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!
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
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Here is a quiz that practises the skills from this page
START QUIZ!
A random variable X is distributed normally with a mean of 10 and a standard deviation of 2
Which graph best represents P(X<12)?
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A random variable X is distributed such that X~N(80 , 10²). Work out the unknown values in the following situations
For a normal distribution, we know that
- about 68% of the values are within one standard distribution away from the mean.
- about 95% of the values are within two standard distributions away from the mean.
- about 97.7% of the values are within three standard distributions away from the mean.
a = b =
| c = d =
| e = f =
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There is no need to a use a graphical calculator in this question. Use the geometry of the shapes and t to work out the missing values
- \(80 \pm 20\)
- \(80 \pm 10\)
- \(80 \pm 30\)
A random variable X is distributed such that X~N(100 , 15²). Work out the probabilities of the following situations
For a normal distribution, we know that about 68% of the values are within one standard distribution away from the mean.
p =
| p =
| p =
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p =
| p =
| p =
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There is no need to a use a graphical calculator in this question.
Use the fact that we know that about 68% of the values are within one standard distribution away from the mean and to work out the areas of each part
The random variable X is normally distributed with mean of 50. The following diagram shows the curve for X.
The area of the shaded region R is 0.19
Work out the following probabilities
P(X < 58) =
P(50 < X < 58) =
P(42 < X < 58) =
P(X < 58) = 1 - 0.19
P(50 < X < 58) = 0.81 - 0.5
P(42 < X < 58) = 1 - 2x0.19 or 0.81 - 0.19
The random variable X is normally distributed with mean of 16. The following diagram shows the curve for X.
The area of the shaded region R is 0.37
Work out the following probabilities
P(X < 11.5) =
P(11.5 < X < 20.5) =
P(X < 20.5) =
P(X < 11.5) = 0.5 - 0.37
P(11.5 < X < 20.5) = 2x0.37
P(X < 20.5) = 0.74 - 0.13
Find all the unknowns below
The random variable X is normally distributed with mean of 10.
Given that P(X < 7) = P(X > a)
a =
The random variable X is normally distributed with mean of 18.
P(X < 16.5) \(\approx\) 0.309
P(16.5 < X < 19.5) \(\approx\)
The random variable X is normally distributed with mean of 100.
P(88 < X < 112) \(\approx\) 0.576
P(X > 112) \(\approx\)
The random variable X is normally distributed with mean of 35.
P(X > 30) \(\approx\) 0.841
P(X > 40) \(\approx\)
A random variable X is normally distributed. Work out the unknown values in the following situations to 3 significant figures
X~N(25 , 3²). P(X < 27) =
X~N(-3 , 2²). P(X > 0) =
Z~N(0 , 1²). P(-0.2 < Z < 0.3) =
All of the above are probability calculations. Use the normal cumulative function on your calculator.
A random variable X is normally distributed. Work out the unknown values in the following situations to 3 significant figures
X~N(100 , 15²). P(X < a) = 0.95, a \(\approx\)
X~N(58 , 8²). P(X > b) = 0.67, b \(\approx\)
Z~N(0 , 1²). P(|Z| < c) = 0.75 c \(\approx\)
All of the above calculations use the inverse normal function.
You could think of the last one as P(-c < Z < c) = 0.75
The heights of adults can be modelled using the normal distribution H~N(165 , 12²).
Calculate the following to 3 significant figures.
Upper quartile height =
Interquartile range of the heights=
90th percentile height =
Upper quartile P(H < Q3) = 0.75, P(H < 173) = 0.75,
Using symmetry, lower quartile, Q1 = 149 Interquartile range = Q3 - Q1 = 173 - 157 = 16
90th percentile P(H < a) = 0.9
A random variable X is distributed such that X~N(14 , 2²). Work out these conditional probabilities to 3 significant figures
P(X >15 / X > 13) \(\approx\)
P(X >15 / 14 < X < 16) \(\approx\)
These are questions about conditional probability
\(P(A/ B)=\frac{P(A \cap B )}{P(B)}\)
P(X >15 / X > 13) = \(\frac{P(X >15)}{P(X > 13)}\approx\)\(\frac{0.3085}{0.6915}\approx 0.446\)
P(X >15 / 14 < X < 16) =\(\frac{P(15<X <16)}{P(14<X< 16)}\approx\)
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
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
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
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
How much of Normal Distribution have you understood?
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