A reliable method
Use z = (x − mean)/SD and keep the sign. Under a normal model, approximately 95% of observations fall within two standard deviations of the mean. Specify the model before applying this rule.
The common trap
Adding a constant shifts the center without changing spread. Multiplying all measurements by a constant multiplies the standard deviation by its absolute value.
Make the connection
A curve height is a density, not a probability at an exact point. Probability is represented by area across an interval; the entire area equals one.
Translation and rescaling are different
Start with mean 31 and SD 3. Adding 3 to each observation gives mean 34 and SD 3. Multiplying the original data by 2 gives mean 62 and SD 6. Doing both, y=2x+7, gives mean 69 and SD 6. Keep the location rule separate from the spread rule.
A score and an area answer different questions
With mean 50 and SD 4, a score of 42 has z=−2. That is a standardized location. Under a normal model approximately 95% lies between 42 and 58; approximately 2.5% lies below 42. Reporting −2 as a probability or 95% for the lower tail answers a different question.
Transfer check
If measurements change from grams to kilograms, mean and SD are both divided by 1000. Show algebraically why z is unchanged. Then explain why a symmetric-looking sample of ten observations is still insufficient to prove that its population is normal.
A dataset has mean 31. Each value is increased by 3. What is the new mean?
- Adding a constant increases every observation equally.
- The mean shifts by the same constant.
- New mean = 31 + 3 = 34.
Compare the other answer choices
Report only the amount of change.
Shift the mean in the opposite direction.
Apply the rule for SD under translation to the mean.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Statistics 2026–27 course framework