Normal Curve Percentages
This page follows the JoeCMath video Process to Find the % in Every Region of the Normal Curve. The video turns the empirical rule into a labeled normal-curve template with the smaller region percentages filled in.
The Starting Template
The video starts with the three normal-curve percentages that come from the empirical rule:
\[P(\mu-\sigma<X<\mu+\sigma)\approx 68\%,\] \[P(\mu-2\sigma<X<\mu+2\sigma)\approx 95\%,\]and
\[P(\mu-3\sigma<X<\mu+3\sigma)\approx 99.7\%.\]Those three facts give the larger bands. The rest of the video is about splitting those bands into the smaller regions that appear between consecutive standard-deviation marks.
Watch this section: introduction at 0:00.
The Center Regions
The first shaded band is the area within one standard deviation of the mean:
\[P(\mu-\sigma<X<\mu+\sigma)\approx 68\%.\]Because the normal curve is symmetric, the mean splits that area into two equal pieces:
\[\frac{68\%}{2}=34\%.\]So the region from $\mu-\sigma$ to $\mu$ is about $34\%$, and the region from $\mu$ to $\mu+\sigma$ is also about $34\%$.
Watch this section: percent between the first standard deviation of the mean at 0:25.
The Next Regions
The next step in the video uses the $95\%$ band. The area within two standard deviations contains the area within one standard deviation plus the two next side regions.
Subtract the known center band:
\[95\%-68\%=27\%.\]That $27\%$ is split evenly between the left and right sides:
\[\frac{27\%}{2}=13.5\%.\]So each region between one and two standard deviations from the mean is about $13.5\%$.
Watch this section: percent between the second standard deviation of the mean at 0:57.
The Outer Regions
The third band uses the $99.7\%$ part of the empirical rule. The area within three standard deviations contains everything inside two standard deviations plus the two outer side regions.
Subtract the $95\%$ band:
\[99.7\%-95\%=4.7\%.\]Then use symmetry again:
\[\frac{4.7\%}{2}=2.35\%.\]So each region between two and three standard deviations from the mean is about $2.35\%$.
Watch this section: percent between the third standard deviation of the mean at 1:35.
The Tail Regions
The final step in the video finds the tiny tails beyond three standard deviations. The full normal curve has area $100\%$, and the empirical rule says about $99.7\%$ is between $\mu-3\sigma$ and $\mu+3\sigma$.
So the two tails together are:
\[100\%-99.7\%=0.3\%.\]By symmetry, the left tail and right tail match:
\[\frac{0.3\%}{2}=0.15\%.\]That gives about $0.15\%$ below $\mu-3\sigma$ and about $0.15\%$ above $\mu+3\sigma$.
Watch this section: percent in tails to infinities at 2:01.
Region Cheat Sheet
The video builds this table one region at a time. Once the pieces are filled in, the normal curve can be read from the middle outward.
| Region | Calculation | Percent |
|---|---|---|
| From $\mu-\sigma$ to $\mu$ | $\dfrac{68\%}{2}$ | $34\%$ |
| From $\mu$ to $\mu+\sigma$ | $\dfrac{68\%}{2}$ | $34\%$ |
| From $\mu-2\sigma$ to $\mu-\sigma$ | $\dfrac{95\%-68\%}{2}$ | $13.5\%$ |
| From $\mu+\sigma$ to $\mu+2\sigma$ | $\dfrac{95\%-68\%}{2}$ | $13.5\%$ |
| From $\mu-3\sigma$ to $\mu-2\sigma$ | $\dfrac{99.7\%-95\%}{2}$ | $2.35\%$ |
| From $\mu+2\sigma$ to $\mu+3\sigma$ | $\dfrac{99.7\%-95\%}{2}$ | $2.35\%$ |
| Below $\mu-3\sigma$ | $\dfrac{100\%-99.7\%}{2}$ | $0.15\%$ |
| Above $\mu+3\sigma$ | $\dfrac{100\%-99.7\%}{2}$ | $0.15\%$ |
If a problem asks for a larger region, add the pieces that cover the shaded part of the curve. For example, the area from $\mu-\sigma$ to $\mu+2\sigma$ is:
\[34\%+34\%+13.5\%=81.5\%.\]Watch this section: normal curve region template at 2:01.
Timestamp Guide
Related Statistics Work
For the broader rule behind these numbers, review the empirical rule notes.
For choosing whether to use z-scores, the empirical-rule template, or quartiles, use the above, below, or between notes.
If a normal-curve question needs a more exact area than the empirical-rule template provides, continue with the z-table notes.
You can also browse the Normal Distribution notes or watch the full JoeCMath normal distribution playlist on YouTube.