A z score table is a reference grid that converts any standardized value into a cumulative probability under the normal curve. Instead of solving an integral every time you need P(Z ≤ 1.96), you look up the z-score in the table and read off 0.9750 in seconds. This single table of normal distribution values unlocks an enormous range of statistical work: confidence intervals, hypothesis tests, percentile calculations, and quality-control limits all depend on it.
The table works for any normal distribution, not just the standard one, because converting a raw measurement to a z-score absorbs the mean and standard deviation of the original variable. Once you have the z-score, a single lookup in the z-score table gives you the answer.
This guide explains exactly what the z score table contains, walks through the row-and-column lookup step by step, works two complete numeric examples, covers right-tail and two-tail calculations, handles negative z-scores, and lists the mistakes that most beginners make. An interactive calculator at the end lets you verify every lookup instantly.
What Is the Z Score Table?
The z score table — also called the standard normal z-table, the normal distribution table, or the z-score table — is a lookup grid for the standard normal distribution: a normal distribution with mean 0 and standard deviation 1, written N(0, 1).
Every cell in the table holds a cumulative left-tail probability, the probability that a standard normal random variable Z takes a value at or below the row-and-column z-value. Written formally:
Table value = P(Z ≤ z) where Z ~ N(0, 1)
For z = 1.96, the table returns 0.9750, meaning 97.50 % of the standard normal distribution lies at or below z = 1.96. For z = -1.96 the table returns 0.0250, meaning only 2.50 % of the distribution falls below that point.
The z score table was computed long before electronic calculators existed, and it remains a standard reference in statistics courses and exams for good reason: working through the lookup by hand builds intuition about the shape and spread of the normal curve that clicking a button on software cannot replace. Knowing how the normal distribution tables work also lets you cross-check software output and quickly catch errors.
The table of normal distribution values has been the foundation of applied statistics for over a century. According to the NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.1 — Normal Distribution, the standard normal distribution is the reference distribution for a wide class of statistical methods, including z-tests, confidence intervals, normal approximations to the binomial, and process capability analysis.
The Standard Normal Distribution: A Quick Recap
Any normally distributed measurement X with population mean μ and population standard deviation σ can be converted into a z-score:
z = (x − μ) / σ
The z-score tells you how many standard deviations x sits above (positive) or below (negative) the mean. A z-score of +2.0 means the value is two standard deviations above average; a z-score of −1.5 means one and a half standard deviations below average.
Once you have the z-score, the normal dist z table — which always describes N(0, 1) — converts that score into a probability. This is why a single z-score table can handle any normal distribution: the standardization step in the formula above absorbs whatever mean and standard deviation the original variable has.
The standard normal curve is symmetric about 0. Its total area equals 1.00. The tails extend infinitely in both directions, but the area beyond z = ±4 is smaller than 0.01 %, so most normal distribution tables list z-values from −3.49 to +3.49, covering more than 99.9 % of possible outcomes.
How to Read the Z Score Table
Understanding the layout of the table is the first skill. The actual lookup takes only seconds once you know which number goes where.
Table Layout
A standard z table normal reference is organized as follows:
- Rows: The z-score to the tenths place — for example, 1.9, 2.0, −0.5. Each row label is the whole number plus the first decimal digit.
- Columns: The second decimal digit — 0.00, 0.01, 0.02, … 0.09. Each column header adds this hundredths amount to the row’s z-value.
- Cell: The cumulative probability P(Z ≤ z).
Combine the row and column to identify z precisely:
| Desired z | Row | Column |
|---|---|---|
| 1.96 | 1.9 | 0.06 |
| −1.25 | −1.2 | 0.05 |
| 0.83 | 0.8 | 0.03 |
| 2.14 | 2.1 | 0.04 |
Step-by-Step Lookup
To find P(Z ≤ 1.96) using the distribution z table:
- Locate the row labeled 1.9 in the leftmost column.
- Move right along that row to the column labeled 0.06 (the hundredths digit).
- Read the cell value: 0.9750.
This single cell value tells you that 97.50 % of the standard normal distribution falls at or below z = 1.96.
To find P(Z ≤ −0.83) using the table of z-score:
- Locate the row labeled −0.8 in the left column.
- Move right to the column labeled 0.03.
- Read the cell value: 0.2033.
About 20.33 % of the distribution lies below z = −0.83.
Reading the Column Headers Correctly
Each column header in the z normal distribution table represents the hundredths digit — not a step size of 0.01. When you see the column header 0.06, that means the second decimal place of z is 6. So row 1.9 combined with column 0.06 gives z = 1.96, not z = 1.9 + 0.06 = 1.96 (the arithmetic happens to match, but the concept is important: you are assembling a z-value digit by digit, not adding increments).
Some older normal distribution tables label columns as single digits (0 through 9) instead of decimals (0.00 through 0.09). The logic is identical — the column digit is always the hundredths place. Always check the header row of your table before reading values.
Worked Example 1: Left-Tail Probability
Problem: Adult heights in a population are normally distributed with mean μ = 70 inches and standard deviation σ = 3 inches. What fraction of the population is shorter than 75.88 inches?
Step 1 — Convert to a z-score
z = (75.88 − 70) / 3 = 5.88 / 3 = 1.96
Step 2 — Look up z = 1.96 in the normal distribution table
Row 1.9, Column 0.06 → P = 0.9750
Interpretation: 97.50 % of people in this population are shorter than 75.88 inches. Only 2.50 % are taller.
The z-value 1.96 appears in countless textbook problems because it is the critical value for a 95 % confidence interval: exactly 95 % of a standard normal distribution lies between −1.96 and +1.96.
Worked Example 2: Negative Z-Score
Problem: Scores on a standardized exam are normally distributed with mean μ = 72 and standard deviation σ = 8. A student scores 62. What percentile is this?
Step 1 — Convert to a z-score
z = (62 − 72) / 8 = −10 / 8 = −1.25
Step 2 — Look up z = −1.25
Row −1.2, Column 0.05 → P = 0.1056
Interpretation: About 10.56 % of students scored below 62. This student is approximately at the 11th percentile.
Negative z-scores always return a cumulative probability below 0.5000 because they lie below the mean. Positive z-scores return probabilities above 0.5000. At exactly z = 0 the table returns 0.5000, confirming that half the distribution falls below the mean — as expected for a symmetric distribution.
Try the Calculator
Enter any z-score below. The calculator returns the cumulative left-tail probability P(Z ≤ z), the right-tail probability P(Z > z), and the area between 0 and z. The results match the values a complete normal distribution table would give.
For a larger display with full options, open the z-table calculator. To explore other statistics tools, visit the calculators hub.
Right-Tail and Two-Tail Probabilities
The standard normal z-table delivers left-tail probabilities P(Z ≤ z) by default. Many real problems require the right-tail probability or the probability that Z falls within a range. Here is how to convert.
Right-Tail Probability
The area to the right of z is the complement of the left-tail area. Because the total area under the curve is 1:
P(Z > z) = 1 − P(Z ≤ z)
Example: What is the probability of observing a z-score above 1.96?
P(Z > 1.96) = 1 − 0.9750 = 0.0250
This 2.50 % right-tail area is the critical threshold for a one-tailed hypothesis test at the α = 0.05 significance level. Observing z > 1.96 would lead to rejecting the null hypothesis in that test.
Two-Tail Probability
A two-tail probability asks what fraction of the distribution falls between −z and +z. Using the symmetry of the standard normal distribution:
P(−z ≤ Z ≤ z) = P(Z ≤ z) − P(Z ≤ −z)
= 2 × P(Z ≤ z) − 1
Example with z = 1.96:
P(−1.96 ≤ Z ≤ 1.96) = 2 × 0.9750 − 1 = 0.9500
Exactly 95.00 % of the standard normal distribution lies within ±1.96 standard deviations of the mean. This symmetric interval is the backbone of the two-sided 95 % confidence interval.
