Common mistake 1 with Two-way table

Quoting row percentages as though they were column percentages. This is the single most common mistake made with these tables and it reverses the claim.

Common mistake 2 with Two-way table

Reading association as causation from a two-way table. The table contains no controls at all.

Common mistake 3 with Two-way table

Percentaging on small cells. 100% of two people is two people.

Common mistake 4 with Two-way table

Using percentages on a table of means. A percentage of a mean is not a quantity; the app refuses it rather than computing it.

EconStatsCraft › Two-way table

Two-way table

One variable down the side, another across the top - Stata's tabulate. Counts, or the mean of a third variable in each cell, as raw figures or as row, column or table percentages. Weighted counts are population totals.

What it needs

Rows
Columns
Measurement - Leave empty for counts.

The intuition

A pivot table cross-tabulates: rows by one variable, columns by another, a summary statistic in the cells. It is the fastest way to see whether two categorical variables are related, and its main hazard is the same one that makes it useful - collapsing to two dimensions hides a third, and the third is often the one doing the work.

When to use it

For any question about two categorical variables at once. It is also the fastest way to check a merge or a filter did what you meant - a cell that should be empty and is not tells you immediately.

Where the estimator comes from

A two-way table of counts n_{ij} has row totals n_{i\cdot}, column totals n_{\cdot j} and grand total n. Under independence the expected count is E_{ij}=n_{i\cdot}n_{\cdot j}/n, and the departure from it is measured by \chi^2=\sum_{i,j}\frac{(n_{ij}-E_{ij})^2}{E_{ij}}, with (I-1)(J-1) degrees of freedom. Because \chi^2 grows with n, effect size is reported separately as Cramer's V, \sqrt{\chi^2/(n\min(I-1,J-1))}, which is bounded between 0 and 1 and does not grow with the sample. Row percentages, column percentages and cell percentages answer three different questions from the same table, and stating which is being shown is not a formatting detail.

The model

\text{row }\%_{ij}=100\times\frac{n_{ij}}{\sum_j n_{ij}}\qquad \text{col }\%_{ij}=100\times\frac{n_{ij}}{\sum_i n_{ij}}

What it assumes

Both variables have few enough categories to read. Two continuous variables make a table with one row per observation; band them first.
For weighted counts, the weight is a population count rather than a relative importance - the cell totals are then population totals.
Every cell has enough observations to support the statistic shown. A two-way table spreads the sample thinly, and a mean computed from two rows sits beside one computed from two hundred with no visual difference.
The two dimensions are not proxies for each other. Cross-tabulating two codings of the same thing fills the diagonal and says nothing.

Statistical properties

Row and column percentages answer different questions. "What share of women chose A" is not "what share of A-choosers were women", and confusing them is the most common misreading of a crosstab.
Chi-squared grows with sample size, so with 100,000 rows a trivial association is significant. Always report Cramer's V or another effect size beside it.
Expected counts below 5 make the chi-squared approximation unreliable. Fisher's exact test is the alternative for small tables.
A two-way table can reverse when a third variable is introduced. The pivot shows what it is asked to show, and cannot warn you about the variable you did not put in it.
Averaging a ratio across cells is not the overall ratio. If the cells hold percentages, the grand total must come from summing the underlying numerators and denominators.
Empty cells are not zeros. A cell with no observations means no data, and displaying it as 0 invites a reading that the value was measured and found to be nothing.

How to read the output

Choose the percentage direction to match the question. Row percentages condition on the row variable, so they answer "given that someone is in this region, how likely are they to be…". Column percentages answer the reverse. They are different numbers and only one of them answers what you asked.
The heat map is there to show the pattern the grid of numbers hides. Look for a row or column that is uniformly darker: that is the association.
A cell of zero is information. It means that combination does not occur, which is often a data-collection fact rather than a fact about the world.
Read the margins before the cells. Most apparent interaction in a two-way table is the product of the two marginal distributions, and comparing cells without them invites reading composition as effect.

A worked reading

A table of purchase by region shows 62% conversion in one region against 41% in another, and chi-squared rejects at p < 0.001 on 20,000 rows. Cramer's V is 0.08. Both statements are true and they say opposite-feeling things: the association is real and it is small, and the p-value is small mostly because the sample is large. Add channel as a third dimension and the regional gap narrows sharply - the first region simply sells more through the higher-converting channel - which the two-way table could not have shown and did not claim to.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Percentages not summing to 100	Cell percentages, not row or column	State which is displayed
Chi-squared significant with a tiny effect	Large sample	Report Cramer's V
Expected counts under 5	Chi-squared unreliable	Use Fisher's exact test, or merge categories
A pattern reversing when a third variable is added	Simpson's paradox	The three-way table is the honest one
Totals row not matching the sum of cells	Missing values excluded per cell	Report the missing count
A category with two observations	Percentages of 50% and 100% look decisive	Show counts alongside percentages

Common mistakes

Quoting row percentages as though they were column percentages. This is the single most common mistake made with these tables and it reverses the claim.
Reading association as causation from a two-way table. The table contains no controls at all.
Percentaging on small cells. 100% of two people is two people.
Using percentages on a table of means. A percentage of a mean is not a quantity; the app refuses it rather than computing it.

Where to read this properly

Agresti, Categorical Data Analysis (2013), ch. 2 and 3.
Cramer (1946), Mathematical Methods of Statistics.
Simpson (1951), JRSS-B 13(2), for the reversal.

All analyses

Data & variables

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Keep rows where

Value

and

Fill with

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Also mark which rows were filled

Rules

Base period = 100

Which part

Percent at each tail

Threshold

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Direction

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Two-way table

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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