Z-Scores

Overview

A z-score is a statistical measure that describes how many standard deviations a data point is from the mean of a data set. It is useful for comparing individual data points within different distributions and identifying outliers.

The formula for calculating the $z$-score is:

$$z = \frac{x - \mu}{\sigma}$$

Where:

• $x$: The data point.
• $\mu$: The mean of the data set.
• $\sigma$: The standard deviation of the data set.

Key Points About Z-Scores

Interpretation:

• A positive $z$-score indicates the data point is above the mean.
• A negative $z$-score indicates the data point is below the mean.
• A z-score of $0$ indicates the data point is exactly at the mean.

Applications:

• Comparisons: Compare scores from different data sets or distributions.
• Outlier Detection: Data points with z-scores beyond $\pm 3$ are often considered outliers.

Standardisation:

• Z-scores transform raw data into a standard normal distribution with a mean of $0$ and a standard deviation of $1$.

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Worked Examples

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Summary

• Z-scores indicate how far a data point is from the mean in terms of standard deviations.
• Formula:

$$z = \frac{x - \mu}{\sigma}$$

• Interpretation:
  • Positive $z$-scores: Above the mean.
  • Negative $z$-scores: Below the mean.
  • Z-scores near 0: Close to the mean.
• Applications:
  • Compare scores across different distributions.
  • Identify outliers ($|z| > 3$).
• $Z$-scores standardise data, enabling analysis on a common scale.