Z-Scores

A z-score tells you how many standard deviations a value is from the mean. Think of it as the universal translator for comparing values from different datasets — it strips away the original units and scales, leaving you with a single number that answers: Is this value typical, or is it unusual?

What Is a Z-Score?

The z-score (also called the standard score) converts any raw value into a number that expresses its distance from the mean in terms of standard deviations.

Population formula:

$z = \frac{x - \mu}{\sigma}$

Sample formula:

$z = \frac{x - \bar{x}}{s}$

Where $x$ is the raw value, $\mu$ (or $\bar{x}$) is the mean, and $\sigma$ (or $s$) is the standard deviation.

Interpreting the sign and size:

Calculating Z-Scores

Example 1: Exam Scores

A student scores 82 on an exam where the class mean is $\mu = 75$ and the standard deviation is $\sigma = 5$.

$z = \frac{x - \mu}{\sigma} = \frac{82 - 75}{5} = \frac{7}{5} = 1.4$

Interpretation: The student scored 1.4 standard deviations above the mean. This is a solid performance — above average, but not unusually high.

Example 2: Two Different Exams

This is where z-scores become truly powerful. Suppose a student takes two exams:

Which performance was better relative to classmates?

You cannot compare the raw scores directly because the exams have different means and different amounts of variability. Convert both to z-scores:

Math z-score:

$z_{\text{math}} = \frac{85 - 78}{4} = \frac{7}{4} = 1.75$

English z-score:

$z_{\text{english}} = \frac{90 - 82}{6} = \frac{8}{6} \approx 1.33$

Answer: Even though the English raw score (90) is higher than the Math raw score (85), the Math performance was relatively better. The student was 1.75 standard deviations above the Math mean, compared to only 1.33 standard deviations above the English mean. In other words, the student outperformed a larger proportion of classmates on the Math exam.

Interpreting Z-Scores

The Empirical Rule (68-95-99.7 Rule)

For data that follows a roughly normal (bell-shaped) distribution, z-scores connect directly to the percentage of data in each region:

This means that if a value has a z-score of 2.5, it falls outside the range that contains 95% of the data — it is in the most extreme 5%. A z-score of 3.1 places a value outside 99.7% of the data — it is extremely rare.

Identifying Unusual Values

A practical rule of thumb:

• $|z| \leq 1$: Typical — the value is within the bulk of the data
• $1 < |z| \leq 2$: Somewhat unusual — the value is in the outer portion of the distribution
• $|z| > 2$: Unusual — fewer than about 5% of values are this far from the mean
• $|z| > 3$: Very unusual — fewer than about 0.3% of values are this extreme

In quality control, manufacturing, and healthcare, a z-score beyond $\pm 2$ or $\pm 3$ often triggers a review or investigation.

Converting Between Raw Scores and Z-Scores

Sometimes you know the z-score and need to find the corresponding raw value. Rearranging the z-score formula:

$x = \mu + z \cdot \sigma$

This says: start at the mean, then move $z$ standard deviations away from it.

Example 3: Finding a Raw Score

Adult male heights in a certain population have $\mu = 170$ cm and $\sigma = 8$ cm. What height corresponds to a z-score of $-1.5$?

$x = \mu + z \cdot \sigma = 170 + (-1.5)(8) = 170 - 12 = 158 \text{ cm}$

Interpretation: A height of 158 cm is 1.5 standard deviations below the mean. This person is shorter than average, but not extremely so — the z-score of $-1.5$ falls within the range that contains about 95% of the population.

Example 4: Setting a Threshold

A company’s shipping times have $\mu = 5.2$ days and $\sigma = 0.8$ days. They want to flag any order that takes unusually long — specifically, any order more than 2 standard deviations above the mean. What is the threshold?

$x = \mu + z \cdot \sigma = 5.2 + 2(0.8) = 5.2 + 1.6 = 6.8 \text{ days}$

Any order taking longer than 6.8 days would be flagged for review.

Z-Scores and Percentiles

Z-scores and percentiles are closely related for normal distributions. Each z-score corresponds to a specific percentile:

For example, a student with a z-score of 1.4 on an exam is at approximately the 92nd percentile — they scored higher than about 92% of the class. A z-score of $-1$ puts someone at about the 16th percentile — only 16% of values are lower.

These conversions require a z-table or calculator for exact values, but the landmarks above help you estimate quickly.

Real-World Application: Nursing — Comparing Lab Results Across Tests

A patient has the following lab results:

Both results deviate from the reference mean, but which is more unusual?

Hemoglobin z-score:

$z_{\text{Hgb}} = \frac{11.2 - 14.0}{1.5} = \frac{-2.8}{1.5} \approx -1.87$

WBC z-score:

$z_{\text{WBC}} = \frac{9.8 - 7.5}{2.0} = \frac{2.3}{2.0} = 1.15$

Comparison: The hemoglobin result deviates more from the mean ($|z| = 1.87$) than the WBC result ($|z| = 1.15$). The hemoglobin is nearly 2 standard deviations below the mean, which is approaching the “unusual” threshold. The WBC is elevated but only about 1 standard deviation above the mean — within the range you would see in a healthy population fairly often.

Clinical implication: The z-scores help a nurse prioritize. While both results deserve monitoring, the hemoglobin is the more concerning finding. A z-score of $-1.87$ means that fewer than about 3% of healthy adults have hemoglobin this low, suggesting possible anemia that warrants further investigation.

Key nursing takeaway: Different lab tests use completely different scales and units. A hemoglobin of 11.2 and a WBC of 9.8 cannot be compared as raw numbers. Z-scores put both on the same scale — standard deviations from the reference mean — making it possible to assess relative severity.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A z-score measures how many standard deviations a value is from the mean: $z = \frac{x - \mu}{\sigma}$
• Positive z-scores indicate values above the mean; negative z-scores indicate values below the mean
• Z-scores allow you to compare values from different distributions by putting them on the same scale
• To convert back to a raw score: $x = \mu + z \cdot \sigma$
• Values with $|z| > 2$ are considered unusual; values with $|z| > 3$ are very unusual
• The empirical rule connects z-scores to percentages: 68% within $\pm 1$, 95% within $\pm 2$, 99.7% within $\pm 3$
• In real-world applications like nursing, z-scores help compare results across tests with completely different units and scales

Return to Statistics for more topics in this section.