Formula Used:
Z = (X - μ) / σ
Where:
X = Raw Score
μ (mu) = Population Mean
σ (sigma) = Population Standard Deviation
Theory:
A Z-score (also known as a standard score) indicates how many standard deviations a particular data point (raw score) is from the mean of its distribution. It's a way to standardize scores on a common scale, allowing for comparison of values from different distributions.
- Purpose:
- To determine if a score is typical or unusual for a given dataset.
- To compare scores from different tests or measurements that might have different means and standard deviations.
- To calculate probabilities of a score occurring within a normal distribution (by referring to a standard normal Z-table or using statistical software).
- Interpretation:
- Z = 0: The raw score (X) is exactly equal to the population mean (μ).
- Positive Z-score: The raw score (X) is above the population mean (μ). For example, Z = 1 means X is 1 standard deviation above the mean.
- Negative Z-score: The raw score (X) is below the population mean (μ). For example, Z = -2 means X is 2 standard deviations below the mean.
- The magnitude of the Z-score indicates how far the raw score is from the mean in terms of standard deviation units. Larger absolute Z-scores represent more unusual or extreme values. Typically, Z-scores beyond ±2 or ±3 are considered significantly different from the mean.
Step-by-step Calculation:
