Cogito
Probability and Statistics · Chapter 6 · Lesson 2
z-Scores
How unusual is one value?
12 problems · about 21 minutes · TEKS P.S.3.B, S-ID.A.4
What this lesson teaches
The student computes z-scores and uses them to compare values from different distributions.
- z = (value − mean) ÷ standard deviation.
- The sign says above or below the mean; the size says how far.
- Because z has no units, values from different distributions become comparable.
Warm Up
Straightforward practice. Get the method working first.
5 problemsValue 90, mean 78, sd 6. What is the z-score?
Answer 2
Why 2.
Why can z-scores compare results from different tests?
Answer A z-score has no units, so both are on the same scale.
Why The units cancel, leaving a common scale.
Value 85, mean 75, sd 5. z-score?
Answer 2
Why 10 ÷ 5.
Value 70, mean 70, sd 4. z-score?
Answer 0
Why Exactly at the mean.
Value 55, mean 60, sd 10. z-score?
Answer -0.5
Why −5 ÷ 10.
Build It Up
The same ideas with more to keep track of.
3 problemsz = 1.5, mean 100, sd 10. What is the value?
Answer 115
Why 100 + 15.
z = −1, mean 50, sd 8. What is the value?
Answer 42
Why 50 − 8.
Which z is more unusual? 1 for z = 0.4, 2 for z = −2.6.
Answer 2
Why Distance from zero, ignoring sign.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the z-score comparison in order.
Answer 1. Subtract the mean from the value. 2. Divide by the standard deviation. 3. Repeat for the other distribution. 4. Compare the two z-scores.
Why The subtraction comes before the division.
Value 92, mean 80, sd 6. z-score?
Answer 2
Why 12 ÷ 6.
The Score: Value 80, mean 70, standard deviation 5. What is the z-score?
Answer 2
Why 2.
The Below: Value 60, mean 70, standard deviation 5. What is the z-score?
Answer -2
Why −2.