cascade

This book looks some basic statistics:

- covariance
- variance
- standard deviation
- correlation coefficient

---

We'll look at the *population* statistics first equation over all:

---
class: inverse, center, middle

# Covariance

---
class: center, middle

### Covariance is a measure of the joint variability of two random variables.

---
class: inverse, center, middle

$$ cov(x,y) = \frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n} $$
]

---

``` r
*datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point()
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_01_output-1.png)
]

---
count: false

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
* geom_xmean()
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_02_output-1.png)
]

---
count: false

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
  geom_xmean() +
* geom_ymean()
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_03_output-1.png)
]

---
count: false

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
  geom_xmean() +
  geom_ymean() +
* geom_xmeandiff()
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_04_output-1.png)
]

---
count: false

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
  geom_xmean() +
  geom_ymean() +
  geom_xmeandiff() +
* geom_ymeandiff()
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_05_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
### `$$\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})$$`
]
 
.panel2-cov_steps-auto[

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
  geom_xmean() +
  geom_ymean() +
  geom_xmeandiff() +
  geom_ymeandiff() +
* geom_xydiffs()
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_06_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
### `$$\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{n - 1}$$`
]
 
.panel2-cov_steps-auto[

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
  geom_xmean() +
  geom_ymean() +
  geom_xmeandiff() +
  geom_ymeandiff() +
  geom_xydiffs() +
* geom_covariance()
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_07_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
### `$$\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{n - 1}$$`
]
 
.panel2-cov_steps-auto[

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
  geom_xmean() +
  geom_ymean() +
  geom_xmeandiff() +
  geom_ymeandiff() +
  geom_xydiffs() +
  geom_covariance() +
* geom_covariance_label()
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_08_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
### `$$\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{n - 1}$$`
]
 
.panel2-cov_steps-auto[

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
  geom_xmean() +
  geom_ymean() +
  geom_xmeandiff() +
  geom_ymeandiff() +
  geom_xydiffs() +
  geom_covariance() +
  geom_covariance_label() +
* labs(title = "Picture of covariance")
```
]
 
.panel3-cov_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cov_steps_auto_09_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
### `$$\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{n - 1}$$`
]
 
.panel2-cov_steps-auto[

``` r
datasaurus_dozen |> filter(dataset == "dino") |>  ggplot() +  aes(x = x, y = y) +   geom_point() +
  geom_xmean() +
  geom_ymean() +
  geom_xmeandiff() +
  geom_ymeandiff() +
  geom_xydiffs() +
  geom_covariance() +
  geom_covariance_label() +
  labs(title = "Picture of covariance") ->
*xy_covariance
```
]
 
.panel3-cov_steps-auto[

]

---
class: inverse, middle, center

# Variance and Standard Deviation

---
class: inverse, center, middle

$$ var(x) = \frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n - 1} $$
]

---
class: inverse, center, middle

$$ \sigma^2 = \frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n - 1} $$
]

---

.panel1-variance_steps-auto[
### `$$\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{n - 1}$$`
]
 
.panel2-variance_steps-auto[

``` r
*xy_covariance
```
]
 
.panel3-variance_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/variance_steps_auto_01_output-1.png)
]

---
count: false

.panel1-variance_steps-auto[
### `$$\frac{\sum_{i=1}^n (x_i-\bar{x})(x_i-\bar{x})}{n - 1}$$`
]
 
.panel2-variance_steps-auto[

``` r
xy_covariance +
* aes(y = x)
```
]
 
.panel3-variance_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/variance_steps_auto_02_output-1.png)
]

---
count: false

.panel1-variance_steps-auto[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n - 1}$$`
]
 
.panel2-variance_steps-auto[

``` r
xy_covariance +
  aes(y = x) +
* labs(title = "Picture of Variance")
```
]
 
.panel3-variance_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/variance_steps_auto_03_output-1.png)
]

---
count: false

``` r
xy_covariance +
  aes(y = x) +
  labs(title = "Picture of Variance") +
* geom_x_sd()
```
]
 
.panel3-variance_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/variance_steps_auto_04_output-1.png)
]

---
count: false

``` r
xy_covariance +
  aes(y = x) +
  labs(title = "Picture of Variance") +
  geom_x_sd() +
* labs(title = "Picture of Variance & Standard Deviation")
```
]
 
