Normal distribution is a continuous probability distribution defined by two parameters μ, the mean of the distribution and σ2 the variance. In real world situations distributions of many measures like height of adults, SAT scores, physical conditions like Blood pressure are approximately normal. A normal variable can assume any sub intervals in the interval (-∞, ∞). Normal distribution is also known as Gaussian distribution named after the German mathematician Karl Gauss who published a work describing it.Normal distribution is considered as the most important probability distribution in statistics.

Normal distribution is a symmetric bell shaped distribution of a continuous random variable. It can be considered as the limiting distribution of the binomial random variable. If we analyze the Histograms of a binomial distribution, it can be seen the histogram approaches the normal shape as the parameter n ( number of trials ) increases.
The normal distribution is defined by the mathematical equation,
y = $\frac{e^{\frac{-(X-\mu )^{2}}{2\sigma ^{2}}}}{\sigma \sqrt{2\pi }}$
where μ is the mean and σ is the standard deviation of the distribution.
The value of the mean μ determines the position of the center of the graph for a specific normal variable, while the value of the variance (σ2) determines spread and the peak of the graph as seen below.

Normal Distribution


Normal Distribution Definition

The Normal distribution curve is bell shaped, centered and symmetric about its mean. This means 50% of the values a normal variable assumes are less than the mean and 50 % are greater than the mean.
Normal Distribution Curve
The probability of a normal assuming values in an interval (a, b) is proportional to the area of below the normal curves between the two intervals. If the total area below the normal curve is taken to be 1, the probabilities can be equated to the corresponding areas in the normal curve.
  • The normal curve is continuous
  • It is Bell shaped.
  • Normal curve is symmetric about the mean
  • The mean, median and mode of a normal distribution are equal and located at the center of the distribution
  • The curve is unimodal
  • The curve never touches the x- axis.
  • The total area under the normal curve is 1 or 100%.
  • 68% of the distribution lies within one standard deviation, 95% within two standard deviations and 99.7% within three standard deviations from the mean. This is known as Empirical rule for normal distributions.
The probabilities for normal distribution are found from the area under normal curve. This means that a table relating the variable values to the corresponding areas is of great use in determining the probabilities real quick. But this would mean that we require separate tables for distributions defined by different parameter values.

In order to make the method of using tables efficient, a standard normal distribution of variable Z is defined with mean = 0 and standard deviation 1. Any normal variable X can be transformed to z scores and the table of areas of Z distribution can be used for all normal variables.
The z-score for any normal variable X with mean μ and standard deviation σ can be found using the formula,
z = $\frac{x-\mu }{\sigma }$
There are two types of normal distribution tables are used for determining the probabilities.

Positive z-score table gives the area between a positive z value and mean. Probability of any interval value can be found using this table and applying the property of symmetry of the normal curve. The picture of the table below shows the area corresponding to z = 2.35 as 0.4906.

Normal Distribution Table

When we say that the area corresponding to z = 2.35 is equal to 0.4906, if means P(0 < Z < 2.35 ) = 0.4906 and not
P( Z = 2.35) = 0.4906, as the normal variables assumes only intervals as values and not any specific numerical value.
Indeed the probability of the variable assuming any numerical value is zero and hence P(Z = 2.35) = 0

Left area table, gives the area to the left of any z-score both positive and negative. This table can also be used for finding any probability applying the property of symmetry of the normal curve.
Suppose the variable X is a multi valued function expressed as a vector function as x = $\begin{bmatrix}
x_{1}\\
.\\
.\\
.\\
x_{n}
\end{bmatrix}$ with μ = E(x) = $\begin{bmatrix}
E(x_{1})\\
.\\
.\\
.\\
E(x_{n})
\end{bmatrix}$ and ∑ = cov(x) = E((x - μ)(x - μ)T).

Then the multivariate normal distribution of X with parameters μ and ∑ is defined as

p(x) = $\frac{1}{(2\pi )^{\frac{N}{2}}|\sum |^{\frac{1}{2}}}.e^{-\frac{(x-\mu )^{T}\sum ^{-1}(x-\mu )}{2}}$


Solved Examples

Question 1: For a normal distribution X, μ = 3.1 and σ = 0.5. Find
  1. P( X > 3.5)
  2. P( 2 < X < 3.5)
  3. P(X < 2)

Solution:
 
For all these the z -scores are found and the probabilities are found by making a suitable sketch and using the standard normal table.
      Normal Distribution Example z = $\frac{x-\mu }{\sigma }$

   = $\frac{3.5-3.1}{0.5}$ = 0.8

P(X > 3.5) = P( Z > 0.8)
                 = P(Z > 0) - P(0 < z < 0.8)
                 = 0.5 - 02881 = 0.2119
 Z-score for X = 2,

 z1 = $\frac{2-3.1}{0.5}$ = -2.2

Z-score for X = 3.5
z2 = 0.8
P(2 < X < 3.5) = P(-2.2 < z < 0.8)
                       = P(-2.2 < z < 0) + P(0 < z < 0.8)
                       = 0.4861 + 0.2881 = 0.7742
 Normal Distribution Problem
       Normal Distribution Problems  P(X < 2) = P(z < -2.2)
               = P(z < 0) - P(-2.2 < z <0)
               = 0.5 - 0.4861 = 0.0139


 

Question 2: The household income in a hypothetical town is normally distributed with an average of 1,500 dollars and standard
     deviation of 400 dollars per moth. Find percentage of Households in the town whose monthly income range between
     1,800 and 2,500 dollars.
Solving Normal Distribution

Solution:
 
μ = 1,500       σ = 400       x1 = 1,800 and  x2 = 2,500   

     z = $\frac{x-\mu }{\sigma }$

     z1 = $\frac{1800-1500}{400}$ = 0.75       and     z2 = $\frac{2500-1500}{400}$ = 2.5

     P(1,800 < X < 2,500) = P(0.75 < Z < 2.5) = P(0 < Z < 2.5) - P (0 < Z < 0.75)

                                                                      = 0.4938 - 0.2734 = 0.2204