The Normal Distribution

statistics year12 standard

The normal distribution | NSW Curriculum Website

• MST-12-S2-10 analyses normally distributed datasets using statistical processes

📖 Prior Knowledge

Content	Prerequisite relationships
Data Analysis	- Describe the shape of a distribution → Recognise a normal distribution
Data Analysis A	- Calculate and compare standard deviation → Calculate a z-score

Normally distributed datasets

• Recognise that a dataset that is normally distributed can be represented by a bell-shaped curve
• Explain that the mean, median and mode are approximately equal for data arising from a random variable that is normally distributed

Calculating z-scores

• Describe the $z$-score as the number of standard deviations that a value is above or below the mean
• Recognise that the set of $z$-scores for data arising from a random variable that is normally distributed has a mean of 0 and standard deviation of 1
• Calculate the $z$-score corresponding to a specific value in a dataset by applying the formula
• $z=\frac{x-\mu }{\sigma }$, where $x$ is a specific value, $\mu$ is the mean and $\sigma$ is the standard deviation
• Use $z$-scores to compare scores from different datasets and justify conclusions in the context of the problem

Probability using z-scores

• Calculate probabilities using $z$-scores and the empirical rule
• Represent probabilities by shading areas under the normal distribution curve
• Use $z$-scores to identify probabilities of events less or more extreme than a given event and solve problems
• Use $z$-scores to make judgements related to outcomes of a given event or sets of data