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:
- $P (X < a) = P (Z < \frac{a - μ}{σ})$
- This will be useful when the mean or variance is unknown
If a value of x is less than the mean then the z-value will be negative
In old textbooks you might see the function $Φ (z)$ this just means $P (Z < z)$

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 the $X ~ N (μ, σ^{2})$ 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 $x (P (X < x) = p or P (X > x) = p)$
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 X
- Try to use lots of decimal places for the z-value 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 x 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 standard deviation 4 minutes.

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

4-3-3-standard-normal-distribution-we-solution

Examiner Tip

These questions are normally given in context so make sure you identify the key words in the question. Check whether your z-values are positive or negative and be careful with signs when rearranging.

Standard Normal Distribution (AQA A Level Maths: Statistics)

Revision Note

What is the standard normal distribution?

Why is the standard normal distribution important?

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?

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1. Statistical Sampling

1.1 Sampling & Data Collection

1.1.1 Sampling & Data Collection

2. Data Presentation & Interpretation

2.1 Statistical Measures

2.1.1 Basic Statistical Measures

2.1.2 Frequency Tables

2.1.3 Standard Deviation & Variance

2.2 Data Presentation

2.2.1 Data Presentation

2.2.2 Box Plots & Cumulative Frequency

2.2.3 Histograms

2.3 Working with Data

2.3.1 Outliers & Cleaning Data

2.3.2 Intrepreting Data

2.3.3 Skewness

2.4 Correlation & Regression

2.4.1 Correlation & Regression

2.5 Further Correlation & Regression (A Level only)

2.5.1 PMCC & Non-linear Regression

2.5.2 Hypothesis Testing for Correlation

3. Probability

3.1 Basic Probability

3.1.1 Calculating Probabilities & Events

3.1.2 Venn Diagrams

3.1.3 Tree Diagrams

3.2 Further Probability (A Level only)

3.2.1 Set Notation & Conditional Probability

3.2.2 Further Venn Diagrams

3.2.3 Further Tree Diagrams

3.2.4 Probability Formulae

4. Statistical Distributions

4.1 Probability Distributions

4.1.1 Discrete Probability Distributions

4.2 Binomial Distribution

4.2.1 The Binomial Distribution

4.2.2 Calculating Binomial Probabilities

4.3 Normal Distribution (A Level only)

4.3.1 The Normal Distribution

4.3.2 Normal Distribution - Calculations

4.3.3 Standard Normal Distribution

4.4 Choosing Distributions (A Level only)

4.4.1 Modelling with Distributions

4.4.2 Normal Approximation of Binomial

5. Hypothesis Testing

5.1 Hypothesis Testing

5.1.1 Hypothesis Testing

5.2 Hypothesis Testing (Binomial Distribution)

5.2.1 Binomial Hypothesis Testing

5.3 Hypothesis Testing (Normal Distribution) (A Level only)

5.3.1 Sample Mean Distribution

5.3.2 Normal Hypothesis Testing

6. Large Data Set

6.1 Large Data Set

6.1 Large Data Set

Author: Dan