Z-Score
Number of standard deviations a value is from the mean.
Formula
Variables
Example
A score of 85 with mean 75, σ = 5: z = (85−75)/5 = 2 (two SDs above average).
How it works
A z-score tells you how far a data value sits from the mean after you measure that gap in units of standard deviation. Subtracting the mean gives the raw distance from average, and dividing by the standard deviation rescales that distance so it can be compared across different datasets and units. A z-score of 0 means the value is exactly average, positive values are above the mean, and negative values are below it.
When to use it
Use this when you want to compare a single value to its distribution, especially across different tests, measurements, or populations. It is also useful in hypothesis testing, standardization, and spotting unusually high or low values.
Worked example, step by step
Suppose a class has mean test score μ = 62 and standard deviation σ = 8, and one student scored x = 78. Step 1: Write the formula: z = (x - μ) / σ Step 2: Substitute the values: z = (78 - 62) / 8 Step 3: Calculate the difference: z = 16 / 8 Step 4: Divide: z = 2 Final answer: z = 2, so the score is 2 standard deviations above the mean.
Common mistakes to avoid
- Using the sample standard deviation formula instead of the z-score formula's σ value from the distribution
- Forgetting to subtract the mean before dividing by the standard deviation
- Mixing up the sign and saying a value below the mean has a positive z-score
Did You Know?
A z-score tells you how many standard deviations a value sits from the mean — the language of "grading on a curve".
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