Visualizing covariance, variance, standard deviation, correlation

# Visualizing covariance, variance, standard deviation, correlation
### <br><br>Gina Reynolds
### <br>

---

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

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

---

]
 
.panel2-cov_steps-auto[
![](covariance_correlation_files/figure-html/cov_steps_auto_1_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
## `$$\mu_x$$`
]
 
.panel2-cov_steps-auto[
![](covariance_correlation_files/figure-html/cov_steps_auto_2_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
## `$$\mu_y$$`
]
 
.panel2-cov_steps-auto[
![](covariance_correlation_files/figure-html/cov_steps_auto_3_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
## `$$x_i-\mu_x$$`
]
 
.panel2-cov_steps-auto[
![](covariance_correlation_files/figure-html/cov_steps_auto_4_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
## `$$y_i-\mu_y$$`
]
 
.panel2-cov_steps-auto[
![](covariance_correlation_files/figure-html/cov_steps_auto_5_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
## `$$\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)$$`
]
 
.panel2-cov_steps-auto[
![](covariance_correlation_files/figure-html/cov_steps_auto_6_output-1.png)
]

---
count: false

.panel1-cov_steps-auto[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n}$$`
]
 
.panel2-cov_steps-auto[
![](covariance_correlation_files/figure-html/cov_steps_auto_7_output-1.png)
]

---
class: inverse, middle, center

# Variance and Standard Deviation

---
class: inverse, center, middle

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

---
class: inverse, center, middle

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

---

]
 
.panel2-variance_steps-user[
![](covariance_correlation_files/figure-html/variance_steps_user_1_output-1.png)
]

---
count: false

.panel1-variance_steps-user[
## `$$\mu_x$$`
]
 
.panel2-variance_steps-user[
![](covariance_correlation_files/figure-html/variance_steps_user_2_output-1.png)
]

---
count: false

.panel1-variance_steps-user[
## `$$\mu_x$$`
]
 
.panel2-variance_steps-user[
![](covariance_correlation_files/figure-html/variance_steps_user_3_output-1.png)
]

---
count: false

.panel1-variance_steps-user[
## `$$x_i-\mu_x$$`
]
 
.panel2-variance_steps-user[
![](covariance_correlation_files/figure-html/variance_steps_user_4_output-1.png)
]

---
count: false

.panel1-variance_steps-user[
## `$$x_i-\mu_x$$`
]
 
.panel2-variance_steps-user[
![](covariance_correlation_files/figure-html/variance_steps_user_5_output-1.png)
]

---
count: false

.panel1-variance_steps-user[
## `$$\sum_{i=1}^n (x_i-\mu_x)^2$$`
]
 
.panel2-variance_steps-user[
![](covariance_correlation_files/figure-html/variance_steps_user_6_output-1.png)
]

---
count: false

.panel1-variance_steps-user[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n}$$`
]
 
.panel2-variance_steps-user[
![](covariance_correlation_files/figure-html/variance_steps_user_7_output-1.png)
]

---
class: inverse, center, middle

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

---
class: inverse, center, middle

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

---

.panel1-sd_steps-user[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n}$$`
]
 
.panel2-sd_steps-user[
![](covariance_correlation_files/figure-html/sd_steps_user_1_output-1.png)
]

---
count: false

.panel1-sd_steps-user[
## `$$\sqrt\frac{\sum_{i=1}^n (x_i-\mu_x)^2}{n}$$`
]
 
.panel2-sd_steps-user[
![](covariance_correlation_files/figure-html/sd_steps_user_2_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[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n}$$`
]
 
.panel2-cor_steps-user[
![](covariance_correlation_files/figure-html/cor_steps_user_1_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[
![](covariance_correlation_files/figure-html/cor_steps_user_2_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[
![](covariance_correlation_files/figure-html/cor_steps_user_3_output-1.png)
]

---
count: false

---

A negative correlation

---

.panel1-cor_steps_neg-user[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n}$$`
]
 
.panel2-cor_steps_neg-user[
![](covariance_correlation_files/figure-html/cor_steps_neg_user_1_output-1.png)
]

---
count: false

.panel1-cor_steps_neg-user[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n*\sigma_x}$$`
]
 
