The standard Normal Distribution: Calculation and Interpretation of Probabilities

The standard normal distribution refers to a normal distribution that has been standardized such that it has a mean of 0 and a standard deviation of 1. The shorthand notation used is:

N~(0, 1)

In the context of statistics and mathematics, standardization is the process of converting an observed value for a random variable into a z-value where:

Z = (observed value – population mean)/standard deviation

= (x – μ)/σ

The z-value, also referred to as the z-score in some books, represents the number of standard deviations a given observed value is from the population mean.

Example 1

The returns on ABC stock are normally distributed where the mean is $0.6 with a standard deviation of $0.2. Calculate the z-scores for a return of $0.1

Solution:

If the return is $0.1, then x = 0.1 (this is our observed value)

Therefore, z = (x – μ)/σ

= (0.1 – 0.6)/0.2

= -2.5 (The return of $0.1 is two and a half standard deviations below the mean.)

We could get the z-score for any other observed value following a similar approach. For instance, the z-score for a return of $1 will be:

Z = (1 – 0.6)/0.2

= 2 (The return of $1 is two standard deviations above the mean)

Calculating Probabilities using z-values Under the Standard Normal Distribution

Using the standard normal distribution table, we can be able to calculate the that a normally distributed random variable Z, with mean equal to 0 and variance equal to 1, is less than or equal to z. i.e. P(Z ≤ z). However, the table does this when we only have positive values of z. Simply put, if the examiner asks you to find the probability behind a given positive z-value, you’d have to look it up directly on the table.

P(Z ≤ z) = θ(z) when z is positive

probability-less-than-a-z-value

Example 2:

Using the data from our first example, suppose you were asked to calculate the probability that the return is less than $1:

Solution

Firstly, you’d be required to calculate the z-value (2 in this case)

P(Z ≤ 2) can be read off directly from the table

You just move down and locate the z-value that lies to the right of “2” i.e. 0.9772

normal-table

Note: The table above is incomplete

P(Z ≤ -z)

If we have a negative z-value, we still can calculate the corresponding probability by noting that:

P(Z  ≤ -z) = 1 – P(Z ≤ z) or

θ(–z) = 1 – θ(z)

This relationship is true when we consider the following facts:

  1. The total area (probability) under the standard normal distribution is 1
  2. The standardized normal distribution is symmetrical about the mean

negative-z-values

Question

Calculate P(Z  ≤ -2.5)

A. 0.9938

B. 0.0062

C. 0.06

Solution

The correct answer is B.

P(Z  ≤ -2.5) = 1 – P(Z  ≤ 2.5)

= 1 – 0.9938

= 0.0062

Reading 10 LOS 10l:

Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.

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