Normal Curve Quartiles
This page follows the JoeCMath video Find $Q_1$ and $Q_3$ FAST. The video shows how the quartiles of a normal curve split the distribution into four equal-area pieces.
The Setup
The video begins with a normal curve centered at the mean. The quartile idea is about area, not height. Each quartile contains one fourth of the area under the curve:
\[25\%,\qquad 25\%,\qquad 25\%,\qquad 25\%.\]So the first goal is to locate the cut points that create those four equal regions.
Watch this section: introduction at 0:00.
First Quartile
The first quartile, $Q_1$, is the left-side cut point. It has $25\%$ of the normal curve to its left:
\[P(X<Q_1)=0.25.\]That also means $75\%$ of the curve is to the right of $Q_1$:
\[P(X>Q_1)=0.75.\]On a standard normal curve, the first quartile is a little less than one standard deviation below the mean. A useful approximation is:
\[Q_1\approx \mu-0.67\sigma.\]Watch this section: first quartile at 0:11.
Third Quartile
The third quartile, $Q_3$, is the right-side cut point. It has $75\%$ of the normal curve to its left:
\[P(X<Q_3)=0.75.\]It also has $25\%$ of the curve to its right:
\[P(X>Q_3)=0.25.\]Because a normal curve is symmetric, $Q_3$ sits the same distance above the mean that $Q_1$ sits below the mean:
\[Q_3\approx \mu+0.67\sigma.\]Watch this section: third quartile at 0:18.
Four Equal Regions
The middle of the normal curve is split by the mean. For a normal distribution, the mean and median are at the same center point, so the two middle quartile regions sit on either side of the mean.
The picture in the video is:
\[P(X<Q_1)=0.25,\] \[P(Q_1<X<\mu)=0.25,\] \[P(\mu<X<Q_3)=0.25,\]and
\[P(X>Q_3)=0.25.\]That is why the interval from $Q_1$ to $Q_3$ contains the middle half of the normal distribution:
\[P(Q_1<X<Q_3)=0.50.\]Watch this section: how the quartiles break up the normal curve at 0:24.
Recognizing Quartile Questions
The last part of the video points out the wording that should make you think about quartiles. If a normal-curve problem asks about $25\%$, $50\%$, $75\%$, or directly uses the word quartile, the four-region picture may be the fastest way to start.
| Problem wording | What it points to |
|---|---|
| Bottom $25\%$ | Values below $Q_1$ |
| Top $25\%$ | Values above $Q_3$ |
| Middle $50\%$ | Values between $Q_1$ and $Q_3$ |
| First quartile | The cutoff where $P(X<Q_1)=0.25$ |
| Third quartile | The cutoff where $P(X<Q_3)=0.75$ |
If the problem gives a mean and standard deviation, the quick calculation is:
\[Q_1\approx \mu-0.67\sigma, \qquad Q_3\approx \mu+0.67\sigma.\]Watch this section: types of normal distribution quartile problems at 0:39.
Timestamp Guide
Related Statistics Work
For the broader normal-curve decision process, use the above, below, or between notes.
If the problem gives a raw value, mean, and standard deviation, review why we convert to z-scores.
If the problem needs table values beyond the quartile cutoffs, continue with the z-table notes.
You can also browse the Normal Distribution notes or watch the full JoeCMath normal distribution playlist on YouTube.