.panel2-cor_steps_neg-user[
![](covariance_correlation_files/figure-html/cor_steps_neg_user_2_output-1.png)
]

---
count: false

.panel1-cor_steps_neg-user[
## `$$\frac{\sum_{i=1}^n (x_i-\mu_x)(y_i-\mu_y)}{n*\sigma_x\sigma_y}$$`
]
 
.panel2-cor_steps_neg-user[
![](covariance_correlation_files/figure-html/cor_steps_neg_user_3_output-1.png)
]

---
count: false

---

# 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
*mysetseed(199402)
```
]
 
.panel2-cov_steps1-auto[

]

---
count: false

```r
mysetseed(199402)  
*create_x_y(relationship = .5)
```
]
 
.panel2-cov_steps1-auto[

```
# A tibble: 20 x 15
       x     y mean_x mean_y     area mean_area quasi_mean_area sd_x_sample
   <dbl> <dbl>  <dbl>  <dbl>    <dbl>     <dbl>           <dbl>       <dbl>
 1  68.2 32.1    53.8   26.9   76.1        269.          4.00          23.4
 2  45.2 23.2    53.8   26.9   30.8        269.          1.62          23.4
 3  26.4 16.7    53.8   26.9  278.         269.         14.6           23.4
 4  90.7 41.3    53.8   26.9  533.         269.         28.0           23.4
 5  58.0 27.0    53.8   26.9    0.420      269.          0.0221        23.4
 6  41.6 20.5    53.8   26.9   78.2        269.          4.11          23.4
 7  67.7 41.2    53.8   26.9  200.         269.         10.5           23.4
 8 -10.4 -8.66   53.8   26.9 2280.         269.        120.            23.4
 9  65.4 30.5    53.8   26.9   42.7        269.          2.25          23.4
10  53.3 24.5    53.8   26.9    1.20       269.          0.0634        23.4
11  58.5 31.1    53.8   26.9   19.8        269.          1.04          23.4
12  45.9 21.8    53.8   26.9   39.9        269.          2.10          23.4
13  73.9 38.5    53.8   26.9  234.         269.         12.3           23.4
14  60.1 23.2    53.8   26.9  -23.1        269.         -1.22          23.4
15  99.0 51.1    53.8   26.9 1094.         269.         57.6           23.4
16  44.2 25.6    53.8   26.9   12.5        269.          0.656         23.4
17  45.0 24.0    53.8   26.9   25.5        269.          1.34          23.4
18  35.9 14.5    53.8   26.9  220.         269.         11.6           23.4
19  39.8 20.6    53.8   26.9   87.5        269.          4.60          23.4
20  66.9 38.6    53.8   26.9  154.         269.          8.12          23.4
# … with 7 more variables: sd_x <dbl>, sd_y_sample <dbl>, sd_y <dbl>,
#   some_x <dbl>, some_x_sample <dbl>, some_y <dbl>, some_y_sample <dbl>
```
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
* data_create_scatterplot()
```
]
 
.panel2-cov_steps1-auto[
![](covariance_correlation_files/figure-html/cov_steps1_auto_3_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
* plot_draw_mean_x()
```
]
 
.panel2-cov_steps1-auto[
![](covariance_correlation_files/figure-html/cov_steps1_auto_4_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
  plot_draw_mean_x() %>%  
* plot_draw_mean_y()
```
]
 
.panel2-cov_steps1-auto[
![](covariance_correlation_files/figure-html/cov_steps1_auto_5_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
  plot_draw_mean_x() %>%  
  plot_draw_mean_y() %>%  
* plot_draw_differences_x()
```
]
 
.panel2-cov_steps1-auto[
![](covariance_correlation_files/figure-html/cov_steps1_auto_6_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
  plot_draw_mean_x() %>%  
  plot_draw_mean_y() %>%  
  plot_draw_differences_x() %>%  
* plot_draw_differences_y()
```
]
 