Probability Between Two Different Z-Scores
When a problem specifies two different z-values z₁ < z₂:
P(z₁ ≤ Z ≤ z₂) = P(Z ≤ z₂) − P(Z ≤ z₁)
Example: P(0.50 ≤ Z ≤ 1.50)
- P(Z ≤ 1.50) = 0.9332
- P(Z ≤ 0.50) = 0.6915
- P(0.50 ≤ Z ≤ 1.50) = 0.9332 − 0.6915 = 0.2417
About 24.17 % of the distribution falls between z = 0.50 and z = 1.50. Subtracting two table lookups is all it takes.
Negative Z-Scores and the Symmetry Property
Many learners feel less confident when looking up negative z-scores in the z normal distribution table. Most modern tables include a full section for negative z-values, and the lookup process is identical — just use the negative row in the left column.
If your table lists only positive z-values, the symmetry of the standard normal curve provides a simple shortcut:
P(Z ≤ −z) = 1 − P(Z ≤ z)
Example: Suppose your table lists P(Z ≤ 1.64) = 0.9495. Then:
P(Z ≤ −1.64) = 1 − 0.9495 = 0.0505
The symmetry property holds because the standard normal curve is perfectly symmetric about zero: any left-tail area below −z equals the right-tail area above +z, which equals 1 minus the left-tail area below +z.
This symmetry also explains an important fact: for a symmetric two-tailed test, you only ever need to look up the positive z critical value, then use symmetry to find the negative counterpart.
Applying the Z Score Table to Any Normal Distribution
The z score table always describes the standard normal distribution N(0, 1), but you can apply it to any normal distribution by standardizing first. The three-step workflow is:
- Identify μ (mean) and σ (standard deviation) for the original variable X.
- Compute
z = (x − μ) / σfor the value of interest. - Look up z in the normal distribution table to obtain P(Z ≤ z), which equals P(X ≤ x) in the original distribution.
This three-step standardization means that a single z-score table covers all normal distributions, no matter their mean or spread. It is why textbooks pair the standardization formula with the z table — together, they solve any normal probability problem.
For further practice with the standardization workflow, the Penn State STAT 414 course, Lesson 3.2 — The Standard Normal Distribution provides additional worked examples and exercises using the same approach.
Common Mistakes When Reading Normal Distribution Tables
Mixing Up Rows and Columns
The single most common error is reading the column as the tenths digit instead of the hundredths digit. The row always carries the tenths place, and the column always carries the hundredths place. Swapping them produces a wrong z-value and therefore a wrong probability. For z = 1.96, row 1.9 and column 0.06 is correct; row 1.6 and column 0.09 gives z = 1.69, a completely different value.
Check yourself: the z-value you are looking up should equal the row label plus the column label. For row 1.9 and column 0.06: 1.9 + 0.06 = 1.96. If that does not equal your target z, you are in the wrong row or column.
Forgetting That the Table Gives Left-Tail Values
All standard normal distribution tables report the cumulative left-tail probability P(Z ≤ z). If a problem asks for the probability above a value (right tail), you must subtract: P(Z > z) = 1 − P(Z ≤ z). Students who skip this step get answers like 0.9750 when the correct answer for the corresponding upper-tail region is 0.0250.
Applying the Table to Non-Normal Variables
The z-score table is valid only when the underlying variable follows a normal distribution (or can be approximated by one). Applying it to a strongly skewed distribution, a discrete count variable, or any other non-normal shape gives incorrect probabilities. Always verify the normality assumption before using any of the normal distribution tables.
Rounding the Z-Score Too Early
If you round z = 1.963 to z = 2.0 before looking it up, you introduce unnecessary error. Most tables support two decimal places, so round to two decimal places (z = 1.96) rather than one (z = 2.0). Rounding to one decimal costs you up to 0.0099 in probability — small but avoidable.
Treating the Table Value as the Probability of a Single Point
In a continuous distribution, the probability that Z equals any exact value is zero. The table gives probabilities over intervals, not at single points. P(Z = 1.96) = 0, while P(Z ≤ 1.96) = 0.9750. When a problem asks for P(Z = 1.96), the correct answer is 0; when it asks for P(Z ≤ 1.96), look it up in the table.
Ignoring the Sign on a Negative Z-Score
A negative z-score must be looked up in the negative section of the table (or converted using the symmetry formula). Looking up z = 1.25 when you have z = −1.25 gives you 0.8944 instead of the correct 0.1056 — a drastic error that changes whether you are in the 89th percentile or the 11th.
