Standard Normal Distribution in Statistics

As you learnt in the previous question, it doesn’t matter what the values of µ and σ are. If you want to find the probability, all you need to know is how far the value of X is from µ, and specifically, what multiple of σ is the difference between X and µ.

Let’s see how you can find this.

As you just learnt, the standardised random variable is an important parameter. It is given by:

$Z=\frac{X-\mu }{\sigma }$

Basically, it tells you how many standard deviations away from the mean your random variable is. As you just saw, you can find the cumulative probability corresponding to a given value of Z, using the Z table:

Alternatively, you can use the following equation to find the cumulative probability:

$F\left(Z\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^Z{e}^{\frac{-t^2}{2}}dt$

You can also use Excel to find the cumulative probability for Z. For example, let’s say you want to find the cumulative probability for Z = 1.5. In the Excel sheet, you will type:

= NORM.S.DIST(1.5, TRUE)

Basically, the syntax is:

= NORM.S.DIST(z, TRUE)

Here, z is the value of the Z score for which you want to find the cumulative probability. TRUE = find cumulative probability, FALSE = find probability density.

Also, you can find the probability without standardising. Let’s say that X is normally distributed, with mean (μ) = 35 and standard deviation (σ) = 5. Now, if you want to find the cumulative probability for X = 30, you would type:

= NORM.DIST(30, 35, 5, TRUE)

Basically, the syntax is:

= NORM.DIST(x, mean, standard_dev, TRUE)

Notice how the value of σ affects the shape of the normal distribution. Use the slider for σ to adjust its value.

As you can see, the value of σ is an indicator of how wide the graph is. This will be true of any graph, not just a normal distribution. A low value of σ means that the graph is narrow, while a high value implies that the graph is wider. This is because a wider graph has more values away from the mean, resulting in a high standard deviation.

Again, some more probability distributions are commonly seen among continuous random variables. They are not covered in this course, but if you want to go through some of them, you can use the links below. (In all the links you can view the respective topic on the lefthand side of the webpage):

1. Exponential Distribution
2. Gamma Distribution
3. Chi-Squared Distribution