1. Definition

  • A z-score (also called standard score) tells you how many standard deviations a data point is from the mean of a distribution.
  • It transforms raw values into a standardized scale, allowing comparisons across different datasets or variables.

2. Formula

For a single observation $x$: $z = \frac{x – μ}{σ}$

Where:

  • $x$ = raw value
  • $μ$ = population mean
  • $σ$ = population standard deviation

If you only have a sample: $z = \frac{x – \bar{x}}{s}$

Where $\bar{x}$ = sample mean, $s$ = sample standard deviation.

3. Interpretation

  • z = 0 → value is exactly at the mean.
  • z = +1 → value is 1 standard deviation above the mean.
  • z = -2 → value is 2 standard deviations below the mean.
  • Extreme z-scores (e.g., |z| > 3) → outliers in many contexts.

4. Example

Example 1 – Exam Score

  • Mean = 70, Standard Deviation = 10
  • Student’s score = 85

$z = \frac{85 – 70}{10} = 1.5$

The student scored 1.5 standard deviations above the mean.

Example 2 – Height

  • Mean height = 170 cm, SD = 8 cm
  • Person = 150 cm

$z = \frac{150 – 170}{8} = -2.5$

This height is 2.5 SD below the mean.

5. Why Z-Scores Are Useful

  • Standardization: Makes values comparable across different scales.
    • Example: Compare math and English scores even if grading scales differ.
  • Probability: In a standard normal distribution (mean 0, SD 1), z-scores correspond to probabilities.
    • e.g., z = 1.96 → 95% of data lies within ±1.96 SD.
  • Hypothesis Testing: Test statistics (z, t, χ²) are based on comparing observed vs expected values.
  • Outlier Detection: Values with |z| > 3 are often considered outliers.

6. Z-Score vs Z-Test

  • Z-score = standardized value of one data point.
  • Z-test = statistical test using z-scores to evaluate hypotheses about means or proportions.

In short:
The z-score tells you how far and in what direction a data point is from the mean, measured in standard deviations. It’s widely used for comparison, probability calculation, and hypothesis testing.