Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Compute squared deviations from a mean.
- Distinguish population variance from standard deviation.
- State why a denominator choice depends on the question.
Before you start
Means, squaring numbers, and square roots.
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
Two fictional machines have recorded completion times A = {8, 10, 12} and B = {2, 10, 18} minutes. Treat each list as the complete population being described. Which is more consistent?
Why this math matters
An average can hide substantial variation. Both processes might promise ten minutes on average while producing very different experiences on individual runs. Squared deviations measure spread without positive and negative differences canceling. Taking a square root returns that measure to the original unit so it can be interpreted beside the mean.
Set up the model
A useful answer starts with clear assumptions:
- Each three-value list is the entire population for this descriptive exercise.
- Every time is measured in minutes using the same procedure.
- These lists alone do not establish how future production runs will behave.
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.
Can two processes have the same average but different reliability?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Two fictional machines have recorded completion times A = {8, 10, 12} and B = {2, 10, 18} minutes. Treat each list as the complete population being described. Which is more consistent?
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.
Compare the centers
mean A = 30/3 = 10; mean B = 30/3 = 10 min
The identical means establish that any difference in this comparison comes from spread rather than center.
Average the squared deviations
variance A = (4 + 0 + 4)/3 = 8/3; variance B = (64 + 0 + 64)/3 = 128/3
We divide by three because the task describes each complete population. The variance units are square minutes.
Return to minute units
SD A = √(8/3) ≈ 1.633 min; SD B = √(128/3) ≈ 6.532 min
B's standard deviation is four times A's. Its variance is sixteen times as large because deviations are squared.
The result
A is more consistent in these records, despite both means being ten minutes.
If the observations were an independent sample used to estimate a larger population's variance, the usual sample variance would divide by n − 1. That change is about the estimator's purpose, not permission to switch denominators until a preferred number appears.
Common mistakes to catch
- Averaging the signed deviations gives zero and misses spread.
- Variance and standard deviation have different units and cannot be substituted interchangeably.
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
Find the population variance and SD of {4, 4, 4}.
Show a hint
Each observation equals the mean.
Reveal answer and explanation
Variance 0; SD 0
Every deviation is zero, so both spread measures are zero.
Practice 2
Using A as a sample, what is the usual sample variance?
Show a hint
Divide its squared-deviation sum eight by n − 1.
Reveal answer and explanation
4 min²
There are three observations, so s² = 8/(3 − 1) = 4 square minutes.
Take the idea with you
When comparing summaries, report center, spread, units, and whether the values describe a population or estimate one.
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.
Up next: When can you count outcomes to find a probability?
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