.panel2-cov_steps1-auto[
![](covariance_correlation_files/figure-html/cov_steps1_auto_7_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
  plot_draw_mean_x() %>%  
  plot_draw_mean_y() %>%  
  plot_draw_differences_x() %>%  
  plot_draw_differences_y() %>%  
* plot_multiply_differences()
```
]
 
.panel2-cov_steps1-auto[
![](covariance_correlation_files/figure-html/cov_steps1_auto_8_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
  plot_draw_mean_x() %>%  
  plot_draw_mean_y() %>%  
  plot_draw_differences_x() %>%  
  plot_draw_differences_y() %>%  
  plot_multiply_differences() %>%  
* plot_take_average_rectangle()
```
]
 
.panel2-cov_steps1-auto[
![](covariance_correlation_files/figure-html/cov_steps1_auto_9_output-1.png)
]

---

```r
*mysetseed(199402)
```
]
 
.panel2-cov_steps2-auto[

```
function (seed, kind = NULL, normal.kind = NULL, sample.kind = NULL) 
{
    kinds <- c("Wichmann-Hill", "Marsaglia-Multicarry", "Super-Duper", 
        "Mersenne-Twister", "Knuth-TAOCP", "user-supplied", "Knuth-TAOCP-2002", 
        "L'Ecuyer-CMRG", "default")
    n.kinds <- c("Buggy Kinderman-Ramage", "Ahrens-Dieter", "Box-Muller", 
        "user-supplied", "Inversion", "Kinderman-Ramage", "default")
    s.kinds <- c("Rounding", "Rejection", "default")
    if (length(kind)) {
        if (!is.character(kind) || length(kind) > 1L) 
            stop("'kind' must be a character string of length 1 (RNG to be used).")
        if (is.na(i.knd <- pmatch(kind, kinds) - 1L)) 
            stop(gettextf("'%s' is not a valid abbreviation of an RNG", 
                kind), domain = NA)
        if (i.knd == length(kinds) - 1L) 
            i.knd <- -1L
    }
    else i.knd <- NULL
    if (!is.null(normal.kind)) {
        if (!is.character(normal.kind) || length(normal.kind) != 
            1L) 
            stop("'normal.kind' must be a character string of length 1")
        normal.kind <- pmatch(normal.kind, n.kinds) - 1L
        if (is.na(normal.kind)) 
            stop(gettextf("'%s' is not a valid choice", normal.kind), 
                domain = NA)
        if (normal.kind == 0L) 
            stop("buggy version of Kinderman-Ramage generator is not allowed", 
                domain = NA)
        if (normal.kind == length(n.kinds) - 1L) 
            normal.kind <- -1L
    }
    if (!is.null(sample.kind)) {
        if (!is.character(sample.kind) || length(sample.kind) != 
            1L) 
            stop("'sample.kind' must be a character string of length 1")
        sample.kind <- pmatch(sample.kind, s.kinds) - 1L
        if (is.na(sample.kind)) 
            stop(gettextf("'%s' is not a valid choice", sample.kind), 
                domain = NA)
        if (sample.kind == 0L) 
            warning("non-uniform 'Rounding' sampler used", domain = NA)
        if (sample.kind == length(s.kinds) - 1L) 
            sample.kind <- -1L
    }
    .Internal(set.seed(seed, i.knd, normal.kind, sample.kind))
}
<bytecode: 0x7fda69ccf8b8>
<environment: namespace:base>
```
]
 
.panel3-cov_steps2-auto[

]

---
count: false

```r
mysetseed(199402)  
*create_x_y(relationship = .5)
```
]
 
.panel2-cov_steps2-auto[

```
function(num = 20, 
                       spread_x = 20,
                       relationship = .1, 
                       noise = 3
                       ){
  
  tibble(x = rnorm(num, sd = spread_x) + 50 ) %>% 
  mutate(y = relationship * x + rnorm(num, sd = noise)) %>% 
  mutate(mean_x = mean(x)) %>% 
  mutate(mean_y = mean(y)) %>% 
  mutate(area = (x - mean_x)*(y - mean_y)) %>% 
  mutate(mean_area = mean(area)) %>% 
  mutate(quasi_mean_area = area/(n() - 1)) %>%
  mutate(sd_x_sample = sd(x)) %>% 
  mutate(sd_x = sd_pop(x)) %>% 
  mutate(sd_y_sample = sd(y)) %>% 
  mutate(sd_y = sd_pop(y)) %>% 
  mutate(some_x = sd_x) %>% 
  mutate(some_x_sample = sd_x_sample) %>% 
  mutate(some_y = mean_area/some_x) %>% 
  mutate(some_y_sample = sd_y_sample) 
  
}
<bytecode: 0x7fda6c9d67c0>
```
]
 
