The normal distribution table — the z-table — is just a long grid of decimals, and the first time you meet it on an exam it is easy to freeze. Reading a normal distribution table is a mechanical skill: convert your value to a z-score, find it in the grid, and interpret the area. This walkthrough takes you through each step with real numbers and the three problem types you will actually be asked.
What the Table Actually Gives You
A standard normal table lists cumulative areas under the curve. For a given z-score, the table entry is the proportion of the distribution that lies to the left of that z-score — everything below it.
The standard normal distribution has a mean of 0 and a standard deviation of 1. The total area under the curve is 1, representing 100% of the data. So a table entry of 0.8413 means 84.13% of values fall below that z-score, and the remaining 15.87% fall above it.
Two facts make the table easier to use. First, because the curve is symmetric, the area below z = −1 equals the area above z = +1. Second, the area at z = 0 is exactly 0.5000 — half the distribution sits below the mean.
Step 1: Convert a Raw Value to a Z-Score
The table only knows z-scores, so any real-world value must first be standardized. The z-score formula is:
z = (x − μ) ÷ σ
Here x is your value, μ is the population mean, and σ is the standard deviation. A z-score tells you how many standard deviations a value sits from the mean — positive above, negative below.
Example: exam scores are normally distributed with μ = 70 and σ = 8. A student scored 82. The z-score is (82 − 70) ÷ 8 = 12 ÷ 8 = 1.50. The student is 1.5 standard deviations above the mean.
Step 2: Find the Z-Score in the Grid
A z-table splits the z-score across its rows and columns. The row gives the z-score to one decimal place; the column gives the second decimal place.
For z = 1.50: find the row labeled 1.5, then move across to the column headed .00. The cell where they meet is the cumulative area. For z = 1.50 that entry is 0.9332.
That means 93.32% of students scored below 82. If the z-score were 1.53, you would use row 1.5 and column .03 instead.
A common slip: some textbooks print a table giving the area between 0 and z rather than the area to the left. Before you trust an answer, check the small diagram printed above your table — the shaded region tells you which area the entries represent. This walkthrough assumes the standard "area to the left" table, which is the most common.
Step 3: Translate the Question Into an Area
Most exam questions are one of three types. The lookup is always the same; the arithmetic afterward depends on the wording.
"Below" or "less than"
This is the table value directly. P(X < 82) with z = 1.50 is simply 0.9332, or 93.32%.
"Above" or "greater than"
The table gives area to the left, so subtract from 1. P(X > 82) = 1 − 0.9332 = 0.0668, or 6.68%. About 6.7% of students scored above 82.
"Between two values"
Find the cumulative area for each z-score, then subtract the smaller from the larger. Suppose you want the proportion scoring between 70 and 82. The z-score for 70 is (70 − 70) ÷ 8 = 0, with area 0.5000. The z-score for 82 is 1.50, with area 0.9332. The area between them is 0.9332 − 0.5000 = 0.4332, or 43.32%.
Handling Negative Z-Scores
A value below the mean produces a negative z-score. A score of 62 gives z = (62 − 70) ÷ 8 = −1.00. Many tables have a separate page for negative z-scores; the area below z = −1.00 is 0.1587.
If your table only lists positive z-scores, use symmetry: the area below z = −1.00 equals the area above z = +1.00, which is 1 − 0.8413 = 0.1587. Same answer, reached through the symmetry of the curve.
Going Backwards: From Area to Z-Score
Sometimes a question gives the percentage and asks for the cutoff value — "what score puts a student in the top 10%?" Now you read the table in reverse. The top 10% means 90% lies below, so search the body of the table for the entry closest to 0.9000. That value sits at roughly z = 1.28. Convert back to a raw score with x = μ + zσ = 70 + (1.28)(8) = 80.24. A score of about 80 or higher lands in the top 10%.
Getting Help
Z-scores and the normal table are the engine behind many other procedures. They power confidence intervals and the critical values in a hypothesis test, so the lookup skill here pays off across the whole course.
Conclusion
Reading a normal distribution table comes down to three reliable steps: standardize your value with z = (x − μ) ÷ σ, locate the z-score using the row for the first decimal and the column for the second, and then translate the question — below, above, or between — into the right area arithmetic. Check whether your table gives area-to-the-left, use symmetry for negative z-scores, and you can run the table backward to turn a percentage into a cutoff value.
