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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?

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    $\begingroup$ You are approaching this the wrong way. A graphical representation is not always the best way to look at things. Variance is just the square of the standard deviation, which you already understand. A better question is: why is the square interesting enough that it has its own name? The answer to that is that variances are additive (while standard deviation is not). $\endgroup$
    – Szabolcs
    Nov 20, 2013 at 17:31
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    $\begingroup$ Thanks for your replies. I do understand that the variance represents the dispersion of the values, and that the standard deviation includes 68.2% of the values in normally distributed, nominal number sets. Therefore, it must be interesting enough beecause it should kind of represent the area of the Gaußian distribution, but I can not calculate this area precisely. That's why I would like to understand that visually. Do you refer to 'additive' because the formula you provided (Bienaymé) doesn't contain the sum of(each value minus the difference to the average)^2 divided by (n-1) part? Thanks $\endgroup$
    – nic
    Nov 21, 2013 at 0:43
  • $\begingroup$ Try this. Very simple and clear graphical explanation. Variance explained, how to calculate, how it related to standard deviation $\endgroup$
    – Sergei
    Jan 3, 2023 at 6:11

2 Answers 2

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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!

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  • $\begingroup$ Could've at least pasted the image. Smh $\endgroup$
    – NoName
    Apr 28, 2020 at 23:26
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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:

enter image description here

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.

enter image description here

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