Normal Distribution in Statistics

You saw how the probability distributions of continuous random variables differ from those of discrete random variables.

But can you think of some examples of continuous distributions? Which is the most commonly used continuous probability distribution? Which distribution occurs most commonly in nature? Let’s hear from Prof. Tricha on this.

Normally distributed data follows the 1-2-3 rule. This rule states that there is a:

1. 68% probability of the variable lying within 1 standard deviation of the mean,

2. 95% probability of the variable lying within 2 standard deviations of the mean, and

3. 99.7% probability of the variable lying within 3 standard deviations of the mean.

This is similar to this case: If you buy a loaf of bread every day and measure it, where the mean weight = 100 g and the standard deviation = 1 g, then:

1. For 5 days every week, the weight of the loaf that you bought that day will be within 99 g (100-1) and 101 g (100+1).

2. For 20 days every 3 weeks, the weight of the loaf that you bought that day will be within 98 g (100-2) and 102 g (100+2).

3. For 364 days every year, the weight of the loaf that you bought that day will be within 97 g (100-3) and 103 g (100+3).

A lot of naturally occurring variables are normally distributed. For example, the heights of a group of adult men would be normally distributed. To try this out, we took the heights of 50 male employees at the upGrad office and then plotted the probability density function using that data.

As you can see, the data is roughly normal.

You can visualise the PDF and CDF for different normal distributions using the interactive app given below. From the various options in the drop-down menu, select 'Normal'. You will then get to see the probability distribution for a normal distribution with µ = 0 and σ = 1. In fact, you can play around with the value of µ and σ to see how that changes the distribution. Using the green slider below the distribution, you can visualise the distribution’s cumulative probability.

In fact, you can select 'Uniform' in the drop-down menu and visualise the CDF and PDF for the uniform distribution too. Be sure to play around with a and b, which give the lowest and highest possible values, respectively, for the random variable.