Make the Sketch Before the Calculation
In Using the Empirical Rule for Normal Areas, you used a sketch to check which normal-curve bands belonged in an estimated area. Here the sketch comes first: before calculating any probability, label the axis, mark the event boundaries, and shade the region described in the question. This makes the probability statement visible and helps prevent a correct calculation from answering the wrong question.
A normal-curve sketch is not meant to show exact heights or exact areas. It is a planning diagram. Its job is to show the center, the scale, the boundary or boundaries, and which side or interval is included. You can use raw values on the horizontal axis, z-scores, or both. When a question gives raw values, showing both scales is especially useful: the raw-value axis preserves the context, while the z-score axis makes each boundary’s position relative to the mean clear.
For a model \(X\sim N(\mu,\sigma)\), put \(\mu\) at the center of the horizontal axis. Then mark values one, two, and, when useful, three standard deviations from the mean. The marks are \(\mu-\sigma\), \(\mu+\sigma\), \(\mu-2\sigma\), and \(\mu+2\sigma\), continuing in the same way. Since the curve is symmetric, equal distances to the left and right of \(\mu\) have matching positions on the sketch.
The z-score formula from Calculating a z-Score locates a raw value \(x\) in standard-deviation units. A negative z-score belongs left of the mean, a positive z-score belongs right of it, and \(z=0\) is the mean. A z-score’s absolute value tells how many standard deviations the value is from the mean. Use those facts to position boundaries accurately; do not decide where a value goes just by looking at whether the raw number itself is positive or negative.
A useful sketch can show two aligned horizontal scales. The top scale gives raw values in the variable’s units, and the bottom scale gives their z-scores. Both axes should have marks at the same horizontal positions. For example, if \(\mu=40\) and \(\sigma=5\), the raw value 45 lines up with \(z=1\), because it is one standard deviation above the mean. This paired-axis method is a new way to check that the raw-value event and its standardized version describe the same region.
A Reliable Sketching Routine
Read the event wording before drawing. “Less than” describes the area to the left of a cutoff; “greater than” describes the area to its right; “between” describes the region between two cutoffs. Then decide whether the event includes the mean, crosses the mean, or lies entirely on one side. This quick classification predicts the basic shape of the shaded part.
Next draw a bell-shaped curve and label its center with \(\mu\), along with the variable and units. Add standard-deviation marks if they help locate the given values. Put each boundary at its correct position: a boundary below the mean goes left of center, and one above the mean goes right. If the question supplies a raw value, calculate its z-score when needed to determine how far from the mean to place it. Do not shade until the boundaries are in place.
Finally, shade the event exactly as stated and write its probability notation beside the sketch. For example, if the question asks for the proportion of measurements less than \(c\), shade the left side of \(c\) and write \(P(X<c)\). Notation and shading should agree. If they do not, stop and correct the setup before looking for an area.
Identify what \(X\) measures, its units, and the values of \(\mu\) and \(\sigma\).
Place \(\mu\) at the center. Add helpful standard-deviation marks, raw-value labels, or a second z-score scale.
Use the wording to identify the cutoff or cutoffs. If needed, use \(z=(x-\mu)/\sigma\) to locate raw values.
Shade the left tail, right tail, or interval that matches the wording, then record the corresponding probability notation.
Confirm that the shaded part is on the right side of each boundary and includes exactly the requested values before calculating its area.
Worked Example: A Right-Tail Event
Worked Example: A Right-Tail Event
A fictional community greenhouse models the height of a variety of young plants as normal, with mean 72 centimeters and standard deviation 8 centimeters. Set up a sketch for the proportion of plants taller than 84 centimeters.
State. Let \(X\) be the height of a young plant, in centimeters. The model is \(X\sim N(72,8)\), and the event is that a plant is taller than 84 centimeters: \(X>84\).
Plan. Draw the mean at the center, locate 84 relative to the mean, and shade to the right of that cutoff because the event says “taller than.”
Do. Calculate the cutoff’s z-score and identify its position:
Thus 84 centimeters is 1.5 standard deviations above the mean. On a paired axis, label the center \(72\) with \(z=0\), and place \(84\) to its right with \(z=1.5\). For orientation, one standard deviation above the mean is \(72+8=80\), so 84 belongs between 80 and the two-standard-deviation mark \(72+2(8)=88\). Shade only the area to the right of 84.
The sketch represents \(P(X>84)\), equivalently the area to the right of \(z=1.5\). It does not represent the area to the left of 84 or the area beyond 80. The request here is to set up the sketch, not to calculate its area; the shaded part identifies what a later normal-area method would calculate.
Conclude. The desired proportion is represented by the right tail beyond 84 centimeters. The diagram and probability statement agree: both include plant heights greater than the cutoff.
Worked Example: An Interval Centered at the Mean
Worked Example: An Interval Centered at the Mean
A fictional machine fills packets with grain. The packet mass is modeled as normal with mean 60 grams and standard deviation 6 grams. Sketch the event that a packet’s mass is between 54 and 66 grams.
State. Let \(X\) be a packet’s mass in grams, with \(X\sim N(60,6)\). The requested event is \(54<X<66\).
Plan. Find how far each endpoint is from the mean, mark both endpoints, and shade the interval between them. Check whether the interval is centered at the mean.
Do. Convert both boundaries to z-scores:
The raw-value axis has 54 to the left of 60 and 66 the same distance to the right. The aligned z-score labels are \(-1\), \(0\), and \(1\). Shade from 54 to 66, including the center of the curve and excluding both outer tails. In probability notation, the shaded region is \(P(54<X<66)\), or, on the standardized scale, \(P(-1<Z<1)\) for a standard normal variable \(Z\).
