Test Yourself

A linear transformation might stretch or shrink the scale of distribution but it does not change the shape of a distribution.

A z-transformation measures the distance of each observation from the mean in units of standard deviation.

A z-score can be converted to a percentile rank.

To convert a value to a z-score we subtract the mean and divide by the standard deviation

Z-scores are useful when you wish to compare two variables with different measurement units. For example, the exam score from an English and Maths exam.

A z-transformation standardises values to have a mean of 1 and SD of 1

Z-scores can be related to the standard normal distribution to work out the probability that a randomly selected observation will have a value as extreme as that observed.

Z-scores can be useful to flag outliers, often observations more than 1 SD from the mean are classified as an outlier.

The standard normal distribution (and hence z-scores) are used to estimate 95% reference ranges

The 68-95-99.7 rule states that when a variable is normally distributed, approximately, 68, 95 and 99.7% of observations will lie with 1 2 or 3 standard deviations of the mean.

If I picked an individual at random from this population, there’s a 95% chance they’d have a blood pressure within this range.

Blood pressure within this range is normal.

5% of smokers would be expected to have a systolic blood pressure outside this range.

95% of smokers in  this population would be expected to have a blood pressure within this range.