Table of Contents

## What is a Normal Distribution?

The most commonly found distribution is known as the Normal Distribution. It takes the shape of the bell curve. The average value has the highest frequency and the frequency of values larger or smaller values than the average taper off in a descending curve.

The Normal Distribution has a peculiar property where the mean = median = mode. The curve is seen to be symmetric and can be found in various statistical measurements. The distribution shown above shows a normal distribution curve for 1500 students and the time they take to tie shoe laces. The mean value can be seen as 50 seconds and the values taper off from fastest to slowest.

## Empirical Rule of Normal Distribution

There is an empirical rule of Normal distribution that can be used for getting quick approximations. This Rule is also commonly known as the 68-95-99 rule. This rule basically states that approximately 68% of the data lies within the first standard deviations from the mean. 95% of data lies between the second standard deviations from the mean. 99% of the data lies between the 3 standard standard deviations from the mean. The figure below shows the divided percentages of data using the empirical rule.

The image above shows a normal distribution for sample of 10,000 sea shells and their size in mm. The distribution has a mean of 60 with a standard deviation of 15.

According to empirical rule, approximately 68% of the shells will have size between 45 and 75 mm.

95% of the shells will have size between 30 and 90mm.

99% of shells will have size between 15 and 105mm.

## Standard Normal Distribution

A standard Normal Distribution has a mean of 0 and standard deviation of 1. This method is required when the statistical problems cannot be approximated using the empirical rule. The formula to convert a normal distribution problem to Standard normal is given below.

In the above formula The notations are as follows:

Z is known as Z value

X is the statistical value for probability calculation

u is the population mean

σ is the Greek letter sigma that is the notation of population standard deviation.

n is the sample size

The Probability of Z from above formula can be found using Z tables.

Click here for Z table

## Practice problems

Normal Distribution empirical rule problem:

Find out the percentage of the following data from above example with mean 60 and standard deviation 15 and sample size 10,000.

- What percentage of shells have size below 75mm ?
- What percentage of shells have size between 15 and 90mm ?
- What percentage of shells have size above 60mm ?
- How many shells have a size between 30 and 60mm ?
- How many shells have size between 45 and 105mm ?
- How many shells have size less than 30mm?

Standard Normal Distribution problem:

A small city has a population of n=50,000 with average age of 40 with a standard deviation of 10. Answer the following questions. (Hint Use Z Table to solve the problems)

- What is the probability that a random person you meet has age less than 42?
- What is the probability that a random person chosen has age between 33 and 64?
- How many people are expected to have age between 25 and 55
- How many people in the town may have age above 75?
- What is the probability that a group of 50 people in a restaurant will have ages between 36 and 45?

Click here to get the step by step solutions to the above problems.

You can find additional problems with solutions for normal distribution here.