The two cutoffs are equally far from the mean, so the sketch should look symmetric around 60. This visual check is helpful: if the drawing places one cutoff farther from the center, one of the z-scores or raw-value positions has been copied incorrectly. A symmetric sketch does not by itself provide an exact area, but it accurately records the event to be calculated.
Conclude. The target is the central interval from 54 to 66 grams, represented by the area between \(z=-1\) and \(z=1\). Neither tail outside the interval belongs to the requested probability.
Worked Example: An Interval That Crosses the Mean Unevenly
Worked Example: An Interval That Crosses the Mean Unevenly
A fictional weather station models afternoon wind speed as normal with mean 50 kilometers per hour and standard deviation 5 kilometers per hour. Sketch the event that wind speed is between 43 and 55 kilometers per hour.
State. Let \(X\) be the afternoon wind speed in kilometers per hour. The model is \(X\sim N(50,5)\), and the event is \(43<X<55\).
Plan. Standardize the lower and upper boundaries separately. Then put them on the appropriate sides of the mean and shade between them, without assuming the interval is centered.
Do. Find each boundary’s z-score:
Place 43 to the left of the mean 50, at \(z=-1.4\). Place 55 to the right, at \(z=1\). Shade the region from 43 through the center to 55. In standardized notation, the same shaded event is \(P(-1.4<Z<1)\). The interval crosses the mean, but it is not symmetric around it: the lower boundary is 1.4 standard deviations below the mean, while the upper one is only 1 standard deviation above.
This example shows why it is important to locate both boundaries rather than shade a memorized “central” region. The sketch includes some area on each side of the mean, but the left portion extends farther from the center. Do not split the area into equal halves merely because the interval crosses the mean; symmetry applies to matching distances on opposite sides, and these distances do not match.
Conclude. The requested probability is the area between \(-1.4\) and \(1\) on the z-score scale. The raw scale identifies the wind-speed interval in context, and the z-score scale confirms the boundaries’ positions.
What the Shading Does—and Does Not—Tell You
For a continuous normal model, the probability of an event is the area under the curve over the values in that event. The total area under the curve is 1, so a shaded part represents a proportion between 0 and 1. The sketch shows the event’s location and shape; it does not supply a precise area by itself. A drawing can make a right tail appear larger or smaller than it really is, so do not estimate exact probabilities by measuring the ink or comparing the picture’s apparent width.
The distinction between width and area matters. A narrow interval near the center can have a different area from an equally wide interval far into a tail because the curve’s height changes across the axis. The sketch remains valuable: it identifies the correct region and can reveal an impossible result, such as a claimed probability for a tail when the diagram shades the center.
For a continuous normal distribution, including or excluding a single endpoint does not change the area. Thus \(P(X<84)\) and \(P(X\leq84)\) are equal for a normal random variable. Still, copy the problem’s wording accurately when writing the event. The endpoint convention matters for interpreting what was asked, even though it does not change the normal probability.
If an area estimate is requested and the boundaries fall at convenient standard-deviation marks, the empirical rule from the previous tutorial can provide an approximation. For more precise areas, a later tutorial will introduce the standard normal table. In either case, the first task remains the same: make sure the shaded region matches the event before calculating.
Common Mistakes and AP Exam Tips
- Shading before locating the cutoff. Draw and label the mean first. Then place each boundary using its value or z-score. This prevents a positive z-score from being mistakenly put on the left.
- Reversing a tail. “Less than” shades to the left; “greater than” shades to the right. Write the event notation beside the sketch to check the direction.
- Shading outside an interval instead of between its bounds. For \(a<X<b\), shade between \(a\) and \(b\), not both outer tails. Compare the interval wording with the diagram before proceeding.
- Assuming an interval is centered. Calculate the two z-scores separately. An interval can cross the mean without extending equally far on both sides.
- Using a raw value as if it were a z-score. Raw values use the variable’s units; z-scores count standard deviations from the mean. Label which scale is shown, or use paired axes.
- Judging probability from apparent width. The shaded area, not just the interval’s width, represents probability. A sketch identifies the region but is not a precision measuring tool.
- Showing a picture without explaining it. A full-credit setup names \(X\) and its model, labels the mean and boundary values, writes the event, and states which region is shaded. If a probability is later calculated, it must refer to that same event and be interpreted in context.
A strong AP response makes the correspondence clear: the words in the question determine the event, the event determines the shaded region, and the shaded region determines the probability to calculate. Even when a calculator or table is used, those setup steps help catch an incorrect tail or interval before it affects the final answer.
Key Takeaway
A normal-curve sketch is a quick way to translate a probability question into a visible region. Put the mean at the center, locate each boundary with raw values or z-scores, and shade only the tail or interval the question describes. Use aligned raw-value and z-score axes when they help, and do not treat the sketch itself as an exact area calculation.
Check Your Understanding
For each situation, identify the event and describe the correct shaded region. Show z-scores when raw-value boundaries need to be located.
- A normal model has mean 35 and standard deviation 4. Sketch the event \(X<31\). Which side of the cutoff is shaded?
- A normal model has mean 120 and standard deviation 10. Sketch the event \(X>135\), and locate 135 on the z-score scale.
- A normal model has mean 18 and standard deviation 3. Sketch the event \(15<X<24\). Does the interval cross the mean, and are its boundaries equally far from it?
- Explain why an interval that crosses the mean is not necessarily symmetric around it.
- For a continuous normal random variable, does \(P(X<50)\) differ from \(P(X\leq50)\)? Explain briefly.