Frequency Table
A frequency table is a chart that lists values or categories alongside how often each one appears in a dataset.
Definition
A frequency table is an organized chart that shows how many times each value or category appears in a set of data, making it easy to see which values are most or least common. It lists each unique value (or class interval for grouped data), its frequency (count), and often its relative frequency (proportion); for continuous data, a grouped frequency table uses intervals called classes. Formally, a frequency table partitions the sample space into $k$ mutually exclusive classes $C_1, \ldots, C_k$ and records counts $n_1, \ldots, n_k$ with sum $n$; it is the input to chi-square goodness-of-fit tests and serves as a discrete approximation to the underlying distribution, though the grouped version loses individual data values, making exact quantile computation impossible.
Example
A survey of $20$ students about their favorite color produces this frequency table: Blue: $8$, Red: $5$, Green: $4$, Yellow: $3$, showing blue is most popular. Test scores grouped into intervals might read: $60$-$69$ (frequency $3$, relative frequency $0.12$), $70$-$79$ (freq $8$, rel freq $0.32$), $80$-$89$ (freq $10$, rel freq $0.40$), $90$-$100$ (freq $4$, rel freq $0.16$), totaling $25$ students. Sturges' rule $k = \lceil 1 + \log_2 n \rceil$ and Scott's rule $h = 3.5 s n^{-1/3}$ (where $h$ is the bin width and $s$ is the sample standard deviation) are principled methods for selecting class intervals that balance resolution and smoothness.
Key Insight
Frequency tables are one of the first steps in understanding data, turning a messy list into a clear, organized summary. Choosing the right number of class intervals matters: too few hides detail, too many creates noise, and a common guideline is $5$-$20$ classes. Chi-square tests compare observed frequency tables to expected ones, with test statistic $\chi^2 = \sum \frac{(O_i-E_i)^2}{E_i}$ following a chi-squared distribution with $k-1-p$ degrees of freedom under the null hypothesis, where $p$ is the number of estimated parameters.