The z-score, also known as the standard score, is a standardized score that measures the distance of a data point from the mean in terms of the number of standard deviations from the mean.

Here is the formula for calculating z-score:

$$z = frac{x – mu}{sigma}$$

where:* z is the z-score* x is the data point* μ is the mean* σ is the standard deviation

Interpretation:

• A z-score of 0 means that the data point is exactly at the mean.
• A z-score of 1 means that the data point is one standard deviation above the mean.
• A z-score of -1 means that the data point is one standard deviation below the mean.
• A z-score of 2 means that the data point is two standard deviations above the mean.
• A z-score of -2 means that the data point is two standard deviations below the mean.

Applications:

• Z-scores are commonly used in statistics to standardize data.
• They are used to compare data points from different distributions with different means and standard deviations.
• Z-scores can be used to identify outliers and data points that are significantly different from the rest of the data.

Examples:

• If a data point has a score of 80, the mean is 70, and the standard deviation is 10, then the z-score is:

$$z = frac{80 – 70}{10} = 1$$

This means that the data point is one standard deviation above the mean.

• If a data point has a score of 50, the mean is 70, and the standard deviation is 10, then the z-score is:

$$z = frac{50 – 70}{10} = -2$$

This means that the data point is two standard deviations below the mean.

Additional Notes:

• Z-scores can be negative.
• The z-score is a dimensionless quantity, meaning that it does not have units.
• The z-score is a standardized measure of deviation from the mean, so it is not affected by the units of the original data.