Normal Distribution

Introduction to normal distribution

You might have come across the normal distribution while studying probability and statistics or hypothesis testing, etc. Moreover, a normal distribution plays a crucial role in day-to-day life. Like in the case, the height of students in a class follows normal distribution; the weight of oil bags produced in a factory follows a normal distribution, IQ scores for a group of persons follow a normal distribution, etc. Here is the synopsis of the normal distribution.

Normal Distribution

The normal distribution is a continuous type distribution, which is often found in nature. The normal distribution has a bell-shaped, symmetric curve about its mean. Let X be a normal random variable with mean µ and standard deviation σ, symbolically we can write this as, X ~ N(µ, σ). We define the random variable Z as follows.

Then Z is called standard normal random variable which has "mean" 0 and "standard deviation" 1, symbolically we can write this as, Z ~ N(0, 1). The above formula is often used to find the Z score; hence, we call it as Z score formula.

Properties of Normal Distribution

Consider the following figure that gives the standard normal curve.

Standard Normal Curve

• The shaded region is the left-tailed probability at Z score = 1.5.
• The Z curve is symmetric around 0.
• The total area under normal is 1. Therefore, the right-tailed probability to the Z score is 1 – (left-tailed probability).
• The right-tailed probability to Z score “1.5” is the same as that of left-tailed probability to the Z score “-1.5” {Because of symmetry}.

There are two types of Z scores table; one with negative Z scores and another with positive Z scores.

Use of normal distribution

In probability and statistics, we use the normal distribution to study some complex random variables like students’ T distribution, chi-square distribution, etc.

In hypothesis testing, we use the normal distribution to test the claim about population mean (µ) if we know population standard deviation (σ) in prior. In this situation, we use standard normal distribution or Z distribution hence we call it as Z test. The formula for the test statistic in the Z test is as follows.

While constructing confidence intervals for the population mean (µ) we use the standard normal distribution. The formula to construct a confidence interval for a population mean is as follows.

Here Z(α/2) is a critical value.