Eleventh Grade · Math · week 33

The normal distribution and z-scores

Use z-scores and the empirical rule to interpret data from a normal distribution.

Many real data sets (heights, test scores, measurement errors) cluster into the familiar bell-shaped normal distribution: symmetric, with most values near the mean and progressively fewer values further away. A z-score standardizes any value by measuring how many standard deviations it sits from the mean: z = (x - mean) / (standard deviation). A z-score of 2 means "two standard deviations above average" -- a way of comparing values across completely different data sets on the same scale.

The empirical rule (sometimes called the 68-95-99.7 rule) gives quick, powerful estimates for any normal distribution: about 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2, and about 99.7% fall within 3. A z-score beyond +-2 is already unusual; beyond +-3 is rare.

📋 For the teacher
Teach it (10–15 min):
  • Present the z-score as a translator: it converts any value, from any normal distribution, into a universal "how many standard deviations from average" scale.
  • Draw a quick bell curve together and mark off the 68/95/99.7 bands by hand -- the picture does most of the explaining.
  • Practice going both directions: z-score from a value, AND value from a z-score, since real problems ask both ways.
  • Discuss what counts as "unusual" using the empirical rule as the benchmark: a z-score beyond 2 is already in the outer 5%, worth noticing.
Talk about it:
  • What does a z-score actually measure?
  • Why is a z-score useful for comparing values from two different data sets?
  • According to the empirical rule, roughly what fraction of a normal distribution lies beyond 2 standard deviations from the mean?

Mastery looks like: They compute a z-score from a value (and a value from a z-score) and use the empirical rule to describe how typical or unusual a result is.

Common stumbles: Subtracting the mean and standard deviation in the wrong order inside the z-score formula; misremembering which percentage (68, 95, or 99.7) goes with which number of standard deviations.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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