Uniform distribution

Uniform means probability is spread evenly, and the word covers two different objects. A discrete uniform distribution has $k$ listable outcomes, each carrying probability $1/k$. A continuous uniform distribution on the interval from $a$ to $b$ has a flat density of height $\frac{1}{b-a}$, so probability is area and every subinterval of the same width carries the same probability. Data can also be called approximately uniform, which is the Fall 2026 course's shape word in topic 1.6 for a graph whose frequencies are all about the same with no prominent peak.

One roll of a fair six-sided die is discrete uniform with $k = 6$, so each face has probability $1/6 \approx 0.167$ and $P(X > 4) = 2/6 \approx 0.333$. A continuous uniform on 0 to 10 has density $1/10 = 0.1$, so $P(3 < X < 7) = 4 \times 0.1 = 0.4$: four units of width out of ten. Both sit centered at their midpoint, 3.5 and 5.

"Every value is equally likely, so $P(X = 3) = 0.1$" is the standard error in the continuous case. The height of the density curve at 3 is 0.1, but the probability of landing exactly on 3 is 0, because a single point has no width and probability here is area. That is why $P(3 < X < 7)$ and $P(3 \le X \le 7)$ are both 0.4 for the continuous uniform, while for the die $P(X > 4) \approx 0.333$ and $P(X \ge 4) = 0.5$ are different numbers.

Flat also does not mean the values sit close together. The continuous uniform on 0 to 10 has standard deviation $10/\sqrt{12} \approx 2.89$, which is real spread. Uniform says every region is equally likely, not that the variable barely varies.

One caution about reading uniformity off a picture. Flatness is a claim about equal-width bins, so unequal bins make an evenly spread variable look bumpy and can hide a genuinely uniform shape. Check the bin widths before you use the word.