Mode
Statistics & ProbabilityThe mode is the value that appears most frequently in a dataset.
Definition
The mode is the value that appears the most in a set of data. A dataset can have one mode, two modes (bimodal), or no mode if all values appear equally.
Example
Shoe sizes in a class: $7$, $8$, $8$, $9$, $9$, $9$, $10$, $10$. The mode is $9$ because it appears $3$ times, more than any other size.
Key Insight
The mode is the only measure of center that works for categories. You can find the most popular flavor, color, or answer, but you cannot average them.
Definition
The mode is the most frequently occurring value in a dataset. For qualitative data, it is the category with the highest frequency. A dataset may be unimodal (one mode), bimodal (two modes), or multimodal. If all values have equal frequency, there is no mode.
Example
Categorical data: survey responses on favorite genre: Action ($45$), Comedy ($30$), Drama ($45$), Horror ($20$). The dataset is bimodal: Action and Drama both appear $45$ times.
Key Insight
The mode corresponds to the peak(s) of the frequency distribution. In continuous data, the mode is the value where the probability density function reaches its maximum.
Definition
The mode of a continuous distribution is the value $x^*$ that maximizes the probability density function: $f'(x^*) = 0$ and $f''(x^*) < 0$. For a sample, the sample mode is not a well-defined statistic for continuous data. Mode estimation for continuous distributions uses kernel density estimation, with the mode identified as the peak of the KDE curve.
Example
A normal distribution $N(\mu, \sigma^2)$ is unimodal with mode $= \mu = \text{mean} = \text{median}$, a special property of symmetric distributions. A beta distribution $\text{Beta}(\alpha, \beta)$ with both $\alpha > 1$ and $\beta > 1$ has mode $= (\alpha-1)/(\alpha+\beta-2)$.
Key Insight
In Bayesian statistics, the mode of the posterior distribution is called the MAP (maximum a posteriori) estimate. It corresponds to penalized MLE and is a common point estimate when the posterior is intractable.