Density Curve vs Histogram

A density curve is a curve $f(x)$ with exactly two requirements. It never drops below the horizontal axis, so $f(x) \ge 0$ everywhere, and the total area between the curve and the axis is exactly 1. Area is the only quantity you read off it: the area above an interval from $a$ to $b$ is the proportion of the distribution lying between them, and for a value drawn at random it is $P(a < X < b)$.

Spread a distribution uniformly over 0 to 0.4. The curve is a flat rectangle whose height is forced by the total-area rule: $1 / 0.4 = 2.5$, since $0.4 \times 2.5 = 1$. The area from 0.1 to 0.3 is then $0.2 \times 2.5 = 0.50$, so exactly half the distribution sits in that stretch.

That height is where the usual mistake lives. "The curve is 2.5 at $x = 0.2$, so a value there has probability 2.5" is wrong twice over. A density value is not a probability. It is a rate, probability per unit along the horizontal axis, with no ceiling: squeeze the same total area of 1 into a narrower interval and the height climbs as far as you like. Only a height multiplied by a width is a probability. The same arithmetic settles the second half. A single point has no width, so it encloses no area, so $P(X = c) = 0$ for any one value $c$, which is why $P(X < c)$ and $P(X \le c)$ agree on a density curve and do not agree for counts.

Heights are still worth comparing to each other. A curve twice as tall at one place as at another says values are packed twice as densely there. What a height never is, on its own, is an answer to a question about chance.

A density curve is a model laid over data, not the data. In the Fall 2026 course the idea is anchored in topic 2.11, The Normal Distribution, where the probability of an interval is the area under the curve over it.

Reading a proportion as an area is not an analogy for calculus, it is calculus: the area under $f$ between $a$ and $b$ is the definite integral of the density, and the total-area-equals-1 rule is the statement that the density integrates to 1 over its whole domain.

A histogram cuts the number line into bins of equal width and draws a bar over each bin whose height is the count of values inside it, or the relative frequency. The bars touch because the axis is a continuous scale rather than a list of labels. Every histogram needs a boundary rule, and the usual one puts a value in the bin whose lower bound it meets or exceeds and whose upper bound it falls below.

Thirty exam scores binned by tens give 9 scores in 70 to 80 and 12 in 80 to 90, and three of those students scored exactly 80. Switch to the other convention, where a value belongs to the bin whose upper bound it equals, and those three move down: the counts become 12 and 9, the tallest bar changes, and so does the shape you would report. Nothing in the drawn picture tells you which rule was used.

The misreading is "twelve students scored 85." The bar over 80 to 90 says twelve students scored somewhere in that interval and nothing more. The individual values have been thrown away, which is also why you cannot read a median off a histogram, only the bin the median lands in.

Equal width is doing real work in the definition. When bins differ in width, height stops encoding frequency fairly: a bin of width 20 holding 12 values and a bin of width 10 holding 9 draw bars of height 12 and 9, so the wide one looks bigger, while the densities are $12 / 20 = 0.6$ and $9 / 10 = 0.9$ values per unit and the narrow bin is the more crowded one. With unequal bins the vertical axis has to carry density and you read area, not height.

A histogram also destroys the order the values arrived in, which is what a time plot keeps, and its shape depends on the bin width you choose. Histograms are topic 1.5.