Frequently Asked Questions
What is the z-score table used for?
The z score table converts a standardized z-score into a cumulative probability — the fraction of a standard normal distribution that falls at or below that value. It is used to find p-values in z-tests, to locate critical values for confidence intervals, to calculate percentile ranks in a normally distributed population, and to solve normal distribution probability problems without performing numerical integration by hand.
How do I find the probability for a z-score that is not listed in the table?
Most tables list z-values to two decimal places, covering steps of 0.01. For a z-score with three or more decimal places (for example, z = 1.963), round to two decimal places (z = 1.96) and use that table entry. The rounding error is at most 0.0005 in cumulative probability, which is usually acceptable. For higher precision, use the interactive z-table calculator above, which computes the exact cumulative normal probability for any z-score you enter.
What does a z-score table value of 0.9750 mean?
It means that 97.50 % of the standard normal distribution falls at or below that z-value. Equivalently, a random draw from N(0, 1) will be less than or equal to z with probability 0.9750. The remaining 2.50 % lies above z in the right tail.
Why are there two types of z-tables?
Some z-tables show the area from 0 to z (the “central” or “body” table), while others show the full cumulative area from −∞ to z (the “left-tail” table used in this article). The left-tail format gives P(Z ≤ z) directly. The 0-to-z format gives P(0 ≤ Z ≤ z) = P(Z ≤ z) − 0.5000 — subtract 0.5 from a left-tail value to get the 0-to-z equivalent. Always check the heading or footnote on the table you are using to confirm which format it provides before reading off values.
What z-score corresponds to the 95th percentile?
Search the body of the z score table for the value closest to 0.9500. You will find P(Z ≤ 1.64) = 0.9495 and P(Z ≤ 1.65) = 0.9505. The 95th percentile z-score is approximately 1.645 by linear interpolation, commonly rounded to 1.64 or 1.65 in applied work. The two-tailed 95 % critical value is ±1.96 (the middle 95 % lies between −1.96 and +1.96).
How do I use the z-score table for a hypothesis test?
For a one-tailed z-test at significance level α, find the z-score where P(Z ≤ z) = 1 − α. For α = 0.05 (one-tailed), look up 0.9500 in the table → z ≈ 1.645. Reject the null hypothesis if your computed test statistic exceeds this critical value.
For a two-tailed z-test at α = 0.05, find z where P(Z ≤ z) = 0.9750 → z = 1.96. Reject if |z| > 1.96. The two-tail critical value always corresponds to 1 − α/2 in the left-tail table.
Can I use the same table for binomial or t-distribution problems?
Not directly. The z score table describes only the standard normal distribution. For small samples or unknown population variance, use the t-distribution table instead (the t-distribution has heavier tails and depends on degrees of freedom). For binomial problems, a normal approximation using the z table is acceptable when both np ≥ 10 and n(1 − p) ≥ 10 — a guideline discussed in most introductory textbooks.
Summary
The z score table operates on three principles that, once learned, make every normal probability problem straightforward:
- Left-tail by default. Every cell in the standard normal z-table gives P(Z ≤ z). Subtract from 1.00 for a right-tail probability. Use subtraction between two lookups for any range.
- Row for tenths, column for hundredths. Rows carry the first decimal digit of z; columns carry the second. Reading a cell at row 1.9 and column 0.06 gives P(Z ≤ 1.96).
- Standardize first. For any normal variable X ~ N(μ, σ), compute z = (x − μ) / σ before consulting the table. One table covers every normal distribution.
Practice a handful of lookups — try z = 1.00 (P = 0.8413), z = 1.64 (P ≈ 0.9495), z = −2.33 (P ≈ 0.0099) — until the row-and-column navigation feels automatic. Then confirm your answers using the calculator above to build confidence in both methods.
To deepen your understanding of the underlying mathematics, the NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.1 — Normal Distribution covers the properties of the standard normal distribution in full. To see how the normal distribution fits into the broader landscape of statistical distributions, visit the probability distribution guide or browse every available tool on the calculators hub.