NSW Syllabuses

Mathematics Standard Stage 6 - Year 12 Standard 2 - Statistical Analysis MS-S5 The Normal Distribution 📎

Outcomes

A student:

• MS2-12-2

analyses representations of data in order to make inferences, predictions and draw conclusions

• MS2-12-7

solves problems requiring statistical processes, including the use of the normal distribution and the correlation of bivariate data

• MS2-12-9

chooses and uses appropriate technology effectively in a range of contexts, and applies critical thinking to recognise appropriate times and methods for such use

• MS2-12-10

uses mathematical argument and reasoning to evaluate conclusions, communicating a position clearly to others and justifying a response

Related Life Skills outcomes: MALS6-2, MALS6-9, MALS6-13, MALS6-14

• Subtopic Focus

• The principal focus of this subtopic is to develop an understanding of the properties of the normal distribution and the value of relative measure in the analysis and comparison of datasets arising from random variables that are normally distributed.
• Students develop techniques to analyse normally distributed data and make judgements in individual cases justifying the reasonableness of their solutions.
• Content

• Students:
• recognise a random variable that is normally distributed, justifying their reasoning, and draw an appropriate ‘bell-shaped’ frequency distribution curve to represent it
• identify that the mean and median are approximately equal for data arising from a random variable that is normally distributed
• calculate the $$z$$-score (standardised score) corresponding to a particular value in a dataset
• use the formula $$z = \frac{x - \bar{x} }{s}$$, where $$\bar{x}$$ is the mean and $$s$$ is the standard deviation
• describe the $$z$$-score as the number of standard deviations a value lies 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
• use calculated $$z$$-scores to compare scores from different datasets, for example comparing students’ subject examination scores
• use collected data to illustrate that, for normally distributed random variables, approximately 68% of data will have $$z$$-scores between -1 and 1, approximately 95% of data will have $$z$$-scores between -2 and 2 and approximately 99.7% of data will have $$z$$-scores between -3 and 3 (known as the empirical rule)
• apply the empirical rule to a variety of problems
• indicate by shading where results sit within the normal distribution, eg where the top 10% of data lies
• use $$z$$-scores to identify probabilities of events less or more extreme than a given event
• use $$z$$-scores to make judgements related to outcomes of a given event or sets of data