Discrete Data
Statistics & ProbabilityDiscrete data consists of countable values with no values possible in between, such as the number of students in a class.
Definition
Discrete data is made up of separate, countable values. You can list the possible values and there are gaps between them.
Example
The number of pets a family has ($0$, $1$, $2$, $3, \ldots$) is discrete. You cannot have $1.7$ pets. The number of goals scored in a soccer game is also discrete.
Key Insight
If you can count it and there is no such thing as "in between" values, it is discrete. Counting words in a sentence gives discrete data.
Definition
Discrete data takes on a countable number of distinct values, often integers. There are no intermediate values possible between consecutive data points. Discrete data is visualized with bar graphs or dot plots (not histograms with continuous bars).
Example
The number of customers entering a store each hour ($0$, $1$, $2, \ldots$) is discrete. A customer count of $3.5$ is impossible. Appropriate models include Poisson (for counts) or binomial (for success/failure counts).
Key Insight
Discrete distributions use probability mass functions (PMFs): $P(X=k)$ for each value $k$. The sum of all PMF values equals $1$.
Definition
A discrete random variable $X$ takes values in a countable set $\{x_1, x_2, \ldots\}$ with probability mass function $p(x_i) = P(X=x_i)$, where $\sum p(x_i) = 1$. Key discrete distributions: Binomial $B(n,p)$, Poisson($\lambda$), Geometric($p$), Negative Binomial. Generating functions (PGF, MGF) facilitate computation of moments.
Example
A Poisson($\lambda=3$) model for customer arrivals gives $P(X=k) = e^{-3} \cdot 3^k / k!$. The mean and variance are both $3$, a hallmark of the Poisson distribution.
Key Insight
The Poisson distribution arises as the limit of Binomial($n,p$) when $n$ is large and $p$ is small with $np = \lambda$ fixed, connecting the two most common count models.