A z-score is calculated to determine how many standard deviations a particular data point is from the population mean. Here's a breakdown of how to calculate it:
Understanding the Z-Score Formula
The formula for calculating a z-score is:
z = (x - μ) / σ
Where:
- z is the z-score.
- x is the raw score or the individual data point you are analyzing.
- μ (mu) is the population mean, representing the average of all data points in the population.
- σ (sigma) is the population standard deviation, which measures the dispersion or spread of the data around the mean.
Step-by-Step Calculation
Here's a simplified approach to calculating a z-score:
- Identify the Raw Score (x): Determine the specific data point you want to analyze.
- Find the Population Mean (μ): Obtain the average of the entire dataset (population).
- Determine the Population Standard Deviation (σ): Calculate how much the data points vary from the mean in the entire dataset.
- Subtract the Mean from the Raw Score: Calculate (x - μ). This indicates how much the raw score deviates from the population mean.
- Divide by the Standard Deviation: Divide the result from step 4 by the population standard deviation (σ). This tells you how many standard deviations the raw score is away from the mean.
Example:
Let's say you have a test score (x) of 85. The population mean (μ) for all test scores is 70, and the population standard deviation (σ) is 10.
- x = 85
- μ = 70
- σ = 10
Using the formula, we have:
z = (85 - 70) / 10
z = 15 / 10
z = 1.5
So, a test score of 85 is 1.5 standard deviations above the mean.
Practical Insights
- Positive Z-Score: A positive z-score means the raw score is above the population mean.
- Negative Z-Score: A negative z-score means the raw score is below the population mean.
- Zero Z-Score: A z-score of zero indicates that the raw score is exactly equal to the population mean.
- Usefulness: Z-scores allow you to compare data points from different distributions.
Summary
The z-score is a standardized score that measures the position of an individual data point within a distribution. It is calculated by subtracting the population mean from the raw score and then dividing by the population standard deviation, as shown in the formula: z = (x - μ) / σ.
