Z-Score Table — Standard Normal Distribution

How to read the table

1. Round your z-value to two decimal places. Say z = 1.27.
2. Find the row that matches the first decimal — for 1.27 that is the row labeled 1.2.
3. Find the column that matches the second decimal — for 1.27 that is the column labeled 0.07.
4. The cell is Φ(z), the area under the standard normal to the LEFT of your z. For z = 1.27, Φ(1.27) = 0.8980 — 89.80% of the distribution is below 1.27.
5. For the area to the RIGHT, subtract from 1: 1 − 0.8980 = 0.1020 (10.20%).
6. For a z between two z-values (two-tailed area), subtract: Φ(1.96) − Φ(−1.96) = 0.9750 − 0.0250 = 0.9500 (95%).

Standard normal table — positive z (0.00 to 3.09)

Cell = Φ(z) = area under the standard normal curve to the left of z. Row is the tenths digit; column is the hundredths digit.

Standard normal table — negative z (−0.00 to −3.09)

Same layout, with negative z-values. Because the normal curve is symmetric, Φ(−z) = 1 − Φ(z) — so if you only have the positive table, you can compute any negative value by subtracting the positive from 1.

Critical values everyone memorizes

The two that get quoted most: 1.96 is the "95% confidence interval" number for a z-test (two-tailed), and 1.645 is the one-tailed 95% cutoff. Both are approximate — 1.96 is really 1.959964… — but the four-decimal table above lands them cleanly.

Percentile to z — the common percentiles

The reverse lookup, pre-computed. A percentile is an area to the left, so the 90th percentile is the z with Φ(z) = 0.90. The lower percentiles are the same numbers with a minus sign, because the curve is symmetric.

The 68-95-99.7 rule is this same table read as intervals: 68.27% of values fall within ±1σ of the mean (Φ(1) − Φ(−1)), 95.45% within ±2σ and 99.73% within ±3σ. It is the quick sanity check for any z you compute — a value beyond ±3 turns up only about 3 times in 1,000.

Worked examples — z to percentile, between two values, and back

All four use the same setup: exam scores are normally distributed with mean μ = 70 and standard deviation σ = 10.

1. Percentile of a score (z → area). A score of 85 gives z = (85 − 70) / 10 = 1.50. Row 1.5, column .00 → 0.9332: the score beats 93.32% of test-takers, roughly the 93rd percentile.
2. Share above a score (right tail). For 90, z = (90 − 70) / 10 = 2.00 → 0.9772. The table gives the area below, so the share above is 1 − 0.9772 = 0.0228: 2.28% score above 90.
3. Share between two scores. For 60 to 85: z = −1.00 → 0.1587 and z = 1.50 → 0.9332. Subtract the smaller from the larger: 0.9332 − 0.1587 = 0.7745, so 77.45% score between 60 and 85.
4. Score for a percentile (area → z, the reverse lookup). The cutoff for the top 20% has 80% below it, so search the body of the table for 0.8000. The nearest cells are 0.7995 (z = 0.84) and 0.8023 (z = 0.85); 0.84 is closer (the exact value is 0.8416). The cutoff score is x = μ + zσ = 70 + 0.84 × 10 = 78.4.

Beyond z = 3.09 — the extreme tails

The four-decimal table flattens out near 0.9990 because the area left over is tiny — but it is not zero, and in quality control, fraud and anomaly detection the tail is the whole point. Past 3.09, read the right-tail column: it is the chance of a value at least that many standard deviations above the mean.

The 4.5 row is where Six Sigma’s famous “3.4 defects per million” comes from: the method assumes a process drifts 1.5σ over the long run, six sigma minus that drift leaves 4.5σ, and the one-sided tail beyond 4.5 is 3.4 in a million. Double the right-tail value for a two-sided question (“more than 3σ from the mean in either direction” is 2 × 0.00135 = 0.0027).

Common mistakes when reading a z-table

Compute a z-score without the table

The z-score itself:
z = (x − μ) / σ

x = the raw value, μ = mean, σ = standard deviation.

Excel and Google Sheets — Φ(z), the area to the left:
=NORM.S.DIST(z, TRUE)   → Φ(z)
=NORM.DIST(x, μ, σ, TRUE) → area to left of x for any normal distribution

Inverse (percentile → z):
=NORM.S.INV(0.95)       → 1.6449 (the one-tailed 95% z-critical)
=NORM.INV(0.95, μ, σ)   → the x-value for any normal distribution

Python (scipy):
from scipy.stats import norm
norm.cdf(1.27)          # 0.8980
norm.ppf(0.95)          # 1.6449

The z-score calculator on this site handles the (x − μ) / σ step for a single value or a whole list, and the descriptive statistics tool computes μ and σ from raw data — both together cover the "I have numbers, tell me the z" pipeline before you even need this table.