A Z-score, also known as a standard score, measures how many standard deviations a data point is from the mean of a distribution. It provides insight into the relative position of a value within a dataset, indicating whether it is above or below the mean and by how much.
Formula:
The Z-score is calculated using the formula:
Z=(X−μ)σZ = \frac{(X – \mu)}{\sigma}Z=σ(X−μ)
Where:
- XXX = Raw score (the data point)
- μ\muμ = Mean of the population
- σ\sigmaσ = Standard deviation of the population
Interpretation:
- A Z-score of 0 indicates that the data point is exactly at the mean.
- A positive Z-score indicates that the data point is above the mean.
- A negative Z-score indicates that the data point is below the mean.
Applications:
- Standardization: Z-scores are used to standardize scores on different scales, allowing for comparison across different datasets.
- Probability and Statistics: In probability theory, Z-scores are used to calculate probabilities and percentiles in a standard normal distribution.
- Outlier Detection: Z-scores help identify outliers in data; typically, data points with Z-scores above 3 or below -3 are considered outliers.