4.6.3 Standardisation of Normal Variables

Standard Normal Distribution

What is the standard normal distribution?

The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1
- It is denoted by $Z$
- $Z ~ N (0, 1^{2})$

Why is the standard normal distribution important?

Any normal distribution curve can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch
Therefore we have the relationship:
- $Z = \frac{X - μ}{σ}$
- Where $X ~ N (μ, σ^{2})$ and $Z ~ N (0, 1^{2})$
Probabilities are related by:
- $P (a < X < b) = P (\frac{a - μ}{σ} < Z < \frac{b - μ}{σ})$
- This will be useful when the mean or variance is unknown
Some mathematicians use the function $Φ (z)$ to represent $P (Z < z)$

z-values

What are z-values (standardised values)?

For a normal distribution $X ~ N (μ, σ^{2})$ the z-value (standardised value) of an x-value tells you how many standard deviations it is away from the mean
- If z = 1 then that means the x-value is 1 standard deviation bigger than the mean
- If z = -1 then that means the x-value is 1 standard deviation smaller than the mean
If the x-value is more than the mean then its corresponding z-value will be positive
If the x-value is less than the mean then its corresponding z-value will be negative
The z-value can be calculated using the formula:
- $z = \frac{x - μ}{σ}$
- This is given in the formula booklet
z-values can be used to compare values from different distributions

Finding Sigma and Mu

How do I find the mean (μ) or the standard deviation (σ) if one of them is unknown?

If the mean or standard deviation of $X ~ N (μ, σ^{2})$ is unknown then you will need to use the standard normal distribution
You will need to use the formula
- $z = \frac{x - μ}{σ}$ or its rearranged form $x = μ + σ z$
You will be given a probability for a specific value of
- $P (X < x) = p$ or $P (X > x) = p$
To find the unknown parameter:
STEP 1: Sketch the normal curve
- Label the known value and the mean
STEP 2: Find the z-value for the given value of x
- Use the Inverse Normal Distribution to find the value of $z$ such that $P (Z < z) = p$ or $P (Z > z) = p$
- Make sure the direction of the inequality for $Z$ is consistent with the inequality for $X$
- Try to use lots of decimal places for the z-value or store your answer to avoid rounding errors
  - You should use at least one extra decimal place within your working than your intended degree of accuracy for your answer
STEP 3: Substitute the known values into $z = \frac{x - μ}{σ}$ or $x = μ + σ z$
- You will be given and one of the parameters (μ or σ) in the question
- You will have calculated z in STEP 2
STEP 4: Solve the equation

How do I find the mean (μ) and the standard deviation (σ) if both of them are unknown?

If both of them are unknown then you will be given two probabilities for two specific values of x
The process is the same as above
- You will now be able to calculate two z -values
- You can form two equations (rearranging to the form $x = μ + σ z$ is helpful)
- You now have to solve the two equations simultaneously (you can use your calculator to do this)
- Be careful not to mix up which z-value goes with which value of x

Worked Example

It is known that the times, in minutes, taken by students at a school to eat their lunch can be modelled using a normal distribution with mean μ minutes and standard deviation σ minutes.

Given that 10% of students at the school take less than 12 minutes to eat their lunch and 5% of the students take more than 40 minutes to eat their lunch, find the mean and standard deviation of the time taken by the students at the school.

4-6-3-ib-aa-sl-finding-mu-sigma-we-solution

DP IB Maths: AA HL

Revision Notes

4.6.3 Standardisation of Normal Variables

Standard Normal Distribution

What is the standard normal distribution?

Why is the standard normal distribution important?

z-values

What are z-values (standardised values)?

Finding Sigma and Mu

How do I find the mean (μ) or the standard deviation (σ) if one of them is unknown?

How do I find the mean (μ) and the standard deviation (σ) if both of them are unknown?

Worked Example

Author: Daniel

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