.panel3-cov_steps2-auto[

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
* data_create_scatterplot()
```
]
 
.panel2-cov_steps2-auto[

```
function(data, background_color = "palegreen4"){

data %>% 
  ggplot() + 
  theme(legend.position = c(.15, .9)) +
  aes(x = x) + 
  aes(y = y) +
  theme(rect = element_rect(fill = background_color)) + 
  theme(text = element_text(color = "white", face = "italic", size = 15)) +
  theme(panel.background = element_rect(fill = background_color)) +
  theme(legend.key = element_blank()) + #element_rect(fill = background_color)) +
  theme(legend.title = element_blank()) +
  theme(axis.text = element_text(color = "white")) +
  theme(axis.ticks = element_line(color = "white")) +
  labs(title = NULL) +
  theme(panel.grid = element_blank()) +
  geom_point(size = 3, pch = 21, col = "white", fill = "white", lwd = 3, alpha = .6) + 
  labs(caption = "Statistical Visualization: Gina Reynolds @EvaMaeRey ")

}
<bytecode: 0x7fda6d7f4c98>
```
]
 
.panel3-cov_steps2-auto[
![](covariance_correlation_files/figure-html/cov_steps2_auto_3_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
* plot_draw_mean_x()
```
]
 
.panel2-cov_steps2-auto[

```
function(plot){
  
  plot + 
  # 1. mean of x
  geom_rug(aes(y = NULL), col = "white") +
  geom_vline(aes(xintercept = mean(x)), 
             lty = "dashed", col = "white")  
  
  
}
<bytecode: 0x7fda6fb08208>
```
]
 
.panel3-cov_steps2-auto[
![](covariance_correlation_files/figure-html/cov_steps2_auto_4_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
  plot_draw_mean_x() %>%  
* plot_draw_mean_y()
```
]
 
.panel2-cov_steps2-auto[

```
function(plot){
  
  plot + 
  # 2. mean of y
  geom_rug(col = "white", aes(x = NULL)) +
  geom_hline(aes(yintercept = mean(y)), 
             lty = "dashed", col = "white")
  
}
<bytecode: 0x7fda6c673368>
```
]
 
.panel3-cov_steps2-auto[
![](covariance_correlation_files/figure-html/cov_steps2_auto_5_output-1.png)
]

---
count: false

```
function(plot){
  
  plot + 
  # difference xi mean x
  scale_color_manual(breaks = c(FALSE, TRUE), 
                     label = c("Negative", "Positive"), 
                     values = c("orchid4", "lightgoldenrod3") )  + 
  geom_segment(aes(col = x > mean(x), 
                   xend = mean(x), yend = y), 
                   arrow = arrow(ends = "first", 
                                 length = unit(0.1, "inches"))) +
  labs(col = "")
  
    
}
<bytecode: 0x7fda6bcca1c8>
```
]
 
.panel3-cov_steps2-auto[
![](covariance_correlation_files/figure-html/cov_steps2_auto_6_output-1.png)
]

---
count: false

```
function(plot){
  
  plot +
  # difference yi mean y
  geom_segment(aes(col = y > mean(y), 
                   xend = x, yend = mean(y)), 
                   arrow = arrow(ends = "first", 
                                 length = unit(0.1, "inches"))) 
    
}
<bytecode: 0x7fda6d6932e0>
```
]
 
.panel3-cov_steps2-auto[
![](covariance_correlation_files/figure-html/cov_steps2_auto_7_output-1.png)
]

---
count: false

```
function(plot){
  
  plot + 
  # multiply differences
  aes(fill = area > 0) +
  geom_rect(aes(xmin = mean(x), ymin = mean(y),  
            ymax = y, xmax = x), 
            alpha = .2  ) +
  labs(fill = "") +
  scale_fill_manual(breaks = c(FALSE, TRUE), 
                    label = c("Negative", "Positive"),  
                    values = c("orchid4", "lightgoldenrod3")) 
  