.panel3-variance_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/variance_steps_auto_05_output-1.png)
]

---
count: false

``` r
xy_covariance +
  aes(y = x) +
  labs(title = "Picture of Variance") +
  geom_x_sd() +
  labs(title = "Picture of Variance & Standard Deviation") ->
*xx_variance_sd
```
]
 
.panel3-variance_steps-auto[

]

---
class: inverse, center, middle

$$ \sigma^2 = \frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n - 1} $$
]

---
class: inverse, center, middle

$$ \sigma = \sqrt\frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n - 1} $$
]

---

.panel1-sd_steps-auto[
#### `$$\frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n - 1}$$`
]
 
.panel2-sd_steps-auto[

``` r
*xx_variance_sd
```
]
 
.panel3-sd_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/sd_steps_auto_01_output-1.png)
]

---
count: false

.panel1-sd_steps-auto[
#### `$$\sqrt\frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n - 1}$$`
]
 
.panel2-sd_steps-auto[

``` r
xx_variance_sd +
* aes(y = y)
```
]
 
.panel3-sd_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/sd_steps_auto_02_output-1.png)
]

---
count: false

``` r
xx_variance_sd +
  aes(y = y) +
* geom_y_sd()
```
]
 
.panel3-sd_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/sd_steps_auto_03_output-1.png)
]

---
count: false

``` r
xx_variance_sd +
  aes(y = y) +
  geom_y_sd() +
* aes(x = x/sd(x),
*     y = y/sd(y))
```
]
 
.panel3-sd_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/sd_steps_auto_04_output-1.png)
]

---
count: false

``` r
xx_variance_sd +
  aes(y = y) +
  geom_y_sd() +
  aes(x = x/sd(x),
      y = y/sd(y)) +
* labs(title = "Picture of Pearson correlation")
```
]
 
.panel3-sd_steps-auto[
![](correlation-covariance-flipbook-rewrite_files/figure-html/sd_steps_auto_05_output-1.png)
]

---

![Creative commons, Wikipedia, M. W. Toews - Own work, based (in concept) on figure by Jeremy Kemp, on 2005-02-09](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/1920px-Standard_deviation_diagram.svg.png)

Creative commons, Wikipedia, M. W. Toews - Own work, based (in concept) on figure by Jeremy Kemp, on 2005-02-09

---
class: inverse, middle, center

# correlation coefficient

---
class: inverse, center, middle

$$ cor(x,y) = \frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n *\sigma_x \sigma_y} $$
]

---

.panel1-cor_steps-user[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n - 1}$$`
]
 
.panel2-cor_steps-user[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cor_steps_user_01_output-1.png)
]

---
count: false

.panel1-cor_steps-user[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n*\sigma_x}$$`
]
 
.panel2-cor_steps-user[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cor_steps_user_02_output-1.png)
]

---
count: false

.panel1-cor_steps-user[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n*\sigma_x\sigma_y}$$`
]
 
.panel2-cor_steps-user[
![](correlation-covariance-flipbook-rewrite_files/figure-html/cor_steps_user_03_output-1.png)
]

---

# Bessel correction: n - 1

*Sample* statistics (Not *population* statistics)

---
class: inverse, center, middle

$$ cov(x,y) = \frac{\sum_{i=1}^n (x_i-\overline{x})(y_i-\overline{y})}{n - 1} $$
]

---

$$ s^2 = \frac{\sum_{i=1}^n (x_i-\overline{x})^2}{n - 1} $$
]

---

$$ s = \sqrt\frac{\sum_{i=1}^n (x_i-\overline{x})^2}{n - 1} $$
]

---

# DRY:  Bonus

## "Don't repeat yourself"

## Writing and using functions

---

``` r
# this is a temporary fix - a 
mysetseed <- set.seed
```

---

r chunk_reveal("cov_steps1")`

---

r chunk_reveal("cov_steps2", display_type = c("code", "func", "output"), widths = c(27, 37, 32))`

---

# DRY next level: writing and using packages

- [A companion guide to Jim Hester’s, 'You can make an R package in 20 minutes'](https://evamaerey.github.io/package_in_20_minutes/package_in_20_minutes)

---

r chunk_reveal("numeric")`

---

.huge {
  font-size: 250%;
}

@media print {
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    display: block;
  }
}
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