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.

[At 0:50, the value of X is incorrectly labeled as 43.5 instead of 43.25]

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

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:

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: