Math Tables
Standard Normal Table
The standard normal table answers one question: for a value that is z standard deviations above the mean, what fraction of the distribution lies below it? The answer has no closed form. It is the integral of the bell curve, and the integral cannot be written with elementary functions, so for two centuries it was tabulated rather than computed.
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The table is still worth having because the numbers are used constantly and the intuitions attached to them are widely misunderstood. One standard deviation captures 68.3% of a normal distribution, two captures 95.4%, and three captures 99.7%. Those three numbers are the entire content of the empirical rule, and every one of them comes from this table.
The most quoted entry is 1.96, which is the z value that leaves 2.5% in each tail and therefore produces a 95% confidence interval. It is not 2. Using 2 instead of 1.96 widens the interval by about 2%, which rarely matters in practice but is the kind of thing that gets flagged in a review.
A warning about applying it: the normal distribution is an assumption, not a fact. Heights and measurement errors are close enough. Incomes, house prices and anything driven by multiplicative growth are not, and forcing them into a z score produces percentiles that look authoritative and are wrong. The last table uses examples where the assumption holds.
Cumulative probability for z from 0.1 to 3.5
| z | Below z | Above z | Outside ±z |
|---|---|---|---|
| 0.1 | 0.539828 | 0.460172 | 0.920344 |
| 0.2 | 0.579260 | 0.420740 | 0.841481 |
| 0.3 | 0.617911 | 0.382089 | 0.764177 |
| 0.4 | 0.655422 | 0.344578 | 0.689157 |
| 0.5 | 0.691462 | 0.308538 | 0.617075 |
| 0.6 | 0.725747 | 0.274253 | 0.548506 |
| 0.7 | 0.758036 | 0.241964 | 0.483927 |
| 0.8 | 0.788145 | 0.211855 | 0.423711 |
| 0.9 | 0.815940 | 0.184060 | 0.368120 |
| 1.0 | 0.841345 | 0.158655 | 0.317311 |
| 1.1 | 0.864334 | 0.135666 | 0.271332 |
| 1.2 | 0.884930 | 0.115070 | 0.230139 |
| 1.3 | 0.903199 | 0.096801 | 0.193601 |
| 1.4 | 0.919243 | 0.080757 | 0.161513 |
| 1.5 | 0.933193 | 0.066807 | 0.133614 |
| 1.6 | 0.945201 | 0.054799 | 0.109599 |
| 1.7 | 0.955435 | 0.044565 | 0.089131 |
| 1.8 | 0.964070 | 0.035930 | 0.071861 |
| 1.9 | 0.971284 | 0.028716 | 0.057433 |
| 2.0 | 0.977250 | 0.022750 | 0.045500 |
| 2.1 | 0.982136 | 0.017864 | 0.035729 |
| 2.2 | 0.986097 | 0.013903 | 0.027807 |
| 2.3 | 0.989276 | 0.010724 | 0.021448 |
| 2.4 | 0.991802 | 0.008198 | 0.016395 |
| 2.5 | 0.993790 | 0.006210 | 0.012419 |
| 2.6 | 0.995339 | 0.004661 | 0.009322 |
| 2.7 | 0.996533 | 0.003467 | 0.006934 |
| 2.8 | 0.997445 | 0.002555 | 0.005110 |
| 2.9 | 0.998134 | 0.001866 | 0.003732 |
| 3.0 | 0.998650 | 0.001350 | 0.002700 |
| 3.1 | 0.999032 | 0.000968 | 0.001935 |
| 3.2 | 0.999313 | 0.000687 | 0.001374 |
| 3.3 | 0.999517 | 0.000483 | 0.000967 |
| 3.4 | 0.999663 | 0.000337 | 0.000674 |
| 3.5 | 0.999767 | 0.000233 | 0.000465 |
| 3.6 | 0.999841 | 0.000159 | 0.000318 |
The distribution is symmetric, so the probability below a negative z equals the probability above the matching positive z. Φ(0) is exactly 0.5 by definition.
