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Practice sheet · Psychological Profiles Path · Lens 1: Math

Scores & base rates

Practice for The Math of Profiles: rating scales, reverse scoring, averages, bell curves, z-scores, correlation, retests and base rates (pp.8–16), plus A1–A2 (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

Reverse a 1–5 item with 6 − answer. Mean = total ÷ number of scores; median = middle value in order. z = (score − mean) ÷ SD. About 68% of scores are within 1 SD of the mean and 95% within 2.

Square a correlation r for a rough share. With a rare trait, count true flags from the small group and false flags from the big group, then divide true flags by all flags.

Warm-up

One step

Scores, flips, percents and z.
1

Scale score

Six 1–5 items are answered 5, 3, 4, 2, 4 and 3, with none reversed. Find the total and the mean (p.8).

2

Flip it

On a 1–5 scale, reverse answers of 1, 4 and 5 (p.9).

3

Retest

In a made-up retest, 18 of 24 people got the same result. What percent is that (p.14)?

4

z-score

A made-up test has mean 50 and SD 8. Find z for a score of 58.

Core

Two or three steps

Say what each number is compared with.
5

Mixed items

Six 1–5 items are answered 4, 2, 5, 3, 1 and 4. Items 2 and 5 are reversed. Find the total after flipping, and the mean to one decimal place.

6

Outlier

Seven made-up scores are 11, 14, 15, 15, 17, 18 and 36. Find the mean and median. Then leave out the 36 and find both again. Which changed more?

Profiles Practice · Psychological Profiles Path · Unizon1/2
Psychological Profiles Path · Unizon · PracticeScores, averages & base rates
7

Bell curve

A made-up scale has mean 100 and SD 15. Between which scores are about 68% of people? About 95%? Of 500 people, about how many are within 1 SD?

8

Compare

Mean 40, SD 6. Find z for scores of 49 and 34. Which score is about the 16th percentile?

9

Correlation

One made-up study finds r = 0.4; another finds r = −0.6. Square each for a rough share. Which link is stronger?

10

Base rate

A made-up test screens 2,000 people; 40 have a rare trait. It catches 75% of them and wrongly flags 5% of the rest. How many are flagged, and what percent of flagged people really have the trait?

Level Up

Equations

A1–A2 (pp.17–18).
11

Undo the formula

A 10-item scale reports P = (r − 10) ÷ 40 × 100. What raw score r gives 85%?

12

Two tools

Survey tool A costs 90 + n dollars for n responses; tool B costs 15 + 2.5n. When are they equal, and at what cost?

13

T-scores

A T-score is T = 50 + 10z. Find z for T = 72. On a made-up test with mean 30 and SD 5, what raw score is that?

14

Pairs

Every pair in a group compares results once, making n(n − 1) ÷ 2 = 66 pairs. Solve for n.

Make your own problem

Your turn to be the test designer

  1. Pick a number from the book: a scale, a mean, an SD, a retest percent or a base rate.
  2. Write a question that uses it with a made-up, harmless questionnaire.
  3. Solve it, and say what the number can’t tell you about a real person.

My problem, and my worked answer:

Profiles Practice · Psychological Profiles Path · Unizon2/2
Psychological Profiles Path · Unizon · PracticeScores, averages & base rates

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.

Answer key

Worked solutions. Scores, groups, tests and prices are made-up examples unless the book cites them.

Warm-up
1
21 · 3.5 5 + 3 + 4 + 2 + 4 + 3 = 21; 21 ÷ 6 = 3.5.
2
5, 2, 1 6 − 1 = 5, 6 − 4 = 2, 6 − 5 = 1.
3
75% 18 ÷ 24 = 0.75.
4
z = 1 (58 − 50) ÷ 8 = 1: about the 84th percentile (p.12).
Core
5
25 · about 4.2 Flip items 2 and 5: 6 − 2 = 4, 6 − 1 = 5. 4 + 4 + 5 + 3 + 5 + 4 = 25; 25 ÷ 6 ≈ 4.2.
6
mean 18, median 15 · mean 15, median 15 126 ÷ 7 = 18. