A standard normal distribution table (often called a z-table) tells you the probability that a value falls below, above, or between specific points on a bell curve. To use one, you convert your data point into a z-score, split that score into its row and column components, and read the corresponding probability from the table. The whole process takes about 30 seconds once you understand the layout.
What the Table Actually Shows
A standard normal distribution is a bell curve with a mean of 0 and a standard deviation of 1. Every value in the table represents an area under this curve, which corresponds to a probability. Most z-tables show the cumulative area to the left of a given z-score. So if you look up z = 1.50 and find 0.9332, that means 93.32% of all values fall below 1.50 standard deviations above the mean.
The total area under the curve equals 1 (or 100%). This simple fact is the key to solving almost every problem with the table, because any area you can’t read directly can be found by subtracting from 1.
How to Calculate a Z-Score
Before you touch the table, you need a z-score. The formula is straightforward: subtract the mean from your data point, then divide by the standard deviation.
z = (x – mean) / standard deviation
Say 15-year-old boys in the United States have an average height of 67 inches with a standard deviation of 2.5 inches. If you want to know where a boy who is 69.5 inches tall falls on the distribution, you’d calculate: (69.5 – 67) / 2.5 = 1.0. His z-score is 1.0, meaning he is exactly one standard deviation above the mean. You’d then use the table to find the probability associated with z = 1.0.
Another example: in 2009, the average SAT math score was 501 with a standard deviation of 116. A student who scored 600 would have a z-score of (600 – 501) / 116 = 0.85. A student who scored 400 would get (400 – 501) / 116 = -0.87. Negative z-scores simply mean the value is below the mean.
Reading the Rows and Columns
The table is organized as a grid. Rows represent the first decimal place of the z-score, and columns represent the second decimal place (the hundredths digit). To look up a z-score, you split it into two parts.
For example, if your z-score is 2.06:
- Row: Find 2.0 along the left side of the table.
- Column: Find 0.06 along the top of the table.
- Intersection: The cell where row 2.0 and column 0.06 meet is your probability. In this case, you’d find 0.9803, meaning about 98% of values fall below a z-score of 2.06.
If your z-score is 1.00, you’d find row 1.0 and column 0.00. If it’s -0.87, you’d find row -0.8 and column 0.07 on a table that includes negative values.
Finding Areas Above a Z-Score
Most tables give you the area to the left (below) a z-score. To find the area to the right (above), subtract the table value from 1.
Going back to the height example: you want the probability that a randomly selected 15-year-old boy is 69.5 inches or taller, which corresponds to z = 1.0. The table gives you 0.8413 for z = 1.0, meaning 84.13% of boys are shorter than 69.5 inches. So the probability of being 69.5 inches or taller is 1 – 0.8413 = 0.1587, or about 15.87%.
Finding Areas Between Two Z-Scores
To find the probability that a value falls between two points, look up both z-scores and subtract the smaller area from the larger one.
Using the SAT example: what’s the probability of scoring between 400 and 600? You already know the z-scores are -0.87 and 0.85. Look up z = 0.85 in the table and get 0.8023. Look up z = -0.87 and get 0.1922. The area between them is 0.8023 – 0.1922 = 0.6101. About 61% of students scored between 400 and 600.
Working With Negative Z-Scores
Some tables only include positive z-scores. If yours does, you can still find probabilities for negative values by using the bell curve’s symmetry. The curve is a perfect mirror image around zero, so the area below z = -2.5 is identical to the area above z = +2.5.
In practice, this means: look up the positive version of your z-score, find its area, and subtract from 1. If you need the area below z = -1.30, look up z = 1.30 (which gives 0.9032) and calculate 1 – 0.9032 = 0.0968. About 9.68% of values fall below -1.30.
For two-tailed problems where you need the combined area in both tails (common in hypothesis testing), find the area for the negative z-score and double it. Both tails are the same size.
Reverse Lookups: Starting With a Probability
Sometimes you start with a probability and need to find the z-score. This is an inverse lookup, and it works by scanning the body of the table instead of the margins.
Say you want to find the SAT score that marks the top 10% of students. The top 10% means 90% of scores fall below this cutoff, so you’re looking for the z-score where the cumulative area is 0.9000. Scan through the table’s interior values until you find the one closest to 0.9000. You’ll land near z = 1.28. Then convert back to a raw score: 501 + (1.28 × 116) = 649.5. A score above roughly 650 put a student in the top 10%.
When the exact probability doesn’t appear in the table, use the closest value. If you find 0.8997 and 0.9015 flanking your target of 0.9000, either pick the closer one or interpolate between them. For most practical purposes, picking the nearest value works fine.
Z-Scores Worth Memorizing
A few z-scores come up so often in statistics that they’re worth knowing by heart, especially for confidence intervals:
- z = 1.645: corresponds to a 90% confidence level (5% in each tail)
- z = 1.96: corresponds to a 95% confidence level (2.5% in each tail)
- z = 2.575: corresponds to a 99% confidence level (0.5% in each tail)
You’ll also see z = 1.0, 2.0, and 3.0 referenced frequently. About 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. These benchmarks, sometimes called the 68-95-99.7 rule, give you quick estimates without even opening the table.
A Full Worked Example
A contractor builds homes in a market where the average price is $246,300 with a standard deviation of $15,000. He wants to target the middle 98% of the market. What price range should he build for?
The middle 98% leaves 1% in each tail. Do a reverse lookup for 0.0100 (the lower tail) and 0.9900 (the upper tail). The z-scores are approximately -2.33 and +2.33. Now convert back to prices:
Lower bound: $246,300 + (-2.33 × $15,000) = $246,300 – $34,950 = $211,350. Upper bound: $246,300 + (2.33 × $15,000) = $246,300 + $34,950 = $281,250. The contractor should build homes priced between roughly $211,350 and $281,250 to cover the middle 98% of the market.
Every z-table problem follows this same pattern: convert to a z-score, look up (or reverse look up) the probability, and convert back to real units if needed. The table itself is just a reference. The skill is knowing which area you need and how to get it from the values the table provides.