Frequency Table
Statistics & ProbabilityA 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. It makes it easy to see which values are most or least common.
Example
A survey of $20$ students about their favorite color produces this frequency table: Blue: $8$, Red: $5$, Green: $4$, Yellow: $3$. The table shows that blue is most popular.
Key Insight
Frequency tables are one of the first steps in understanding data. They turn a messy list into a clear, organized summary.
Definition
A frequency table organizes data by listing 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.
Example
Test scores grouped into intervals: $60$-$69$ (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 $0.16$). Total: $25$ students.
Key Insight
Choosing the right number of class intervals is important. Too few hides detail; too many creates noise. A common guideline is $5$-$20$ classes, or using Sturges' rule: $k = 1 + \log_2 n$.
Definition
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. The grouped frequency table loses individual data values, making exact quantile computation impossible.
Example
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
Chi-square tests compare observed frequency tables to expected ones. The test statistic $\chi^2 = \sum \frac{(O_i-E_i)^2}{E_i}$ follows a chi-squared distribution with $k-1-p$ degrees of freedom under the null hypothesis (where $p$ is the number of estimated parameters).