}
<bytecode: 0x7fda6e4e5698>
```
]
 
.panel3-cov_steps2-auto[
![](covariance_correlation_files/figure-html/cov_steps2_auto_8_output-1.png)
]

---
count: false

```r
mysetseed(199402)  
create_x_y(relationship = .5) %>%  
  data_create_scatterplot() %>%  
  plot_draw_mean_x() %>%  
  plot_draw_mean_y() %>%  
  plot_draw_differences_x() %>%  
  plot_draw_differences_y() %>%  
  plot_multiply_differences() %>%  
* plot_take_average_rectangle()
```
]
 
.panel2-cov_steps2-auto[

```
function(plot, title = "Picture of Covariance"){
  
  plot +      
  # Average areas 
  geom_rect(aes(
           xmin = mean(x),
           ymin = mean(y),
           ymax = mean(some_y) + mean(y),
           xmax = mean(some_x) + mean(x)
           ),
            color = "orange",
          linetype = "dotted",
           fill = "orange",
            lwd = 1.5,
            alpha = .2) +  #BREAK
  labs(title = title)
  
}
```
]
 
.panel3-cov_steps2-auto[
![](covariance_correlation_files/figure-html/cov_steps2_auto_9_output-1.png)
]

---

# 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
*cars
```
]
 
.panel2-numeric-auto[

```
   speed dist
1      4    2
2      4   10
3      7    4
4      7   22
5      8   16
6      9   10
7     10   18
8     10   26
9     10   34
10    11   17
11    11   28
12    12   14
13    12   20
14    12   24
15    12   28
16    13   26
17    13   34
18    13   34
19    13   46
20    14   26
21    14   36
22    14   60
23    14   80
24    15   20
25    15   26
26    15   54
27    16   32
28    16   40
29    17   32
30    17   40
31    17   50
32    18   42
33    18   56
34    18   76
35    18   84
36    19   36
37    19   46
38    19   68
39    20   32
40    20   48
41    20   52
42    20   56
43    20   64
44    22   66
45    23   54
46    24   70
47    24   92
48    24   93
49    24  120
50    25   85
```
]

---
count: false

```r
cars %>%  
* ggplot()
```
]
 
.panel2-numeric-auto[
![](covariance_correlation_files/figure-html/numeric_auto_2_output-1.png)
]

---
count: false

```r
cars %>%  
  ggplot() +  
* aes(x = speed, y = dist)
```
]
 
.panel2-numeric-auto[
![](covariance_correlation_files/figure-html/numeric_auto_3_output-1.png)
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
* geom_point()
```
]
 
.panel2-numeric-auto[
![](covariance_correlation_files/figure-html/numeric_auto_4_output-1.png)
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
*visualization
```
]
 
.panel2-numeric-auto[

]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
*cars$speed
```
]
 
.panel2-numeric-auto[

```
 [1]  4  4  7  7  8  9 10 10 10 11 11 12 12 12 12 13 13 13 13 14 14 14 14 15 15
[26] 15 16 16 17 17 17 18 18 18 18 19 19 19 20 20 20 20 20 22 23 24 24 24 24 25
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
* mean(cars$speed)
```
]
 
.panel2-numeric-auto[

```
 [1] -11.4 -11.4  -8.4  -8.4  -7.4  -6.4  -5.4  -5.4  -5.4  -4.4  -4.4  -3.4
[13]  -3.4  -3.4  -3.4  -2.4  -2.4  -2.4  -2.4  -1.4  -1.4  -1.4  -1.4  -0.4
[25]  -0.4  -0.4   0.6   0.6   1.6   1.6   1.6   2.6   2.6   2.6   2.6   3.6
[37]   3.6   3.6   4.6   4.6   4.6   4.6   4.6   6.6   7.6   8.6   8.6   8.6
[49]   8.6   9.6
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
*x_diff
```
]
 
.panel2-numeric-auto[

]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

*cars$dist
```
]
 
.panel2-numeric-auto[

```
 [1]   2  10   4  22  16  10  18  26  34  17  28  14  20  24  28  26  34  34  46
[20]  26  36  60  80  20  26  54  32  40  32  40  50  42  56  76  84  36  46  68
[39]  32  48  52  56  64  66  54  70  92  93 120  85
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
* mean(cars$dist)
```
]
 
