The Normal Distribution

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In order to understand the normal distribution, you must understand the concept of the mean and standard deviation.

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Mean

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The mean, also known as the average, is the value you get when you divide the sum of a series of observations by the total quantity of observations.

For example:

• Let’s say we have 4 observations: 5,5,5,5
• The average of these 4 observations is = (5+5+5+5)/4 = 20/4 = 5

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Standard Deviation

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The standard deviation is a measure used in statistics to evaluate how widely dispersed observations are from the mean.

• Lets say we have 5 observations: 1,1,2,1,1
• The mean of these observations is = (1+1+2+1+1)/5 = 1.2
• For each observation, subtract the mean & square the result:
  • observation 1: 1 - 1.2 = -.2^2 = .04
  • observation 2: 1 - 1.2 = -.2^2 = .04
  • obseration 3: 2 - 1.2 = -.2^2 = .64
  • obseration 4: 1 - 1.2 = -.2^2 = .04
  • observation 5: 1 - 1.2 = -.2^2 = .04
  • The mean of these squared differences = (.04+.04+.64+.04+.04)/5 = 0.168
  • The square root of 0.168 = .40

The standard deviation for these 5 observations is .40.

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Normal Distribution

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A Normal distribution refers to a set of observations where the highest concentration of observations occurs around the mean(average), with fewer observations occuring towards the tails (far from the mean).

In order to be considered a normal distribution:

• 68% of observed values should fall within 1-standard deviation from the mean
• 95% of observed values should fall within 2-standard deviation from the mean
• 99% of observed values should fall within 3-standard deviation from the mean

Normal Distribution, Mean, & Standard Deviation Visualized

Below is a visualization of a normal distribution.

• The blue line indicates the mean value.
• The red bars indicate 1 standard deviation above/below the mean.

library(ggplot2)
library(dplyr)
library(knitr)
groupA <- c(rnorm(1500, mean = 0))  #Produce list of random values w/ mean of 0 & normal distribtion
normal_hist_A <- data.frame(groupA)
ggplot(normal_hist_A)+ 
  geom_histogram(aes(x=groupA),fill="salmon") +
    geom_vline(aes(xintercept=mean(normal_hist_A$groupA)), color="blue")+
    geom_vline(aes(xintercept=(mean(normal_hist_A$groupA)+sd(normal_hist_A$groupA))),color="red")+
    geom_vline(aes(xintercept=(mean(normal_hist_A$groupA)-sd(normal_hist_A$groupA))),color="red")+
    theme_minimal()+  xlab("observed value")+ ylab("number of observations")