Z Table (Standard Normal Distribution Table)

How to Use a Z Table

Row 1.2 + column 0.03 → z = 1.23 → P(Z ≤ 1.23) ≈ 0.8907.

A different textbook may print the area between 0 and z instead; for z = 1.23 that entry is approximately 0.3907, and you add 0.5 to obtain the left-tail probability.

1. Write the z-score to two decimal places for a conventional table lookup. For example, use z = 1.23.
2. Find row 1.2 in the left column.
3. Move across to column 0.03.
4. Read the intersection: 0.8907 in a four-decimal left-tail table.
5. Interpret the value: approximately 89.07% of the standard normal distribution lies below z = 1.23.

How to Read a Negative Z Table

For a negative z-score, use the negative table or symmetry. To look up z = −1.25, find row −1.2 and column 0.05; the minus sign applies to the entire magnitude 1.25. The left-tail probability is P(Z ≤ −1.25) ≈ 0.1056. Equivalently, use 1 − P(Z ≤ 1.25).

How to Read a Z Table for Positive Values

For z = 1.96, use row 1.9 and column 0.06. The standard cumulative table gives P(Z ≤ 1.96) ≈ 0.9750. Subtracting from 1 gives the right-tail area, approximately 0.0250.

Understanding Z Table Areas: Left Tail, Right Tail and Two Tails

A standard normal table may report different areas, so always read its heading. A left-tail table gives Φ(z) = P(Z ≤ z). A right-tail probability is 1 − Φ(z). For a positive z, the area from the mean to z is Φ(z) − 0.5; for a negative z it is 0.5 − Φ(z). The two-tailed area beyond equally extreme positive and negative thresholds is 2 × min[Φ(z), 1 − Φ(z)].

Z Score to Percentile Table

A cumulative Z-table value can be read as a percentile for a standard normal model. Multiply the left-tail probability by 100. For example, z = 1.23 corresponds to approximately the 89.07th percentile, and z = −1.25 corresponds to approximately the 10.56th percentile. A percentile is a cumulative position, not the percentage of outcomes above the z-score.

Percentile = 100 × Φ(z)

How to Find a P-Value from a Z Table

First identify the alternative hypothesis. For a right-tailed standard-normal test, use p = 1 − Φ(z). For a left-tailed test, use p = Φ(z). For a two-sided test with symmetric alternatives, use p = 2 × [1 − Φ(|z|)]. For z = 1.96, the right-tail area is about 0.0250 and the two-sided area is about 0.0500. These tail areas become hypothesis-test p-values only under a valid standard-normal reference test.

Using a Z Table for Confidence Intervals

Confidence-interval calculations often use a critical z value such as 1.645 for a central 90% interval, 1.960 for 95%, or 2.576 for 99%. The critical value multiplies an appropriate standard error to obtain a margin of error. Use a z critical value only when the interval's statistical assumptions justify the normal reference distribution.

Z Table for AP Statistics and Classroom Problems

Many statistics courses use a printed standard normal reference sheet, sometimes called Table A. Use the table heading to check whether its cells show cumulative left-tail probability or area from the mean. Then find the row for the ones and tenths of the z-score and the column for its hundredths. You can use the interactive highlight here to practice the same lookup steps without changing the underlying probability.

The label "Table A" is textbook-specific; not every publication assigns the same letter to a Z table.

Z Table vs. T Table: Does a Z Table Have Degrees of Freedom?

A standard normal Z table does not use degrees of freedom. Student's t critical values depend on degrees of freedom, so their reference table has additional rows or parameters. If your problem gives a t statistic and df, use the T Distribution Table rather than looking up the value in a Z table.

For df-based cutoffs, use the T Distribution Table.

Worked Z Table Examples

Row 1.9 and column 0.06 meet at approximately 0.9750. Thus the area below 1.96 is 97.50%, and the area above it is 2.50%.

Use row −1.2 and column 0.05, or use symmetry from positive 1.25. The area to the left is approximately 0.1056 and the area to the right is approximately 0.8944.

For −1 ≤ Z ≤ 1, subtract Φ(−1) from Φ(1) using unrounded values: 0.841344746 − 0.158655254 ≈ 0.682689492, or approximately 68.27%. Individual four-decimal table cells are only rounded references.

If a normally distributed measurement has mean 70, standard deviation 10 and observed value 85, its z-score is (85 − 70)/10 = 1.5. The left-tail Z-table probability is Φ(1.5) ≈ 0.9332.

Printable Z Table for Classroom Reference

For a paper reference, use your browser’s Print command to print the positive or negative standard normal table. The printed heading must still identify the probability convention and preserve legible row and column labels. The online lookup remains available for more precise values.

Formula and Assumptions

For a normal variable with mean μ and standard deviation σ > 0, the standardized value is z = (x − μ)/σ. The Z table describes Z ~ N(0,1). The cumulative probability Φ(z) is the area under the standard normal density to the left of z. The printed grid rounds z to hundredths and often probabilities to four decimals; the underlying mathematical model is continuous.

The table does not establish that a real dataset is normally distributed. Exact-value probabilities for a continuous normal random variable are zero; meaningful probabilities correspond to intervals or tails.