Graphical representation of variance I'm learning statistics and I can imagine pretty well what the standard deviation looks like (image here).
But, knowing that the standard deviation is the square root of the variance, I just can't figure out what that looks like.
Can anybody provide me with an illustration or a plot to help me understand that?
 A: I found this paper to be helpful. Hopefully you will find it useful too: 
https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9639.2010.00426.x
Paraphrasing what's explained in the article, if you understand SD, you can imagine that the squared deviation (x - mean of x)^2 can be represented graphically as a square (area = length of side ^2), where each side is the deviation from the mean for that observation. The numerator in the formula for variance is the sum of all n squared deviations, also known as SS, which graphically is the sum of the areas of all these squares. The sum of all these squares itself can also be represented graphically as a square, made up by smaller rectangles, all with the same width (square root of SS), but different heights (given by the squared deviation for that observation divided by the square root of SS). Again, the area of this bigger square is also SS. If you then convert this bigger square into (n-1) equal squares all with the same area, you get variance, which is the area of each one of these smaller square (aka SS/(n-1), or the formula for variance).
Interestingly, the length of each side of these variance squares will be equal to the standard deviation! 
A: One method that is helpful to is to visualize the center point of the data, the mean, and how much each raw data point's distance varies along that mean. One way of achieving this is simply drawing lines that start from where the mean is (where the data on average is centrally located) and finishing those lines where the raw value is.
I show how using R below. First we can load the tidyverse package for plotting, then create some x values and an index (for the order of raw data points). Then we plot the data on a scatterplot that draws segments from the mean to the raw data points.
#### Load Library ####
library(tidyverse)

#### Create X Value and Index ####
x <- c(10,30,20,40,50,10,10,20,50,40)
df <- data.frame(Index = 1:10,
                 X = x)
  
#### Plot Variation from Mean ####

df %>% 
  ggplot(aes(x=Index,
             y=X))+
  geom_point()+
  geom_segment(aes(xend = Index,
                   yend = mean(x)))+
  geom_hline(yintercept = mean(x),
             color = "red")+
  theme_classic()+
  scale_x_continuous(n.breaks = 10)

Which you can see below:

We can see that the fluctuation in distance for each data point here is not the same. Some are very close to the mean ($x = 30$), while others are quite far from the mean ($x = 10$ and $x = 50$). Something you could also consider is drawing grid lines where each standard deviation below/above the mean is located to get a sense of where the bulk of the data should be contained within this variation.
#### Add SD Lines ####
df %>% 
  ggplot(aes(x=Index,
             y=X))+
  geom_point()+
  geom_segment(aes(xend = Index,
                   yend = mean(x)))+
  geom_hline(yintercept = mean(x),
             color = "red",
             size=1)+
  geom_hline(yintercept = mean(x) - 1*sd(x),
             color = "blue",
             linetype = "dashed")+
  geom_hline(yintercept = mean(x) + 1*sd(x),
             color = "blue",
             linetype = "dashed")+
  geom_hline(yintercept = mean(x) - 2*sd(x),
             color = "blue",
             linetype = "dashed")+
  geom_hline(yintercept = mean(x) + 2*sd(x),
             color = "blue",
             linetype = "dashed")+
  theme_classic()+
  scale_x_continuous(n.breaks = 10)

We can see that all of the data is contained within 2 SD above and below the mean and we have a fair amount of data hovering below or above the 1 SD grid lines.

