The Normal Model and the Standard Normal
Working with Bell Curves
The normal distribution
A normal random variable is symmetric and fully described by its mean and standard deviation . Probabilities are areas under the bell curve, and values are standardized using the z-score.
Standardization
| Quantity | Formula | Meaning |
|---|---|---|
| z-score | Number of standard deviations from the mean | |
| Inverse transform | Convert a standardized value back to the original scale | |
| Standard normal | Normal distribution with mean 0 and standard deviation 1 |
Interpretation of z-scores
A positive means a value is above the mean; a negative means it is below the mean.
Solving normal probability problems
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Step 1: Identify and .
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Step 2: Convert the raw value to a -score.
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Step 3: Use the standard normal distribution to find the area or percentile.
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Step 4: Convert back if the question asks for a raw cutoff.
When the normal model is useful
Strong fit
- Symmetric data
- Unimodal shape
- No extreme skew or heavy outliers
Weak fit
- Highly skewed data
- Discrete counts with small values
- Clearly bounded or multimodal data
What is the z-score of a value equal to the mean?
If , then .
Correct answer:
Why do probabilities under a normal curve correspond to areas?
For continuous distributions, probabilities are areas under the density curve.
Correct answer: Because the distribution is represented by a density curve whose total area equals 1.