A z-score is a measure of how many standard deviations a data point is away from the mean. Generally, a z-score outside of the range of -2 to +2 is considered abnormal or unusual. This means that a data point with a z-score less than -2 or greater than +2 is significantly different from the average.
Here's why this range is considered abnormal:
- Empirical Rule: The empirical rule states that approximately 68% of data points fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations.
- Outliers: Data points with z-scores beyond the -2 to +2 range are considered outliers. These values can significantly influence statistical analyses and may require further investigation.
Examples:
- A student scores 1.5 standard deviations above the average on a standardized test, their z-score is +1.5, which is within the normal range.
- A patient's blood pressure is 2.5 standard deviations above the average, their z-score is +2.5, which is considered abnormal.
Practical Insights:
- Data Analysis: Z-scores help identify unusual data points that might be errors or require further investigation.
- Comparison: Z-scores allow for comparing data points from different datasets with different means and standard deviations.
- Decision Making: Z-scores can be used to make informed decisions based on the probability of a data point occurring.
