You calculate a z-score, often mistakenly referred to as an "AZ score," using a specific formula to standardize a raw data point relative to its population's distribution.
Understanding the Z-Score Formula
The z-score measures how many standard deviations a raw score is away from the population mean. The formula, as noted in the reference, is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the individual raw score
- μ (mu) is the population mean
- σ (sigma) is the population standard deviation
Step-by-Step Calculation
Here's how to calculate a z-score:
- Identify the raw score (x): This is the data point you are analyzing.
- Determine the population mean (μ): This is the average value of the entire population.
- Determine the population standard deviation (σ): This measures how spread out the data is in the population.
- Subtract the population mean from the raw score: (x - μ)
- Divide the result by the population standard deviation: (x - μ) / σ. The resulting number is the z-score.
Example:
Let's say:
- A student's test score (x) is 85.
- The class average (μ) is 70.
- The standard deviation of the test scores (σ) is 10.
Using the formula:
z = (85 - 70) / 10
z = 15 / 10
z = 1.5
This means the student's score is 1.5 standard deviations above the mean.
Practical Insights:
- Positive Z-Score: Indicates a value above the population mean.
- Negative Z-Score: Indicates a value below the population mean.
- Z-Score of 0: Means the value is equal to the population mean.
- Standardizing Data: Z-scores allow for comparisons across different distributions.
| Component | Description |
|---|---|
| z | Z-Score |
| x | Raw Score |
| μ | Population Mean |
| σ | Population Standard Deviation |
Key Takeaway:
The z-score, calculated as (x - μ) / σ, provides a standardized measure of how far a raw score is from the mean of its population.
