Cogito
Probability and Statistics · Chapter 6 · Lesson 3
Normal Probabilities and Percentiles
From a z-score to a proportion.
12 problems · about 21 minutes · TEKS P.S.3.B, S-ID.A.4
What this lesson teaches
The student finds proportions and percentiles for normally distributed data.
- Area to the left of a value gives the proportion below it.
- z = 0, 1 and −1 mark roughly the 50th, 84th and 16th percentiles.
- Reverse the z formula to turn a percentile back into a value.
Warm Up
Straightforward practice. Get the method working first.
5 problemsMean 100, sd 15, z = 1. What is the value?
Answer 115
Why 115.
What does the area to the left of a value represent?
Answer The proportion of data below that value.
Why The proportion below, which is its percentile.
z = 0. Percentile?
Answer 50
Why The centre.
z = 1. Approximate percentile?
Answer 84
Why 50 + 34.
z = −1. Approximate percentile?
Answer 16
Why 50 − 34.
Build It Up
The same ideas with more to keep track of.
3 problems60% lies below a value. What proportion lies above, as a decimal?
Answer 0.4
Why 1 − 0.6.
Mean 200, sd 20, z = 2. What is the value?
Answer 240
Why 200 + 40.
Areas below two values are 0.84 and 0.16. What proportion lies between them?
Answer 0.68
Why 0.84 − 0.16.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each z-score with its approximate percentile.
Answer z = −1 → 16th; z = 0 → 50th; z = 1 → 84th
Why The 68% within one standard deviation splits evenly either side.
Mean 70, sd 5, z = −2. What is the value?
Answer 60
Why 70 − 10.
The Median: What percentile does z = 0 correspond to?
Answer 50
Why The 50th.
The Value: Mean 500, sd 100, z = 1. What is the value?
Answer 600
Why 600.