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Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Convert measurements into z-scores.
- Interpret an area under a normal density.
- Distinguish individual weights from averages of several weights.
Before you start
Subtraction, division, standard deviation, and probability as area.
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
A fictional package's fill weight X follows a normal distribution with mean 500 g and standard deviation 4 g. What fraction of packages fall between 492 g and 508 g?
Why this math matters
Standardization expresses distances from a mean in comparable units. A value eight grams below a mean may be close in one process and extreme in another. Under a normal model, standardized distances also connect to known probability areas. The model must be justified separately; converting values to z-scores does not make arbitrary data normal.
Set up the model
A useful answer starts with clear assumptions:
- The package weights follow the stated normal model with known parameters.
- The probability calculation describes individual packages, not a guaranteed tolerance for every package.
- The practice about an average uses sixteen independent packages from this same distribution.
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.
What does two standard deviations mean for package weights?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A fictional package's fill weight X follows a normal distribution with mean 500 g and standard deviation 4 g. What fraction of packages fall between 492 g and 508 g?
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.
Standardize the lower endpoint
z₁ = (492 − 500)/4 = −2
The lower bound is two standard deviations below the mean. The units cancel because both numerator and denominator are grams.
Standardize the upper endpoint
z₂ = (508 − 500)/4 = 2
The upper bound is equally far above the mean, making the requested interval symmetric.
Use the standard normal area
P(492 ≤ X ≤ 508) = P(−2 ≤ Z ≤ 2) ≈ 0.9545
A normal table or calculator supplies the area. The familiar 95% description is a rounded approximation to this probability.
The result
About 95.45% of individual package weights fall within the interval under this normal model.
About 4.55% lie outside, split equally between the two tails. This is a statement about a measurement distribution, not a 95% confidence interval for an unknown mean. Those are different uses of probability.
Common mistakes to catch
- Using 4 as a variance instead of a standard deviation changes the standardization.
- Applying normal areas to nonnormal data without justification can give misleading probabilities.
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 P(X < 494) in the same model?
Show a hint
The standardized cutoff is −1.5; its normal lower-tail area is about 0.0668.
Reveal answer and explanation
Approximately 6.68%
(494 − 500)/4 = −1.5, so use P(Z < −1.5).
Practice 2
For an average of sixteen independent package weights, what are its standard deviation and P(499 ≤ average ≤ 501)?
Show a hint
The average has SD 4/√16.
Reveal answer and explanation
1 g; approximately 68.27%
The average is normal with mean 500 and SD 1, so the bounds are one standard deviation on either side.
Take the idea with you
Always identify whether your random quantity is one measurement, a sum, or an average before selecting its standard deviation.
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: What does a 95% confidence interval actually promise?
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