Normal Distribution

# Normal Distribution
### Dr. Dogucu

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<div class="my-footer"> 
 Copyright &copy; <a href="https://mdogucu.ics.uci.edu">Dr. Mine Dogucu</a>. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div>

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## pdf

`$X \sim \mathcal{N}(\mu, \sigma^2)$`

`$f(x) = \frac{1}{\sigma \sqrt{2\pi}}{e^{-(x-\mu)^2/2\sigma^2}}$`

`$E(X) = \mu$`

`$Var(X) = \sigma^2$`

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class:middle

## Characteristics

<img src="slide-3-normal_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" />
]

`$P(X<\mu) = 0.5$`

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## Parameters

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68.26895% of the data are within 1 standard deviation of the mean.

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95.44997% of the data are within 2 standard deviations of the mean.

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99.73002% of the data are within 3 standard deviations of the mean.

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### Example

Midterm exam scores are normally distributed with a mean of 80 and standard deviation of 5 points.

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`$f(78)$` =

```r
dnorm(x = 78, mean = 80, sd = 5)
```

```
## [1] 0.07365403
```

`$f(92)$` =

```r
dnorm(x = 92, mean = 80, sd = 5)
```

```
## [1] 0.004478906
```

---

## Calculating Probability

What is the probability that a student scores below 73 points?

]

```r
pnorm(q = 73, mean = 80, sd = 5)
```

```
## [1] 0.08075666
```

<br>

If a student scores 73 points, the student's percentile rank is 8.0756659%

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## Calculating Probability

What is the probability that a student scores above 78 points?

```r
1 - pnorm(q = 78, mean = 80, sd = 5)
```

```
## [1] 0.6554217
```

```r
pnorm(q = 78, mean = 80, sd = 5, 
      lower.tail = FALSE)
```

```
## [1] 0.6554217
```

---

## How many standard deviations away from the mean?

Krishna scored 85 points on the midterm. How many standard deviations is Krishna's score away from the mean?

`$\frac{85-80}{5} = 1$`

Hira scored 77.5 points on the midterm. How many standard deviations is Hira's score away from the mean?

`$\frac{77.5-80}{5} = -0.5$`

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class:center middle

## Z-score

$$Z = \frac{X-\mu}{\sigma} $$

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## Calculating z-scores

We can calculate Z-scores regardless of the distribution of the data. In other words, data do not need to be normally distributed. Z-scores only show how many standard deviations a value is away from the mean.

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## Interpreting z-scores

If we know that the data follow a normal distribution then we can use z-scores for probabilistic interpretations.

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### Normal Distribution vs. Standard Normal (z) Distribution

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.pull-right[
<img src="slide-3-normal_files/figure-html/unnamed-chunk-16-1.png" style="display: block; margin: auto;" />
]

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class: middle

__Normal Distribution__

```r
pnorm(q = 90, mean = 80, sd = 5)
```

```
## [1] 0.9772499
```

]

__Standard Normal (z) Distribution__

```r
pnorm(q = 2, mean = 0, sd = 1)
```

```
## [1] 0.9772499
```
]

---

## SAT scores - z scores

Let X represent the SAT scores and `$X \sim N(1100, 200^2)$`

What is the Z-score of a student who scores 1500?

`$z = \frac{1500-1100}{200} = 2$`

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## SAT scores - probability

What is the percentile rank of a student who scores 1500?

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## SAT scores - probability

What is the percentile rank of a student who scores 1500?

```r
pnorm(q = 1500, 
      mean = 1100, 
      sd = 200)
```

```
## [1] 0.9772499
```

If you know the order of arguments:

```r
pnorm(1500, 1100, 200)
```

```
## [1] 0.9772499
```

]

--
.pull-right[

```r
pnorm(2, 
      mean = 0, 
      sd = 1)
```

```
## [1] 0.9772499
```

The default `mean` is 0 and `sd` is 1.

```r
pnorm(2)
```

```
## [1] 0.9772499
```
]

---

What percentage of test takers score between 800 and 1400?

`$P(800<X<1400) =P(X<1400) - P(X<800)$`

```r
pnorm(q = 1400, mean = 1100, sd = 200) -
  pnorm(q = 800, mean = 1100, sd = 200) 
```

```
## [1] 0.8663856
```

---

## Known probability unknown x

If College Board wanted to send a congratulatory email to the top 10% in the SATs, above what SAT score should they consider sending the email to?

--
.pull-left[
<img src="slide-3-normal_files/figure-html/unnamed-chunk-26-1.png" style="display: block; margin: auto;" />
]

```r
qnorm(p = 0.90, 
      mean = 1100, 
      sd = 200) 
```

```
## [1] 1356.31
```
]