Quick Answer
A standard normal distribution z-table converts a z-score into cumulative probability under the normal curve. First calculate z = (x - μ) / σ, then use the row for the first decimal and the column for the second decimal. A left-tail table gives P(Z <= z).
- Use a z-table only after converting the raw value to a z-score.
- For a left-tail table, z = 1.25 means about 0.8944 of values are at or below that score.
- For a right-tail probability, subtract the left-tail value from 1.
- For a probability between two z-scores, subtract the smaller cumulative probability from the larger one.
Definitions
What a Z-Table Shows
A student or analyst usually reaches for a z-table after asking a concrete question: "If this score is 1.25 standard deviations above average, what percentile is it?" The table answers by mapping standardized distance to normal-curve area.
Most classroom z-tables are left-tail cumulative tables. They report the area from negative infinity up to the z-score. NIST defines the normal model by its mean and standard deviation, and the standard normal version fixes those values at 0 and 1 so one table can serve every normally distributed variable after standardization.
Standard normal cumulative probability
| Question | Operation with a left-tail z-table | Example |
|---|---|---|
| What percent is below z? | Read Φ(z) directly | z = 1.25 -> Φ(z) = 0.8944 |
| What percent is above z? | Compute 1 - Φ(z) | 1 - 0.8944 = 0.1056 |
| What percent is between a and b? | Compute Φ(b) - Φ(a) | Φ(1.25) - Φ(-0.50) |
| What percentile is x? | Convert x to z, then read Φ(z) | x = 86, μ = 76, σ = 8 -> z = 1.25 |
Convert Raw Data to a Z-Score
The table does not accept raw scores, weights, delivery times, or lab readings. It accepts standardized values. Use the z-score calculator when you want the arithmetic checked, or use the formula below when the mean and standard deviation are already known.
Z-score formula
Start with the raw value
Use the right center and spread
Standardize the distance
Read the table
Worked Example: Exam Score Percentile
When we sanity-check z-table explanations for students, we use this concrete class exam dataset: the instructor reports μ = 76 and σ = 8 for a roughly bell-shaped score distribution. One student scored 86 and wants to know whether that is merely above average or high enough to land near the top tenth of the class.
Standardize the score
On a left-tail z-table, find row 1.2 and column 0.05. The table value is 0.8944. That means about 89.44% of scores are at or below 86 under the normal model. The right-tail area is 1 - 0.8944 = 0.1056, so about 10.56% are above 86.
Decision from the numbers
| Item | Value | Interpretation |
|---|---|---|
| Raw score | 86 | The student's observed result |
| Mean | 76 | Class average used as the center |
| Standard deviation | 8 | One typical spread unit in score points |
| Z-score | 1.25 | 1.25 standard deviations above the mean |
| Left-tail probability | 0.8944 | About the 89.44th percentile |
| Right-tail probability | 0.1056 | About 10.56% above this score |
Check with a calculator
Left-Tail, Right-Tail, and Between-Two-Values Tables
The weakest part of most z-table explanations is that they say "look up the value" without naming the table type. Replace that vague instruction with this concrete substitution: before reading any number, identify whether your printed table reports left-tail area, right-tail area, or area between zero and z.
| Table type | What the cell reports | How to use z = 1.25 |
|---|---|---|
| Left-tail cumulative | P(Z <= z) | Read 0.8944 directly |
| Right-tail | P(Z >= z) | Read about 0.1056 directly, or compute 1 - 0.8944 |
| Mean-to-z | Area between 0 and z | Read 0.3944, then add 0.5000 for a positive z |
| Two-tail critical value | Cutoff for split tail probability | Use a critical value table or calculator, not a percentile lookup table |
This distinction matters in hypothesis testing. A two-sided 5% z-test uses 1.96 because 2.5% sits in each tail, not because 95% sits below 1.96. For the testing context, review Hypothesis Testing with Standard Deviation and Confidence Intervals.
Decision Checklist
- Distribution shape:Use a z-table when the normal model is reasonable. If the data are strongly skewed or heavy-tailed, check normality before trusting normal tail areas.
- Known parameters:Use z methods when the population standard deviation is known or the normal approximation is justified. If sigma is estimated from a small sample, t methods may be more appropriate.
- Tail direction:Write the probability statement first: P(Z <= z), P(Z >= z), or P(a <= Z <= b). This prevents left-tail and right-tail reversals.
- Rounding:Round z consistently. For z = 1.254, using 1.25 gives 0.8944 while 1.26 gives 0.8962.
- Interpretation:Translate probability into the user's decision: percentile, tail risk, cutoff, or evidence against a null hypothesis.
Common Mistakes
Using raw x in the table
Forgetting negative z-scores
Mixing percentile and tail area
Applying normal rules blindly
When Not to Use a Z-Table
Do not use a z-table just because a dataset has a mean and standard deviation. The normal model must be a defensible approximation for the question. The empirical rule is a fast diagnostic for normal-shaped data, while Standard Deviation and Normal Distribution explains how spread controls the curve.
| Situation | Better choice | Reason |
|---|---|---|
| Small sample, sigma unknown | t distribution | Extra uncertainty from estimating standard deviation |
| Strong skew or outliers | Percentiles, IQR, or robust z-scores | Normal tail areas may understate real tail risk |
| Binary counts or proportions | Binomial or normal approximation for proportions | The raw outcome is not normally distributed |
| Need exact software output | Calculator or statistical software | Tables round z and probabilities |
Further Reading
Sources
References and further authoritative reading used in preparing this article.