Frequency Tables

Precise definition

A frequency table lists values or categories with their counts. Relative frequency is $f_i/n$ and percentage is $100f_i/n$. For ordered categories or numerical classes, cumulative frequency adds counts up to and including a boundary.

Notation and mathematical language

For grouped quantitative data, class intervals must be mutually exclusive and collectively cover the intended range. Notation such as $[10,20)$ includes 10 but excludes 20. Class width, boundaries, and treatment of endpoints must be explicit.

Conceptual picture

A table is a lossless summary for categorical counts but grouping numerical data sacrifices exact values within intervals. Cumulative columns reveal thresholds and approximate percentiles, while relative columns allow fair comparison across different sample sizes.

Conditions and key results

Cumulative frequency is meaningful only for ordered categories. Unequal-width classes require density rather than raw frequency heights in a histogram. Missing observations must be included in the denominator or explicitly excluded according to a stated rule.

A reliable strategy

1. Define the unit, variable, valid categories or interval boundaries, and denominator.
2. Tally each observation exactly once and include missing or other categories transparently.
3. Compute relative and, when meaningful, cumulative frequencies.
4. Check totals, endpoint assignments, and whether grouping is fine enough for the intended interpretation.

Fully worked example

Interpretation and application

Frequency tables support survey summaries, quality control, and distribution plots. Group choices can conceal clusters, gaps, or extreme observations, so retain raw data and document bins.

Common mistakes

Verification and reasonableness

• Sum counts to $n$ and proportions to 1, allowing only rounding tolerance.
• Retally several raw observations at boundaries.
• Recover counts from relative frequencies and compare with integer tallies.

Practice

1. Counts 4,6,10 have what total?
2. What is the relative frequency of count 6 out of 20?
3. Can grouped data determine every exact raw value?

Further deduction

When class widths differ, frequency density is $f_i/\text{width}_i$. Histogram area, not height alone, then represents frequency. A table should include widths or boundaries so the correct density can be reconstructed. Using raw count heights with unequal widths visually overweights broad intervals.

A two-way frequency table cross-classifies two categorical variables. Row and column totals give marginal distributions; dividing cells by a row total gives a conditional distribution within that row. Comparing conditional percentages, not raw cells, assesses association. Differences may be descriptive or inferential, but they do not establish causation without an appropriate design and assumptions.

Expected counts for a two-way table under independence are $E_{ij}=(\text{row total}_i)(\text{column total}_j)/n$. Comparing observed and expected counts helps locate association, while a chi-square statistic aggregates squared discrepancies scaled by expectation. The approximation requires adequate expected counts and independent observations; a large cell percentage alone does not certify significance. Report conditional proportions and sample size alongside any test because practical importance is not encoded by the p-value.

Include explicit categories for missing, unknown, or not applicable values when those counts matter; silently dropping them changes denominators. A table title should identify the population and period, and totals should reconcile with the documented number of records after exclusions.

Related topics

• Bar Graphs
• Histograms
• Percentiles