Definition of a Z-score
A Z-score, also known as a standard score, quantifies the number of standard deviations a data point is from the mean of a dataset. It's a fundamental statistical measure used to standardize and compare individual data points from different distributions by converting them to a common scale.
How Z-scores are Calculated
The formula for calculating a Z-score is Z = (X - μ) / σ, where 'X' is the individual data point, 'μ' (mu) represents the population mean, and 'σ' (sigma) denotes the population standard deviation. A positive Z-score indicates the data point lies above the mean, while a negative Z-score signifies it is below the mean.
Practical Example of a Z-score
Consider a class where the average test score (mean) is 75, with a standard deviation of 5. If a student achieves a score of 85, their Z-score would be calculated as (85 - 75) / 5 = 2. This result means the student's score is 2 standard deviations above the class average.
Importance and Applications
Z-scores are invaluable in statistics for several reasons: they help identify outliers, enable meaningful comparisons of results from different tests or datasets, and allow for the determination of the probability of a score occurring within a normal distribution. By standardizing scores, Z-scores provide clarity and facilitate advanced data analysis.