How to Read Statistical Tables (Z, T, F, χ²)

Statistical tables look intimidating — grids of tiny numbers with no obvious starting point. This guide walks you through reading each of the four main tables, step by step, with annotated examples.

Why Tables Still Matter

Even with calculators and software, statistical tables are essential:

• Exams — most statistics courses require table lookups
• Quick checks — faster than opening software for a single value
• Building intuition — seeing how critical values change with df and α deepens understanding

All four tables below are available as free interactive tools on this site.

How to Read a Z-Table

The Z-Table gives you the area under the standard normal curve to the left of a given z-score.

Reading a positive z-value

Find P(Z < 1.96):

1. Go down the left column to find 1.9
2. Go across the top row to find .06
3. The intersection gives you: 0.9750

This means 97.5% of the standard normal distribution falls below z = 1.96.

Key patterns to remember

z-score	Area to the left	Common use
1.645	0.9500	90% confidence (one-tailed 5%)
1.960	0.9750	95% confidence (two-tailed 5%)
2.326	0.9900	98% confidence
2.576	0.9950	99% confidence

Negative z-values

For negative z-scores, use symmetry: P(Z < −1.96) = 1 − P(Z < 1.96) = 1 − 0.975 = 0.025.

Our interactive Z-Table handles both positive and negative values — just enter your z-score.

Use it with

→ Z-Score Calculator — convert raw scores to z-scores, then look them up

How to Read a T-Table

The t-Table gives critical t-values based on degrees of freedom (df) and significance level (α).

Finding a critical value

Find the critical t-value for df = 15, α = 0.05, two-tailed:

1. Go down the left column to find df = 15
2. Go across to the column for α = 0.05 (two-tailed) — some tables label this as α/2 = 0.025
3. Read the value: 2.131

This means: If your calculated |t| > 2.131, reject H₀ at the 5% significance level.

One-tailed vs. two-tailed

Tables are labeled differently. Here’s the mapping:

You want	Two-tailed α	One-tailed α	They’re the same column
5% two-tailed	0.05	—	α/2 = 0.025
5% one-tailed	—	0.05	Same as 10% two-tailed

Tip: If your table only shows one-tailed values, double the significance level to get two-tailed (or halve it to go from two-tailed to one-tailed).

What if my df isn’t listed?

If your df falls between listed values (e.g., df = 47 and the table jumps from 40 to 50):

• Conservative approach: Use the smaller df (40) — this gives a larger critical value, harder to reject H₀
• Interpolation: Estimate between the two values
• Best approach: Use the T-Test Calculator for the exact value

Use it with

→ T-Test Calculator — compute t-statistics for one-sample, paired, or independent tests → Degrees of Freedom Reference — calculate df for any test type

How to Read a Chi-Square Table

The Chi-Square Table gives critical χ² values based on degrees of freedom and significance level.

Finding a critical value

Test whether observed frequencies differ from expected, with df = 3 and α = 0.05:

1. Go down the left column to df = 3
2. Go across to the α = 0.05 column
3. Read: 7.815

If your computed χ² > 7.815, reject H₀.

Calculating df for chi-square

Test type	df formula
Goodness of fit	k − 1 (k = number of categories)
Test of independence	(r − 1)(c − 1) (r = rows, c = columns)

Example: A 3×4 contingency table → df = (3−1)(4−1) = 6

Important note: chi-square is always one-tailed

The chi-square distribution is always right-tailed — you only reject if your test statistic is large enough. There’s no “two-tailed” chi-square test.

Use it with

→ Chi-Square Table — interactive lookup with instant interpolation → Bartlett’s Test Calculator — uses chi-square to test equal variances

How to Read an F-Table

The F-Table is the trickiest because it has two degrees of freedom: df₁ (numerator) and df₂ (denominator).

Finding a critical value

ANOVA with 3 groups, 30 total observations, α = 0.05:

• df₁ (between groups) = k − 1 = 3 − 1 = 2
• df₂ (within groups) = N − k = 30 − 3 = 27

1. Find the sub-table for α = 0.05
2. Go across the top to df₁ = 2
3. Go down the left to df₂ = 27
4. Read: 3.35

If your computed F > 3.35, reject H₀ (at least one group mean is significantly different).

Practical tips for F-tables

• df₁ is always the smaller df (numerator) — it goes across the top
• df₂ is always the larger df (denominator) — it goes down the left
• Different significance levels (0.10, 0.05, 0.025, 0.01) are usually in separate tables or sub-tables
• Like chi-square, the F-test is right-tailed only

Use it with

→ F-Table — interactive lookup for any df₁, df₂ combination

Quick Reference: Which Table When?

Your test	Table to use	You need
Z-test	Z-Table	z-score → probability
One-sample, paired, or independent t-test	t-Table	df + α → critical t
Chi-square test	Chi-Square Table	df + α → critical χ²
ANOVA / F-test	F-Table	df₁ + df₂ + α → critical F

Tips for Exam Success

1. Know your table before the exam — practice looking up 5-10 values so the layout is familiar
2. Memorize the z-scores for 90%, 95%, and 99% — they come up constantly (1.645, 1.96, 2.576)
3. Always identify one-tailed vs. two-tailed before looking anything up
4. Circle your df and α on the table to avoid reading the wrong row or column
5. Check your answer makes sense — critical values increase as α decreases and decrease as df increases (for t and χ²)

Related Reading

• When to Use Z-Test vs T-Test vs Chi-Square
• Hypothesis Testing: A Beginner’s Complete Guide
• Degrees of Freedom Reference
• How to Choose the Right Statistical Test