Empirical Rule
This page follows the JoeCMath video The Empirical Rule Finally Makes Sense (68-95-99.7). The video explains why the same three normal-curve percentages appear again when the mean and standard deviation change.
Main Idea
The empirical rule applies to normal distributions. In the video, Joe starts with a bell-shaped normal curve centered at its mean. Because the curve is symmetric, half of the distribution is to the left of the mean and half is to the right.
The rule describes how much area is within one, two, and three standard deviations of the mean:
\[P(\mu-\sigma<X<\mu+\sigma)\approx 0.6827\] \[P(\mu-2\sigma<X<\mu+2\sigma)\approx 0.9545\] \[P(\mu-3\sigma<X<\mu+3\sigma)\approx 0.9973.\]That is why the rule is often rounded to $68\%$, $95\%$, and $99.7\%$.
Watch this section: what the empirical rule tells us at 0:00.
The Normal Curve Formula
The video then connects the rule to the normal curve itself. For a normal distribution, the mean is written as $\mu$ and the standard deviation is written as $\sigma$.
The probability density function has the form
\[f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}.\]The important point in the video is not to memorize the formula first. It is to notice what changes: $\mu$ moves the center of the curve, and $\sigma$ changes how spread out the curve is.
Watch this section: normal distribution probability density function at 1:05.
Why Intervals Matter
Joe pauses to point out that this is a continuous distribution. For a continuous probability density function, the probability at one exact point is not what we measure. Instead, probability is found over an interval.
If $X$ is normally distributed and $f(x)$ is its density function, then
\[P(a<X<b)=\int_a^b f(x)\,dx.\]So the empirical rule is really a shortcut for interval areas. It tells you the area under the curve between $\mu-\sigma$ and $\mu+\sigma$, between $\mu-2\sigma$ and $\mu+2\sigma$, and between $\mu-3\sigma$ and $\mu+3\sigma$.
Watch this section: probability at a point vs probability over an interval at 2:20.
Example 1: $\mu=0$, $\sigma=1$
The first full calculation uses the standard normal distribution:
\[\mu=0,\qquad \sigma=1.\]Within one standard deviation of the mean means going from $0-1$ to $0+1$, so the interval is
\[-1<X<1.\]The video writes that area as an integral and gets
\[\int_{-1}^{1} f(x)\,dx=0.6827.\]Then the same setup is repeated for two and three standard deviations:
\[\int_{-2}^{2} f(x)\,dx=0.9545\]and
\[\int_{-3}^{3} f(x)\,dx=0.9973.\]Those are the precise versions of the familiar $68$-$95$-$99.7$ percentages.
Watch this section: integrals for $\mu=0$ and $\sigma=1$ at 3:11.
Changing The Standard Deviation
Next, the video keeps the mean at $0$ but changes the standard deviation to $\frac{1}{2}$.
When $\sigma$ gets smaller, the normal curve becomes tighter around the mean. The intervals also become narrower:
\[0-\frac{1}{2}<X<0+\frac{1}{2}\]for one standard deviation,
\[0-2\left(\frac{1}{2}\right)<X<0+2\left(\frac{1}{2}\right)\]for two standard deviations, and
\[0-3\left(\frac{1}{2}\right)<X<0+3\left(\frac{1}{2}\right)\]for three standard deviations.
Even though the interval endpoints changed, the areas are still $0.6827$, $0.9545$, and $0.9973$. The rule is tied to standard-deviation distance from the mean, not to one fixed set of $x$-values.
Watch this section: changing the standard deviation at 5:01.
Changing The Mean
The video then changes the mean and standard deviation together:
\[\mu=50,\qquad \sigma=10.\]Now one standard deviation from the mean runs from $50-10$ to $50+10$, so the interval is
\[40<X<60.\]Two standard deviations gives
\[30<X<70,\]and three standard deviations gives
\[20<X<80.\]The center moved and the curve spread out, but the matching areas stayed the same:
\[68.27\%,\qquad 95.45\%,\qquad 99.73\%.\]Watch this section: changing the mean to $50$ and standard deviation to $10$ at 6:35.
What Stays The Same
The final comparison in the video uses another distribution with
\[\mu=1000,\qquad \sigma=25.\]Joe’s point is that you do not need to recalculate the same integrals from scratch every time. As long as the distribution is normal, the area within one, two, and three standard deviations of the mean follows the same pattern.
| Distribution | Within $1\sigma$ | Within $2\sigma$ | Within $3\sigma$ |
|---|---|---|---|
| $\mu=0$, $\sigma=1$ | $-1<X<1$ gives $68.27\%$ | $-2<X<2$ gives $95.45\%$ | $-3<X<3$ gives $99.73\%$ |
| $\mu=0$, $\sigma=\frac{1}{2}$ | $-\frac{1}{2}<X<\frac{1}{2}$ gives $68.27\%$ | $-1<X<1$ gives $95.45\%$ | $-\frac{3}{2}<X<\frac{3}{2}$ gives $99.73\%$ |
| $\mu=50$, $\sigma=10$ | $40<X<60$ gives $68.27\%$ | $30<X<70$ gives $95.45\%$ | $20<X<80$ gives $99.73\%$ |
| $\mu=1000$, $\sigma=25$ | $975<X<1025$ gives $68.27\%$ | $950<X<1050$ gives $95.45\%$ | $925<X<1075$ gives $99.73\%$ |
The empirical rule saves time because it lets you jump straight to these interval percentages once you know the mean and standard deviation.
Watch this section: why the areas stay the same at 7:30.
Timestamp Guide
Related Statistics Work
If you need to turn a raw value into standard-deviation units first, review why we convert to z-scores.
If you want the smaller regions inside the empirical-rule template, continue with the normal curve percentages notes.
If the value is not exactly one, two, or three standard deviations from the mean, continue with the z-table notes for more specific normal-curve areas.
You can also browse the Normal Distribution notes or watch the full JoeCMath normal distribution playlist on YouTube.