Standard Statistical analysis – Random variables

This lesson comprises one (1) master class focusing on:

• Normally distributed random variables
• z-scores
• Empirical rule
• Using z-scores to compare data sets
• Using z-scores to calculate probabilities

Content:

MS-S5

• 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 \( z=\frac{x− \bar{\mu}}{\sigma} \), where \( \bar{\mu} \) is the mean and \( \sigma \) 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 statistical tables to determine probabilities
  • use technology to determine probabilities
• Use z-scores to make judgements related to outcomes of a given event or sets of data