Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.
Compare which value is relatively higher in its own normal distribution using z-scores
Problem
Compare score \(85\) from a distribution with mean \(80\) and standard deviation \(5\) to score \(110\) from a distribution with mean \(100\) and standard deviation \(20\). Compute both \(z\text{-}\text{scores}\) and identify the relatively higher value.
Big Picture
What this problem is really about
Raw distances from different means are not comparable when the distributions have different spreads. Standardize each observation using its own mean and standard deviation, then place both results on the shared z scale. The larger z-score is relatively higher within its distribution, regardless of which observation has the larger raw value or raw difference.
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