Normal Distribution

The normal distribution is an extremely useful statistical distribution. Data for heights and weights of adults and objects produced by machines, for example, tend to follow the normal (bell-shaped) distribution. More importantly, if we take large enough samples of any data, the means of the samples can be modelled by a normal distribution. Psychologists use the normal distribution to classify people's test results, including intelligence tests. Don't be a dummy in your IB exam and make sure you understand this topic!


Key Concepts

On this page, you should learn about

  • the normal distribution and its curve
  • probability calculations with the normal distribution
  • standardization of the normal variables
  • finding mean and/or variance using inverse normal calculations

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

The random variable X is normally distributed with a mean of 100 and a standard deviation of 15.

a) Write down P(X < 120)

b) Write down P(95 < X < 105)

c) P(X < c) = 0.6. Write down the value of c

d) Find the interquartile range of X

Hint

Full Solution

Question 2

The weights of female adult penguins found on a remote island are normally distributed with a mean of 3.15 kg. It is found that 81% of these penguins weigh more than 3 kg. The weights of the penguins have a standard deviation of \(\sigma\).

a) Find the standardized value for 3kg.

b) Hence, find the value of \(\sigma\).

It is estimated that there are 4000 female adult penguins on the island. Penguins are considered underweight if they weigh less than 2.9 kg.

c) Find the estimated number of underweight female penguins on the island.


Hint

Full Solution

Question 3

It is known that 2% of Filips lightbulbs have a life of less than 1000 hours and 10% have a life less than 1100 hours. It can be assumed that lightbulb life is normally distributed with a mean of \(\mu\) and a standard deviation of \(\sigma\)

a) Find the value of \(\mu\) and the value of \(\sigma\).

b) Find the probability that a randomly selected Filips lightbulb will have a life of at least 1200 hours.


Hint

Full Solution

Question 4

The amount of caffeine, X in mg, found in a cup of tea can be modelled by the normal distribution \(X\sim N(26,5)\)

a) Find the probability that a randomly selected cup of tea contains at least 27 mg of caffeine.

b) 10 cups of tea are randomly selected. Find the probability that more than 5 of these cups has at least 24 mg of caffeine.


Hint

Full Solution

MY PROGRESS

How much of Normal Distribution have you understood?