A z-score tells how far a data value is from the mean, measured in standard deviations. This makes raw values easier to interpret because the number describes position within a distribution, not just size. Z-scores matter when two data sets use different units, scales, or spreads.
They help students compare test scores, measurements, and observations fairly.
Key Facts
- z = (x - μ) / σ for a population value.
- z = (x - x̄) / s for a sample value.
- A positive z-score means the value is above the mean, and a negative z-score means it is below the mean.
- A z-score of 0 means the value is exactly equal to the mean.
- The absolute value |z| tells the distance from the mean in standard deviations.
- For a normal distribution, about 68% of values are within z = -1 to z = 1, about 95% are within z = -2 to z = 2, and about 99.7% are within z = -3 to z = 3.
Vocabulary
- Z-score
- A standardized value that tells how many standard deviations a data point is from the mean.
- Mean
- The average value of a data set, found by adding all values and dividing by the number of values.
- Standard deviation
- A measure of how spread out data values are from the mean.
- Standardization
- The process of converting raw data values to a common scale so they can be compared.
- Normal distribution
- A symmetric bell-shaped distribution where values near the mean are most common.
Common Mistakes to Avoid
- Using the raw score instead of x - μ in the numerator is wrong because a z-score measures distance from the mean, not the original value itself.
- Forgetting the sign of the z-score is wrong because positive and negative values show whether the data point is above or below the mean.
- Comparing raw values from different distributions is wrong when the means or standard deviations differ, because the same raw score can represent different relative positions.
- Using variance instead of standard deviation in the formula is wrong because z-scores are measured in standard deviation units, not squared units.
Practice Questions
- 1 A quiz score is 84 in a class with mean μ = 76 and standard deviation σ = 4. Find the z-score and interpret it.
- 2 Student A scored 72 on a test with mean 60 and standard deviation 8. Student B scored 85 on a test with mean 78 and standard deviation 5. Which student performed better relative to their class?
- 3 Two data values have z-scores of -1.5 and 1.2. Explain which value is farther from its mean and what the signs tell you about their positions.