Z-Scores and the Standard Normal Table: How to Calculate and Look Up Probabilities

A z-score tells you how many standard deviations a value is from the mean. That is the entire concept. If a student scored 85 on an exam with a mean of 75 and a standard deviation of 5, their z-score is (85 minus 75) divided by 5 = 2.0. They scored exactly 2 standard deviations above average. A z-score of zero means the value equals the mean. Positive z-scores sit above the mean; negative z-scores sit below. The formula is z = (x minus mu) divided by sigma for populations, or z = (x minus x-bar) divided by s for samples. Z-scores are powerful because they create a common scale. Comparing a score of 85 on one exam to a score of 720 on another is meaningless — different exams have different scales. But if the 85 has a z-score of 2.0 and the 720 has a z-score of 1.5, you immediately know the 85 is a relatively stronger performance. This is exactly how standardized tests like the SAT and GRE report scores — they convert raw performance into a standard scale so that different test forms can be compared directly.

When you calculate a z-score, you are converting a value from any normal distribution (with any mean and any standard deviation) to the standard normal distribution, which has mean = 0 and standard deviation = 1. This is important because there is only one standard normal distribution, and its probabilities have been tabulated in the z-table. Instead of needing a separate probability table for every possible combination of mean and standard deviation, you need just one table — the z-table — because every normal distribution can be converted to the same standard normal distribution through z-scores. The standard normal distribution follows all the properties of normal distributions. It is symmetric and bell-shaped. The empirical rule applies: about 68% of z-scores fall between -1 and 1, about 95% between -2 and 2, and about 99.7% between -3 and 3. These percentages are exact for the standard normal distribution and approximate for other normal distributions. Any z-score beyond 2 in absolute value is unusual (roughly the most extreme 5%). Any z-score beyond 3 is rare (roughly the most extreme 0.3%). This gives you a quick mental framework for evaluating whether a value is typical or extreme without even looking at the table.

The standard normal table gives the cumulative probability to the left of a z-score — that is, P(Z is less than or equal to z). Most tables are organized with the ones and tenths digits of z down the left column and the hundredths digit across the top row. To look up z = 1.37: find the row for 1.3 and the column for 0.07. The intersection gives P(Z is less than or equal to 1.37) = 0.9147, meaning 91.47% of the standard normal distribution falls below z = 1.37. Three common lookup patterns cover almost every problem you will encounter. Left-tail probability (area to the left): look up z directly. P(Z less than 1.37) = 0.9147. Right-tail probability (area to the right): subtract the table value from 1. P(Z greater than 1.37) = 1 minus 0.9147 = 0.0853. Between two z-values: look up both z-scores and subtract. P(negative 0.50 less than Z less than 1.37) = P(Z less than 1.37) minus P(Z less than negative 0.50) = 0.9147 minus 0.3085 = 0.6062. For negative z-scores, the table works the same way. P(Z less than negative 1.37) = 0.0853. Notice this equals the right-tail probability for positive 1.37 — that is the symmetry of the standard normal distribution in action. If your table only covers positive z-values, use the symmetry property: P(Z less than negative z) = P(Z greater than z) = 1 minus P(Z less than z). Snap a photo of a z-table problem and StatsIQ walks through the lookup, showing which row, which column, and which arithmetic operation matches your specific question.

Example 1: Heights of adult women are normally distributed with mean 64 inches and SD 2.8 inches. What proportion of women are taller than 69 inches? Step 1: z = (69 minus 64) divided by 2.8 = 5 divided by 2.8 = 1.79. Step 2: look up z = 1.79 in the table. P(Z less than 1.79) = 0.9633. Step 3: we want the right tail. P(Z greater than 1.79) = 1 minus 0.9633 = 0.0367. About 3.67% of women are taller than 69 inches. Example 2: Exam scores are normally distributed with mean 72 and SD 8. What percentage of students scored between 60 and 80? Step 1: z for 60 = (60 minus 72) divided by 8 = negative 1.50. z for 80 = (80 minus 72) divided by 8 = 1.00. Step 2: P(Z less than 1.00) = 0.8413. P(Z less than negative 1.50) = 0.0668. Step 3: P(negative 1.50 less than Z less than 1.00) = 0.8413 minus 0.0668 = 0.7745. About 77.45% of students scored between 60 and 80. Example 3 (reverse): What score marks the 90th percentile on the same exam? Step 1: find the z-score where P(Z less than z) = 0.90. From the table, z is approximately 1.28. Step 2: convert back to the original scale. x = mean plus z times SD = 72 plus 1.28 times 8 = 72 plus 10.24 = 82.24. A score of about 82 marks the 90th percentile. The reverse formula x = mean plus z times SD is just the z-score formula solved for x — it lets you go from probabilities back to raw values.

Z-scores do not just describe individual data points. In inferential statistics, z-scores describe how far a sample statistic is from the hypothesized population parameter, measured in standard errors. The test statistic for a z-test is z = (x-bar minus mu-zero) divided by (sigma divided by the square root of n), where mu-zero is the hypothesized mean. This z-score tells you how many standard errors your sample mean is from the value claimed by the null hypothesis. The standard error (sigma divided by the square root of n) replaces the standard deviation because you are now dealing with a sampling distribution, not individual observations. Thanks to the Central Limit Theorem, the sampling distribution of x-bar is approximately normal for large samples, so the same z-table you use for individual values also gives p-values for hypothesis tests. A z-score of 2.15 for a test statistic means your sample mean is 2.15 standard errors above the hypothesized value. Looking up the right tail: P(Z greater than 2.15) = 0.0158. For a one-tailed test, the p-value is 0.0158. For a two-tailed test, the p-value is 2 times 0.0158 = 0.0316. The connection between individual z-scores and test-statistic z-scores is the single most important conceptual bridge in introductory statistics. Both ask the same question — how far is this value from what we expected, measured in standard deviations (or standard errors)? — and both use the same table. Once you see this parallel, z-scores stop being an isolated topic and become the thread that connects descriptive statistics to inference.

Mistake 1: Subtracting in the wrong order. The z-score formula is (x minus mean) divided by SD, not (mean minus x). Getting this backward flips the sign, which flips the tail you look up in the table and gives you the complement of the correct answer. Always subtract the mean from the value, not the value from the mean. Mistake 2: Using the z-table for non-normal data. The z-table is the probability table for the standard normal distribution. If your data is not approximately normal, converting to z-scores is still valid (it tells you how many SDs from the mean), but looking up probabilities in the z-table gives wrong answers. The empirical rule and z-table probabilities only apply to normal (or approximately normal) distributions. For skewed data, the percentage between z = -1 and z = 1 could be anything — not necessarily 68%. Mistake 3: Confusing the area to the left with the area to the right. The most common table gives P(Z less than z), which is the left-tail area. If the problem asks for the probability of exceeding a value, you need the right tail: 1 minus the table value. Read the problem carefully before deciding which area you need. Mistake 4: Forgetting to convert back to the original scale. After finding a z-score that corresponds to a desired percentile, many students report the z-score as the answer. The question almost always asks for the value in original units (test score, height, weight). Use x = mean plus z times SD to convert back. Mistake 5: Rounding z-scores too aggressively. Rounding z = 1.645 to z = 1.6 before looking up the table changes the probability from 0.9500 to 0.9452 — a difference that matters for critical values and margin of error calculations. Keep at least two decimal places in your z-score before using the table.