What is a density curve? Area equals proportion

What a density curve is

A density curve is a smooth curve drawn over a range of values with two properties. It never dips below the horizontal axis, and the total area between the curve and the axis is exactly 1.

That total of 1 is the whole design. It stands for 100% of the values in the distribution, so every piece of that area is a share of the whole. Shade a region, and the number you get back is a proportion.

A density curve is a model of a distribution, not the data itself. Real data has gaps, ties, and rough edges, and the curve smooths past them so you can compute with clean geometry instead of counting dots.

That trade is worth making because it turns "what fraction of values fall between 6 and 9" into an area problem. Once the question is an area, you can answer it with a rectangle, a table, or a calculator.

Area over an interval is the proportion

Pick two values $a$ and $b$, shade the region under the curve between them, and that area is the proportion of the distribution lying between $a$ and $b$. If you draw one value at random from the distribution, the same area is the probability that the value lands in that interval, written $P(a < X < b)$.

Three rules follow directly from the total area being 1.

• Every area is between 0 and 1, and the area over the full range is exactly 1.
• The area to the right of a value is 1 minus the area to its left: $P(X > c) = 1 - P(X < c)$.
• A single value has no width, so it has no area: $P(X = c) = 0$.

That last rule surprises people, and it has a practical payoff. Since the endpoint contributes nothing, $P(X < c)$ and $P(X \leq c)$ give the same answer on a density curve, which is not true for discrete counts like a binomial setting.

The complement rule is the one you will reach for most, since tables give areas to the left.

From a histogram to a density curve

The bridge from data to curve is a histogram with its heights rescaled. Draw the bars so that each height is the relative frequency divided by the bar width, and the total area of the bars becomes exactly 1.

Here is a five-bar example with bars of width 2 and relative frequencies 0.10, 0.20, 0.40, 0.20, and 0.10. The rescaled heights are 0.05, 0.10, 0.20, 0.10, and 0.05, and the total area is $2(0.05 + 0.10 + 0.20 + 0.10 + 0.05) = 2(0.5) = 1$.

Now collect more data and narrow the bars. The jagged top settles toward a smooth shape, and that limiting shape is the density curve. The area interpretation carries across unchanged, because it was already true of the bars.

This is why the shape vocabulary from describing distributions still applies. A right-skewed histogram becomes a right-skewed curve, and a bell-shaped histogram becomes a bell-shaped curve. For the difference between a histogram and a bar graph, see histogram vs bar graph.

The uniform density curve, the simplest case

A uniform density curve is a flat horizontal segment: every value in the range is equally likely, so the curve has the same height everywhere. That makes it a rectangle, and rectangles are where you should first see the area idea work.

Over the interval from $a$ to $b$, the width is $b - a$. The height is forced, because width times height has to equal 1:

Every area you need is then width times height. Suppose your bus arrives at a uniformly random time in the 15 minutes after you reach the stop, so $a = 0$ and $b = 15$ and the height is $1/15 \approx 0.0667$.

• Waiting less than 4 minutes: width 4, so the area is $4 \times \frac{1}{15} = \frac{4}{15} \approx 0.2667$.
• Waiting between 5 and 12 minutes: width 7, so the area is $\frac{7}{15} \approx 0.4667$.
• Waiting more than 10 minutes: width 5, so the area is $\frac{5}{15} \approx 0.3333$.

Check the whole thing: the full rectangle has area $15 \times \frac{1}{15} = 1$, as required. Notice also that the height 0.0667 is not any of those probabilities.

The normal curve, the case the AP course builds on

The normal curve is the density curve you will use most. It is continuous, unimodal, bell-shaped, and symmetric, and it is pinned down by two numbers: the mean $\mu$ (mu) and the standard deviation $\sigma$ (sigma).

The rule from the previous sections still holds, and the Fall 2026 course states it directly: the probability that a value falls in an interval is the area under the normal curve over that interval, and the total area under the curve is 1. What changes is that the region is no longer a rectangle, so you cannot get its area by multiplying.

Three tools give you normal areas.

1. The empirical rule for boundaries at whole standard deviations: about 68% of the area lies within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.
2. A standard normal table, after standardizing the boundary with $z = \frac{x - \mu}{\sigma}$. See how to find a z-score.
3. Technology, such as the normal distribution calculator, for any boundary at all.

Changing $\sigma$ changes the shape without changing the total. A smaller standard deviation makes the curve taller and more concentrated around the mean, a larger one makes it shorter and more spread out, and the area under it stays exactly 1 either way.

The height of the curve is not a probability

The height at a point tells you how densely values are packed near that point. It is a rate, probability per unit along the horizontal axis, and you have to multiply it by a width before it becomes a probability.

The cleanest proof is a narrow uniform curve. Spread a distribution uniformly over the interval from 0 to 0.25 and the height must be $\frac{1}{0.25} = 4$, since $0.25 \times 4 = 1$. A height of 4 cannot be a probability, because probabilities never exceed 1.

Normal curves do the same thing. Shrink $\sigma$ and the peak climbs without any ceiling at 1, while the area underneath stays fixed at 1.

So when a question asks for a chance, look for a shaded region, not a point on the curve. On a sketch, mark the boundary on the horizontal axis and shade to one side of it, then read off the area.

Where the mean and median sit on a density curve

Both measures of center have a geometric reading once the distribution is a curve.

• The median is the value that cuts the area in half. Half the area lies to its left and half to its right, so the area to the left of the median is 0.5.
• The mean is the balance point. If you cut the region out of a sheet of stiff card, it would balance on a knife edge placed at the mean.

For a symmetric curve the two land in the same place, at the center of symmetry. On a normal curve that is $\mu$, which is also the peak. On a uniform curve from 0 to 15, both are $(0 + 15)/2 = 7.5$, and you can verify the median directly: the area to the left of 7.5 is $7.5 \times \frac{1}{15} = 0.5$.

Skew pulls them apart. On a right-skewed curve the long right tail drags the balance point outward, so the mean typically sits to the right of the median; on a left-skewed curve the mean typically sits to the left. The median moves less because the area far out in a tail is thin, which is the same resistance argument that makes the median the safer center for skewed data.

A useful consequence: on a right-skewed curve, the median is not at the peak and the mean is not at the peak either. Only symmetry puts center, median, and peak in one spot.

Mistakes to avoid

• Reading a height as a probability. Multiply by a width, or find an area some other way.
• Forgetting that the total is 1 when a question asks for an "at least" or "more than" area. Subtract the area to the left from 1.
• Applying the empirical rule to a curve that is not normal. On the uniform curve from 0 to 15 the standard deviation is $15/\sqrt{12} \approx 4.3301$, so the interval within 1 standard deviation of the mean runs from 3.1699 to 11.8301. Its area is $8.6603/15 \approx 0.5774$, about 57.7% rather than 68%. That formula for a uniform standard deviation is intro-college content and is not one of the AP formulas, but the point stands without it.
• Assuming every density curve is normal. Curves can be uniform, right-skewed, or bimodal. The chi-square distributions are a family of density curves that take positive values and are skewed right, with the skew easing as the degrees of freedom grow.
• Treating a density curve as the data. The curve is a model, and a poor fit is possible; check the histogram before trusting normal-curve areas.