorm(x)
```

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

```
qqnorm(x2)
```