Math With AmarA C A D E M Y
All math animations

Grade 11 · Rescaling data

Rescaling data: Reflect and translate

Rescaling data: investigate reflect and translate with scale a = -1; shift b = 2.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Rescaling data: Reflect and translate. Current scale: 1. Current mean: 0. Population standard deviation: 1.414. Population variance: 2Transform every observation together-220.535.5transformed valueTeal: transformed population · dashed: original
All five values are equally weighted and population variance divides by five, not four. Playback shows intermediate affine maps; it is not new measured data. When the current scale is zero, every observation coincides and variance is zero.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Current scale
1
Current mean
0
Population standard deviation
1.414
Population variance
2

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

Applying one affine transformation to every observation separates shifts in location from changes in spread. The fixed population −2, −1, 0, 1, 2 has mean zero and variance two. A negative scale reverses order, but standard deviation remains nonnegative because it measures distance. This investigation starts with Scale a = -1; Shift b = 2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

Y=aX+b; mean(Y)=b; SD(Y)=|a|√2

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Start with the five equally weighted observations and their zero mean. The starting case is “Reflect and translate.”

  2. STEP 2

    Follow the changing quantity

    Playback interpolates the scale from one to a and the shift from zero to b. Read the current transformed values rather than treating intermediate frames as the final transform.

  3. STEP 3

    Explain and test the result

    Compare the final mean and population standard deviation. Adding a constant preserves spread, while multiplying scales standard deviation by the absolute multiplier.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Scale a = -1; Shift b = 2. Pause the timeline at 20%. Given current scale = 0.6, calculate current mean, population standard deviation, population variance. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

At progress 0.2, scale=1+0.2((-1)−1)=0.6 and shift=0.2(2)=0.4. The original mean is 0 and variance is 2, so the new mean is 0.4 and variance is 2(0.6)²=0.72. Results: Current mean: 0.4; Population standard deviation: 0.849; Population variance: 0.72. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Statistics

Predict how rescaling measurements changes their spread

Transform a mean and standard deviation without recalculating every observation.