.panel2-numeric-auto[

```
 [1] -40.98 -32.98 -38.98 -20.98 -26.98 -32.98 -24.98 -16.98  -8.98 -25.98
[11] -14.98 -28.98 -22.98 -18.98 -14.98 -16.98  -8.98  -8.98   3.02 -16.98
[21]  -6.98  17.02  37.02 -22.98 -16.98  11.02 -10.98  -2.98 -10.98  -2.98
[31]   7.02  -0.98  13.02  33.02  41.02  -6.98   3.02  25.02 -10.98   5.02
[41]   9.02  13.02  21.02  23.02  11.02  27.02  49.02  50.02  77.02  42.02
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
*y_diff
```
]
 
.panel2-numeric-auto[

]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

*x_diff
```
]
 
.panel2-numeric-auto[

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

x_diff %>%  
* `*`(y_diff)
```
]
 
.panel2-numeric-auto[

```
 [1] 467.172 375.972 327.432 176.232 199.652 211.072 134.892  91.692  48.492
[10] 114.312  65.912  98.532  78.132  64.532  50.932  40.752  21.552  21.552
[19]  -7.248  23.772   9.772 -23.828 -51.828   9.192   6.792  -4.408  -6.588
[28]  -1.788 -17.568  -4.768  11.232  -2.548  33.852  85.852 106.652 -25.128
[37]  10.872  90.072 -50.508  23.092  41.492  59.892  96.692 151.932  83.752
[46] 232.372 421.572 430.172 662.372 403.392
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

x_diff %>%  
  `*`(y_diff) %>%  
* sum()
```
]
 
.panel2-numeric-auto[

```
[1] 5387.4
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

x_diff %>%  
  `*`(y_diff) %>%  
  sum() %>%  
* `/`(nrow(cars) - 1)
```
]
 
.panel2-numeric-auto[

```
[1] 109.9469
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

x_diff %>%  
  `*`(y_diff) %>%  
  sum() %>%  
  `/`(nrow(cars) - 1)

# or use the function
*library(magrittr)
```
]
 
.panel2-numeric-auto[

```
[1] 109.9469
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

x_diff %>%  
  `*`(y_diff) %>%  
  sum() %>%  
  `/`(nrow(cars) - 1)

# or use the function
library(magrittr)  
*cars
```
]
 
.panel2-numeric-auto[

```
[1] 109.9469
```

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

x_diff %>%  
  `*`(y_diff) %>%  
  sum() %>%  
  `/`(nrow(cars) - 1)

# or use the function
library(magrittr)  
cars %$%  
* cov(dist, speed)
```
]
 
.panel2-numeric-auto[

```
[1] 109.9469
```

```
[1] 109.9469
```
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

x_diff %>%  
  `*`(y_diff) %>%  
  sum() %>%  
  `/`(nrow(cars) - 1)

# or use the function
library(magrittr)  
cars %$%  
  cov(dist, speed)

*visualization
```
]
 
.panel2-numeric-auto[

```
[1] 109.9469
```

![](covariance_correlation_files/figure-html/numeric_auto_19_output-1.png)
]

---
count: false

```r
cars %>%  
  ggplot() +  
  aes(x = speed, y = dist) +  
  geom_point() ->  
visualization

# 'by-hand' calculation
cars$speed -  
  mean(cars$speed) ->  
x_diff

cars$dist -  
  mean(cars$dist) ->  
y_diff

x_diff %>%  
  `*`(y_diff) %>%  
  sum() %>%  
  `/`(nrow(cars) - 1)

# or use the function
library(magrittr)  
cars %$%  
  cov(dist, speed)

visualization +  
* annotate(geom = "text",
*          x = 7,
*          y = 100,
*          label = paste0("cov = ",
*                         cars %$%
*                           cov(dist, speed) %>%
*                           round(1)))
```
]
 
.panel2-numeric-auto[

```
[1] 109.9469
```

![](covariance_correlation_files/figure-html/numeric_auto_20_output-1.png)
]

---

.huge {
  font-size: 250%;
}

@media print {
  .has-continuation {
    display: block;
  }
}
</style>