Learn with Amar
Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Transform a mean and standard deviation without recalculating every observation.
- Justify the conclusion "SD(Y)=|3|SD(X)=6 and Var(Y)=36" using the stated assumptions.
Before you start
Means, deviations, and variance.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
If X has mean ten and standard deviation two, find the mean and standard deviation of Y=3X−5.
Why this math matters
Transform a mean and standard deviation without recalculating every observation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The mean and variance exist.
- The same transformation is applied to every observation.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Predict how rescaling measurements changes their spread
PausedQuestion: Start with the question. Paused.
Question
Start with the question
If X has mean ten and standard deviation two, find the mean and standard deviation of Y=3X−5.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
E[Y]=3E[X]−5=25
Mean follows the same affine transformation as each observation.
Work through the mathematics
Y−E[Y]=3(X−E[X])
Subtracting the transformed mean cancels the shift.
Check the conclusion
SD(Y)=|3|SD(X)=6 and Var(Y)=36
Scaling multiplies deviations, while shifting does not alter spread.
The result
SD(Y)=|3|SD(X)=6 and Var(Y)=36
Scaling multiplies deviations, while shifting does not alter spread.
Common mistakes to catch
- Variance scales by the square of the multiplier.
- A negative multiplier cannot produce a negative standard deviation.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the standard deviation of 100−X?
Show a hint
Use the absolute scale factor.
Reveal answer and explanation
Two
Reflection changes signs of deviations but not their squared sizes.
Practice 2
What is the mean of X+7?
Show a hint
Add the constant to the mean.
Reveal answer and explanation
Seventeen
Every observation and the mean shift by seven.
Take the idea with you
Convert both the center and spread of a dataset when its measurement unit changes.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
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