The Normal Probability Distribution is defined as a continuous distribution in which the mean, median, and mode are equal, leading to what we know as the bell curve as shown below.

\[ f(x)=\frac{1}{\sigma\sqrt(2\pi)}*e^\frac{-1}{2}(\frac{x-\mu}{\sigma})^2 \] This is the mathematical function that represents the normal distribution where:

• • σ = the standard deviation (or the spread graphically)
• • μ = the mean (or the center graphically)
• • π = 3.14159…
• • e = 2.71828…

The Normal Distribution is a probability function. This means that whatever the values of the mean and standard deviation are, the function must integrate to 1.

\[ \int_{-\infty}^{\infty} \frac{1}{\sigma\sqrt(2\pi)}*e^\frac{-1}{2}(\frac{x-\mu}{\sigma})^2 dx=1 \] This is because the total area under the curve of a Probability Density Function represents the probability that a random variable’s value is somewhere in the sample space. So the area under the curve represents all possibilities which will add to 1.

The normal curve is so useful because it can easily be visually analyzed. This is because regardless of what the values of μ or σ are, the probabilities of outcomes within 1, 2, or 3 standard deviations of the mean are always approximately:

• 1 Standard Deviation (μ-σ and μ+σ): 68%
• 2 Standard Deviations (μ-2σ and μ+2σ): 95%
• 3 Standard Deviations (μ-3σ and μ+3σ): 99.7%

Almost all values fall within 3 standard deviations of the mean.

The probability of being within 1, 2, or 3 standard deviations of the mean is shown graphically below:

As shown previously, the usefulness of the normal distribution is its probability accuracy no matter the value of the mean and standard deviation. Meaning that 68% of observations still fall within 1 standard deviation of the mean, 95% within 2, etc. I’ll demonstrate this by changing the mean and standard deviation and showing the graph of the new distribution.

#Instead of using mean=0 and stdev=1 shown below
mean <- 0
sd <- 1
x_values <- seq(mean - 4 * sd, mean + 4 * sd, length.out = 1000)
y_values <- dnorm(x_values, mean = mean, sd = sd)
data <- data.frame(x = x_values, y = y_values)
#We'll try mean=3, stdev=2
mean <- 3
sd <- 2
x_values <- seq(mean - 4 * sd, mean + 4 * sd, length.out = 1000)
y_values <- dnorm(x_values, mean = mean, sd = sd)
data <- data.frame(x = x_values, y = y_values)

This is the new normal distribution with μ=3 and σ=2:

It is important to understand the Standard Normal Distribution Curve and its’ properties to be able to efficiently digest normal distribution qualities. As shown by the previous 2 graphs, the bell curve will always be highest around the mean, showing that occurrences most frequently lie within 1 standard deviation of the mean, and decrease in frequency the further from the mean the occurrence is.