Grade 11 · Rescaling data · 55 of 600
Rescaling data · Scale a=0.5; Shift b=-0.5
Use the completed transformation, at the end of the animation. Find the transformed mean, population variance, and population standard deviation. Givens: Scale a=0.5; Shift b=-0.5.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Treat −2,−1,0,1,2 as a five-observation population. Transform every value by z=(0.5)x+(-0.5). Use the completed transformation, at the end of the animation. Find the transformed mean, population variance, and population standard deviation.
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A starting point
The original population mean is zero and variance is two. A shift changes mean, and a scale multiplies variance by its square.
Work through the reasoning
Step 1
Identify the model and target
Treat −2,−1,0,1,2 as a five-observation population. Transform every value by z=(0.5)x+(-0.5). Use the completed transformation, at the end of the animation. The governing relation is Y=aX+b; mean(Y)=b; SD(Y)=|a|√2. The original population mean is zero and variance is two. A shift changes mean, and a scale multiplies variance by its square.
Step 2
Substitute and calculate
At progress 1, scale=1+1((0.5)−1)=0.5 and shift=1(-0.5)=-0.5. The original mean is 0 and variance is 2, so the new mean is -0.5 and variance is 2(0.5)²=0.5.
Step 3
Check the mathematical meaning
The transformed observations are -1.5, -1, -0.5, 0, 0.5. Their mean is -0.5. Variance uses division by 5, and standard deviation is |0.5|√2, never negative. Current scale: 0.5; Current mean: -0.5; Population standard deviation: 0.707; Population variance: 0.5. Decimal values are rounded, so use unrounded intermediate values.
The answer
At progress 1, scale=1+1((0.5)−1)=0.5 and shift=1(-0.5)=-0.5. The original mean is 0 and variance is 2, so the new mean is -0.5 and variance is 2(0.5)²=0.5. The transformed observations are -1.5, -1, -0.5, 0, 0.5. Their mean is -0.5. Variance uses division by 5, and standard deviation is |0.5|√2, never negative. Animation check: Current scale: 0.5; Current mean: -0.5; Population standard deviation: 0.707; Population variance: 0.5. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. Treat −2,−1,0,1,2 as a five-observation population. Transform every value by z=(0.5)x+(-0.5). Use the completed transformation, at the end of the animation. The requested state occurs at 100% playback; use the exact target stated in the question for your calculation.
Y=aX+b; mean(Y)=b; SD(Y)=|a|√2
03 · Reflect and transfer
Explain what changes and why.
Why does reversing the sign of the scale preserve variance while reversing the ordering of the observations?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.