What happens when the measurement scale changes?
You will be able to: Predict how a linear transformation changes center and spread.
What happens when the measurement scale changes?
A timer reports 2,4,6 minutes. Converting to seconds multiplies every value by 60; adding a 3-minute setup time instead shifts every value by 3.
A useful starting point: How far are values from their mean? →
Words and symbols before equations
- Transformation
- A rule applied to every data value.
- Shift a
- A constant added to all observations.
- Scale b
- A constant multiplying all observations.
- Absolute value abs(b)
- The nonnegative size of a multiplier.
What this picture assumes
Base scores 2,4,6; new score = shift + multiplier × old score. A fixed common axis −25 to 35 shows movement. Negative multipliers reverse order.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- n=3; mean 4, median 4, sample s 2, IQR 4, range 4 score units.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For y=a+bx, the mean transforms to a+b x̄. Adding a moves all observations equally, so it does not change distances between them.
Multiplication stretches distances by abs(b). Thus standard deviation, range and IQR scale by abs(b), while variance scales by b². For b>0, all percentiles transform in the same order.
If b<0, the order reverses: a right tail becomes a left tail, and lower and upper quartiles exchange roles. Spread still cannot be negative. If b=0, every transformed value equals a and spread is zero.
A worked example, step by step
Times have mean 4 minutes and sample s=2 minutes. Convert to seconds and add 30 seconds to every time.
- Define y=30+60x, with y in seconds.
- Mean becomes 30+60×4=270 seconds.
- Standard deviation becomes 60×2=120 seconds.
- The 30-second addition shifts the center but does not increase spread.
Adding a constant does not add it to standard deviation. Multiplying variance uses the square of the scale factor.
If all scores gain 5 points, what happens to s?
Compare with an explanation
It stays unchanged because each value’s distance from the mean is unchanged.
Predict. Change one thing. Explain.
Change the shift and multiplier separately. Watch the dot positions and summaries. Predict the result when the multiplier becomes negative or zero.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
n=3; mean 4, median 4, sample s 2, IQR 4, range 4 score units.
| Summary | Value |
|---|---|
| Data | 2, 4, 6 |
| Mean | 4 |
| Median | 4 |
| Sample variance | 4 squared units |
| Sample s | 2 |
| Q₁ / Q₃ | 2 / 6 |
Base scores 2,4,6; new score = shift + multiplier × old score. A fixed common axis −25 to 35 shows movement. Negative multipliers reverse order.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the data values, graph scales, summary statistics or study-design conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA dataset has mean 10 and s=3. For y=7−2x, find the new mean, standard deviation and variance; explain the negative multiplier.
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Compare with the answer and four-point rubric
- 1 point: Mean is 7−2×10=−13.
- 1 point: s is 2×3=6.
- 1 point: Variance is 36.
- 1 point: Negative scaling reverses order; it does not produce negative variability.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Does a shift affect IQR?
No.
RECALL 2How does a multiplier affect s?
Multiply s by the multiplier’s absolute value.
RECALL 3What happens at multiplier zero?
All values become the shift constant, with zero spread.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What happens when the measurement scale changes?
- mean(y)=a+b mean(x).
- s(y)=abs(b)s(x).
- variance(y)=b²variance(x).
Remember: Adding a constant does not add it to standard deviation. Multiplying variance uses the square of the scale factor.
Conditions: Base scores 2,4,6; new score = shift + multiplier × old score. A fixed common axis −25 to 35 shows movement. Negative multipliers reverse order.
Refresh Kid · AP Statistics Unit 1 · Objectives 1.7.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 1.7, objectives 1.7.C. Framework effective Fall 2026, checked September 17, 2026. Unit 1 includes one-variable data and data collection; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Quartile calculations state a median-of-halves convention. Outlier screens identify values to investigate, not data to discard. Random selection and random assignment have different inferential roles. These lessons introduce design and descriptive reasoning; formal inference comes in later units.
The Organic Chemistry Tutor companion title and destination were checked; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original sampling model uses self-hosted Three.js with its MIT license. It shows labeled units in four groups; camera rotation does not change the sampling procedure. Quantitative graphs remain 2D to avoid perspective distortion. Complete labeled diagrams, selected IDs and explanations remain available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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