Normal distribution

• A normal distribution curve is a type of probability distribution for continuous random variables.
• Examples:
  • The distribution of height
  • The distribution of IQ
• Properties of normally distributed curve are as follows:
  • It is symmetrical on both sides of its mean $\mu$
  • The mean lies at the middle of the curve. Mean = Median = Mode
  • The total area under the curve is equal to 1 (since it is probability density function)

• Probability density function for Normal distribution is:
  $\begin{align*} & f(x) = \dfrac{e^{-\frac{(x-\mu)^2}{(2{\sigma}^2)}}}{\sigma \cdot \sqrt{2\pi}} \\ & where, \\ & \mu = mean \\ & \sigma = \text{standard deviation} \\ & e = 2.7 \end{align*}$

1-2-3 rule of normal distribution

• The area of the curve lying within 1 standard deviation from the mean i.e between $\mu - \sigma$ and $\mu + \sigma$ is 0.68 or 68%,
• The probability of a continuous random variable that will lie within 1 standard deviation from mean is 0.68 or 68%,
• The probability of a continuous random variable that will lie within 2 standard deviation of the mean is 0.95 or 95%,
• The probability of a continuous random variable that will lie within 3 standard deviation of the mean is 0.997 or 99.7%.
• Graph for the same above:
  

Standardized Normal distribution and Z-score

• Standardized Normal distribution is a special type of Normal distibution where $\mu = 0$ and $\sigma = 1$.
• The standardized normal distribution is used to compare between differnt normal distributions.
• A normal distribution can be converted into standardized normal distribution with the help of Z-score.
• $$\text{Z-score} = \dfrac{x - \mu}{\sigma}$$
• For example, for a normal distribution with μ= 35 and σ = 5, the normal distribution curve and the standard normal distribution curve will look like this:
• Z-score can be used to:
  • Calculate the probability of the occurence of a particular random variable
    It is done with the help of Z-table. Excel can also be used with function NORMDIST.
  • Compare normal distributions
    Example: Suppose that the marks obtained by the students of a class are normally distributed.
    • In the mid-term exam, the mean score was 50 out of 100, and the standard deviation was 10;
    • And in the end-term, the mean score was 60, and the standard deviation was 20.

    A student Ram scored 70 in the mid-term exam and 72 in the end-term exam. In which exam was his relative performance better?
    Soln: To answer above question, we can use Z-score to compare scores.

    • Z-score for Mid-term = (70 - 50)/10 = 2
    • Z-score for End-term = (72 - 60)/20 = 0.6

    Looking at the Z-scores, we can conclude that Ram’s relative performance was better in the mid-term exam compared to end-term exam.