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

Read the heading before the number

A statistical table converts a statistic or probability into a reference value. The same distribution can be printed using cumulative probability, upper-tail probability, or central area. A correct number from the wrong column answers the wrong question.

Before looking up anything, write down: distribution, degrees of freedom, requested region, and tail convention. Then identify whether you want a probability, a percentile, or a critical value.

Z-table: cumulative area to the left

The site’s z-table reports Φ(z) = P(Z ≤ z). For z = 1.23, read row 1.2 and column 0.03 to get 0.8907.

• Above 1.23: 1 − 0.8907 = 0.1093.
• Below −1.23: by symmetry, 0.1093.
• Between −1.23 and 1.23: 0.8907 − 0.1093 = 0.7814.

If your printed table reports the area from zero to z, its entry for 1.23 is 0.3907 instead. Neither is wrong; the headings specify different areas.

A raw value must be standardized with the appropriate mean and SD. A sample mean uses its standard error instead of an individual-outcome SD. Standardization alone does not justify a normal probability model.

T-table: split total α for a two-sided test

The printed t-table uses upper-tail probabilities. For a two-sided 5% test with df = 15, use the 0.025 upper-tail column and find 2.1314. Reject for t below −2.1314 or above +2.1314.

For a one-sided upper-tail 5% test, use the 0.05 column and find 1.7531. Decide the alternative before observing the result; do not choose the easier cutoff afterward.

Degrees of freedom come from the design: n − 1 for one-sample inference, pairs − 1 for paired inference, and a different formula for Welch’s independent-group test.

Chi-square: positive discrepancies, upper tail

For categorical tests, χ² sums squared observed-versus-expected discrepancies. The chi-square table gives an upper-tail cutoff. A 2×3 independence table has df = (2 − 1)(3 − 1) = 2; the 5% cutoff is 5.9915.

Do not halve α or double the upper-tail p-value for this categorical test. The statistic already incorporates departures in both directions through squaring. Normal-population variance intervals are a different use of the distribution and can require both tails.

F-table: keep numerator and denominator df in order

For classical one-way ANOVA with three groups and nine total observations, numerator df = 2 and denominator df = 6. The F-table gives an upper-tail 5% cutoff of 5.1433.

Swapping the df changes the distribution. ANOVA rejects for large F; there is no ± cutoff. A two-sided variance-ratio test uses different tail handling from the ANOVA mean test.

A quick decision checklist

If you need…	Check first
A normal probability	Left, right, between, or outside area?
A t cutoff	Which design supplies df, and how many tails?
A categorical χ² cutoff	Which df formula and adequate expected counts?
An ANOVA F cutoff	Numerator k − 1, denominator N − k?

A table may only bracket a p-value. Do not report a critical value as if it were a probability. Retain precision in intermediate calculations and disclose when a printed row is substituted for an unavailable df.

Practice

You need a central 95% normal interval. Is its positive cutoff 1.6449 or 1.9600?

Show the explanation

Use 1.9600: the central region leaves 2.5% in each tail. The value 1.6449 is the 95th percentile, leaving 5% in only the upper tail.

Continue learning

The reference pages above include full grids, worked examples, and distribution references. Use the formula sheet for a compact summary or choose a method before looking up a cutoff.