z values for the usual confidence levels
| Confidence | z (two-tailed) | Cumulative Φ(z) | Outside the interval |
|---|---|---|---|
| 80% | 1.281552 | 0.900000 | 0.100000 |
| 90% | 1.644854 | 0.950000 | 0.050000 |
| 95% | 1.959964 | 0.975000 | 0.025000 |
| 98% | 2.326348 | 0.990000 | 0.010000 |
| 99% | 2.575829 | 0.995000 | 0.005000 |
| 99.9% | 3.290527 | 0.999500 | 0.000500 |
The 95% figure is 1.96, which is where the familiar rule of thumb comes from. The exact value is 1.959964, and using 2 instead widens the interval by about 2%.
Turning real measurements into z scores
| Measurement | Value | Mean | Standard deviation | z | Percentile |
|---|---|---|---|---|---|
| Adult height, 170 cm | 170 | 175 | 7 | -0.714 | 23.8 |
| Adult height, 185 cm | 185 | 175 | 7 | 1.429 | 92.3 |
| Exam score, 85 | 85 | 75 | 10 | 1.000 | 84.1 |
| Exam score, 62 | 62 | 75 | 10 | -1.300 | 9.7 |
| IQ score, 130 | 130 | 100 | 15 | 2.000 | 97.7 |
| IQ score, 85 | 85 | 100 | 15 | -1.000 | 15.9 |
| Package weight, 510 g | 510 | 500 | 20 | 0.500 | 69.1 |
| Systolic BP, 145 | 145 | 120 | 12 | 2.083 | 98.1 |
A z score is the distance from the mean measured in standard deviations. An IQ of 130 is two standard deviations above the mean, which puts it at the 97.7th percentile.
How to Use These Tables
- Reading the table: for z = 1.5 the cumulative probability is 0.933193, so a value 1.5 standard deviations above the mean is at the 93.3rd percentile.
- Negative z: the table is symmetric, so the probability below z = −1.5 is 1 − 0.933193 = 0.066807, the 6.7th percentile.
- Confidence intervals: multiply the z value by the standard error. A mean of 50 with a standard error of 2 gives a 95% interval of 50 ± 1.96 × 2, or 46.1 to 53.9.
- Comparing across scales: convert both measurements to z scores first. A 130 IQ and a 700 GMAT are both roughly two standard deviations above their means.
- Checking a claim: if a result is more than three standard deviations from the expected value, the probability of that under the model is below 0.3%, which usually means the model is wrong rather than the result being remarkable.
Common Questions
Why is the 95% confidence interval 1.96 and not 2?
Because 1.96 leaves exactly 2.5% in each tail. Using 2 leaves 2.28% in each tail, which makes the interval about 2% wider. The difference is small but it is the correct value.
What does a z score of 0 mean?
That the value equals the mean. The cumulative probability is exactly 0.5, so the value sits at the 50th percentile.
How do I read a negative z score?
Use the symmetry of the distribution. The probability below −z equals one minus the probability below +z. For z = −2 the answer is 1 − 0.977250 = 0.022750.
Can I use this table for any data?
Only if the data is roughly normal. Heights, measurement errors and test scores usually are. Incomes, prices and counts are skewed, and a z score computed on them will give a percentile that is systematically wrong.
What is the difference between a z score and a t score?
A t score is the same idea but accounts for estimating the standard deviation from a small sample. As the sample grows the t distribution converges to the normal, so above about 30 observations the tables agree to two decimals.
How much of the distribution is within one standard deviation?
68.27%. Within two it is 95.45%, and within three it is 99.73%. These are the three numbers behind the empirical rule.
More Math Tables
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods — normal distribution — National Institute of Standards and Technology
- Abramowitz and Stegun, Handbook of Mathematical Functions, equation 7.1.26 — National Institute of Standards and Technology
- ISO 3534-1 — statistics, vocabulary and symbols, probability — International Organization for Standardization
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