Conditional Relative Frequency Table: How to Build One

A conditional relative frequency table turns the counts in a two-way table into proportions inside one group: each cell is divided by the total of its own row or its own column, not by the grand total. The choice of row or column decides which question the table answers, and it is the step most often done in the wrong direction.

What a conditional relative frequency table shows

A table that summarises two categorical variables is called a contingency table (OpenIntro, Introduction to Modern Statistics, “Exploring categorical data”). Row proportions are the counts divided by their row totals; column proportions are the counts divided by the corresponding column totals. OpenIntro notes that both can be thought of as conditional proportions.

The word “given” marks the condition. OpenStax (Introductory Statistics 2e, section 3.4, Contingency Tables) puts it this way: for P(cell phone user | violation) = 25/70, the sample space is reduced to the drivers who had a violation. A conditional relative frequency is the same idea written as a proportion of the table: restrict to one row or one column, then divide by that group’s total. For how the marginal and joint totals are defined, see how to calculate marginal probability and the joint probability formula.

The formula

For a table with rows i and columns j, let n_ij be the count in a cell, n_i· the row total and n_·j the column total.

• Row conditional relative frequency: n_ij / n_i·. Answers “among members of row i, what share are in column j?”
• Column conditional relative frequency: n_ij / n_·j. Answers “among members of column j, what share are in row i?”

Self-check: in a row conditional table every row sums to 1; in a column conditional table every column sums to 1. After rounding to three decimals (the convention OpenIntro uses in its tables), a total may read 0.999 or 1.001 because of rounding; that is not an error. For the underlying probability version, see how to calculate conditional probability.

Row or column: which table should you build?

OpenIntro advises that it is usually most useful to condition on the explanatory variable. Put the variable you think of as the group or the cause in the rows and build a row conditional table; then each row is directly comparable with the others. Build the column version only when the question is the other way round (“of those who chose R, what share work in the field?”). The two tables use the same counts and give different numbers, so state which one you computed.

Worked example 1: row and column tables from the same counts

The counts below are illustrative, not survey data. They cross a student’s program (rows) with the tool they use most for analysis (columns).

Program	Spreadsheet	R	Python	Row total
Lab-based	36	24	20	80
Field-based	24	36	60	120
Column total	60	60	80	200

Row conditional relative frequencies (each count divided by its row total; rounded to three decimals):

Program	Spreadsheet	R	Python	Row sum
Lab-based (n = 80)	36/80 = 0.450	24/80 = 0.300	20/80 = 0.250	1.000
Field-based (n = 120)	24/120 = 0.200	36/120 = 0.300	60/120 = 0.500	1.000

Reading: 0.250 is the share of lab-based students who use Python most. Column conditional relative frequencies (each count divided by its column total):

Program	Spreadsheet (n = 60)	R (n = 60)	Python (n = 80)
Lab-based	36/60 = 0.600	24/60 = 0.400	20/80 = 0.250
Field-based	24/60 = 0.400	36/60 = 0.600	60/80 = 0.750
Column sum	1.000	1.000	1.000

Reading: 0.250 appears again in the Python column, but it now means that a quarter of Python users are lab-based. Same cell, same count of 20, different question. Mixing up the two readings is the common error.

Worked example 2: OpenIntro loan data

OpenIntro’s Table 4.1 cross-classifies 10,000 loan applications by application type and homeownership (see Sources):

Application type	Rent	Mortgage	Own	Total
Joint	362	950	183	1,495
Individual	3,496	3,839	1,170	8,505
Total	3,858	4,789	1,353	10,000

Row proportions (OpenIntro Table 4.3): for individual applications, 3,496 / 8,505 = 0.411 rent, 3,839 / 8,505 = 0.451 mortgage and 1,170 / 8,505 = 0.138 own. So 0.451 is the share of individual applicants who have a mortgage. For joint applications the row is 0.242, 0.635 and 0.122.

Column proportions (Table 4.4): for mortgage holders, 3,839 / 4,789 = 0.802 applied individually and 950 / 4,789 = 0.198 applied jointly. So 0.802 is the share of mortgage holders who applied individually. The two numbers, 0.451 and 0.802, come from the same cell and answer different questions.

Reading the table: do the groups differ?

Compare the conditional distributions with each other. In example 1 the row distributions differ clearly (Python 0.250 versus 0.500), so program and tool preference look related in these illustrative counts. In OpenIntro’s loan data, the column rates of joint applications differ across rent, mortgage and own (0.094, 0.198 and 0.135), which OpenIntro describes as evidence that application type and homeownership may be associated.

A useful reference point is the overall (marginal) distribution. In example 1, 60 of 200 students (0.300) use R most. Both rows show 0.300 for R, equal to that marginal share, so being lab- or field-based makes no difference to R use here, while the other two columns do differ. This comparison describes the sample; it is not a significance test. For a test, see the chi-square test, and for a strength-of-association measure see Cramér’s V. For a general reading walkthrough, see how to read a contingency table.

How to compute it in Excel, R and Python

Each method is plain division by the right total. Put the counts from example 1 in B2:D3, row totals in E2:E3 and column totals in B4:D4.

Excel

Row conditional:    =B2/$E2     (fill right and down)
Column conditional: =B2/B$4     (fill right and down)
Rounding:           =ROUND(B2/$E2, 3)

If you start from raw records instead of a count table, build the counts first; see crosstabs in Excel.

R

tbl <- matrix(c(36, 24, 24, 36, 20, 60), nrow = 2,
              dimnames = list(Program = c("Lab", "Field"),
                              Tool = c("Spreadsheet", "R", "Python")))
round(prop.table(tbl, 1), 3)   # row conditional: each row sums to 1
round(prop.table(tbl, 2), 3)   # column conditional: each column sums to 1

The second argument of prop.table is the margin: 1 for rows, 2 for columns. See also crosstabs in R.

Python (pandas)

import pandas as pd
counts = pd.DataFrame([[36, 24, 20], [24, 36, 60]],
                      index=["Lab", "Field"],
                      columns=["Spreadsheet", "R", "Python"])
row_cond = counts.div(counts.sum(axis=1), axis=0).round(3)
col_cond = counts.div(counts.sum(axis=0), axis=1).round(3)
# From raw data: pd.crosstab(df["program"], df["tool"], normalize="index")  # or "columns"

Sources and rounding

Example 1 uses illustrative counts created for this page. Example 2 reproduces counts and proportions from OpenIntro, Introduction to Modern Statistics (Tables 4.1, 4.3 and 4.4). All proportions are rounded to three decimals, so rows or columns can sum to 0.999 or 1.001. Sources checked 2026-10-29:

• OpenIntro, Introduction to Modern Statistics, “Exploring categorical data”: https://openintro-ims.netlify.app/explore-categorical
• OpenStax, Introductory Statistics 2e, section 3.4 Contingency Tables: https://openstax.org/books/introductory-statistics-2e/pages/3-4-contingency-tables
• R documentation, proportions and prop.table (base): https://stat.ethz.ch/R-manual/R-devel/library/base/html/proportions.html
• pandas documentation, pandas.crosstab (the normalize argument): https://pandas.pydata.org/docs/reference/api/pandas.crosstab.html

FAQ

What is conditional relative frequency?

It is a cell count divided by the total of its row or its column rather than by the grand total. It tells you the share of one group that falls in a given category, such as the share of individual applicants who have a mortgage (0.451 in OpenIntro’s data). For the single-variable version, see relative frequency and frequency distributions.