Above Below Or Between
This page follows the JoeCMath video % Above, Below, or Between? (Normal Distribution). The video is a short overview of how to choose a method when a normal curve question asks for an area above a value, below a value, or between two values.
Main Idea
The video starts with the kind of decision students make before doing any arithmetic: what information did the problem give, and what region of the normal curve is being shaded?
The three region types are:
\[\text{below a value},\qquad \text{above a value},\qquad \text{between two values}.\]Once the region is clear, the method depends on the labels in the problem. A z-score problem usually sends you to a z-table. A standard-deviation problem can often use the empirical rule. A quartile problem can use the $25\%$ chunks of the normal curve.
Watch this section: set up to normal distribution problem at 0:00.
Z-Score Approach
The first method in the video is the z-score approach. This is the flexible method for normal curve area questions because it turns the original value into a standard normal value:
\[z=\frac{x-\mu}{\sigma}.\]After that, the z-table gives the left-tail area below the z-score. From there, the region controls the last step:
\[P(X<a)=\text{left-tail area at }a,\] \[P(X>a)=1-P(X<a),\]and
\[P(a<X<b)=P(X<b)-P(X<a).\]That is the heart of the above, below, or between decision. Below uses the table value directly. Above uses the complement. Between subtracts two left-tail areas.
Watch this section: z-score approach to solving problem at 0:15.
Standard-Deviation Template
The second method in the video is the standard-deviation normal distribution template. This is useful when the marked values are exactly one, two, or three standard deviations from the mean.
For a normal distribution, the empirical-rule areas are:
\[P(\mu-\sigma<X<\mu+\sigma)\approx 68\%,\] \[P(\mu-2\sigma<X<\mu+2\sigma)\approx 95\%,\]and
\[P(\mu-3\sigma<X<\mu+3\sigma)\approx 99.7\%.\]The template also gives the smaller slices. For example, the area from the mean to one standard deviation above the mean is about $34\%$, and the area from one to two standard deviations above the mean is about $13.5\%$.
So if a shaded region is built from standard-deviation marks, the video points you toward the template instead of forcing a z-table calculation.
Watch this section: using standard deviation normal distribution template at 0:41.
Quartile Template
The third method in the video is the quartile normal distribution template. Quartiles split the distribution into four equal area pieces:
\[25\%,\qquad 25\%,\qquad 25\%,\qquad 25\%.\]That means
\[P(X<Q_1)=25\%,\] \[P(Q_1<X<Q_3)=50\%,\]and
\[P(X>Q_3)=25\%.\]The video also connects the quartile marks to the standard normal scale. For a normal distribution, the first quartile is close to
\[z=-0.67,\]and the third quartile is close to
\[z=0.67.\]So when the problem names $Q_1$, $Q_3$, or quartiles directly, the area may be available from the quartile structure before opening a z-table.
Watch this section: using quartile normal distribution template at 1:14.
Choosing The Method
The video is really about choosing the shortest honest path. Start by naming the region, then use the information the problem gives.
| Problem clue | Method to try | What to do |
|---|---|---|
| A raw value $x$, mean $\mu$, and standard deviation $\sigma$ | Z-score approach | Convert with $z=\dfrac{x-\mu}{\sigma}$, then use the z-table. |
| A mark like $\mu+\sigma$ or $\mu-2\sigma$ | Standard-deviation template | Use the empirical-rule areas and slices. |
| A mark like $Q_1$, $Q_2$, or $Q_3$ | Quartile template | Use the $25\%$ sections of the distribution. |
| A region above a value | Complement | Find the area below first, then subtract from $1$. |
| A region between two values | Difference of left tails | Find both left-tail areas, then subtract. |
Watch this section: normal curve area overview at 0:00.
Timestamp Guide
Related Statistics Work
If the problem gives raw data values, start with why we convert to z-scores.
If you need table values for above, below, or between regions, use the z-table notes.
If the marks are one, two, or three standard deviations from the mean, review the empirical rule notes.
For the smaller $34\%$, $13.5\%$, $2.35\%$, and $0.15\%$ regions inside the empirical-rule template, use the normal curve percentages notes.
You can also browse the Normal Distribution notes or watch the full JoeCMath normal distribution playlist on YouTube.