Z-Scores (Leaving Cert Mathematics): Revision Notes

Z-Scores

What are Z-Scores?

A Z-score is a statistical measure that indicates how many standard deviations a data point is from the mean of a dataset. It standardises data, allowing comparison across different datasets or distributions.

Formula

The formula for calculating a $Z$-score is:

$$Z = \frac{X - \mu}{\sigma}$$

Where:

• $Z$: Z-score.
• $X$: the data value.
• $\mu$: the mean of the dataset.
• $\sigma$: the standard deviation of the dataset.

Key Points about Z-Scores:

• A $Z$-score of 0 means the data point is equal to the mean.
• A positive $Z$-score indicates a value above the mean.
• A negative $Z$-score indicates a value below the mean. $Z$-scores are fundamental in determining probabilities and areas under the normal distribution curve.

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Applications of Z-Scores

1. Comparing Data Points: Standardising values using $Z$-scores allows you to compare data from different distributions.

2. Calculating Probabilities: Z-scores are used with $Z$-tables to find the cumulative probability of a value in a normal distribution.

3. Identifying Outliers: Data points with $Z$-scores less than -3 or greater than 3 are typically considered outliers.

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Worked Examples

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Summary

• $Z$-Score Formula:

$$Z = \frac{X - \mu}{\sigma}$$

• $Z$-scores standardise data and indicate how far a value is from the mean in terms of standard deviations.
• Positive $Z$-scores are above the mean; negative $Z$-scores are below the mean.
• Applications include comparing datasets, calculating probabilities, and identifying outliers.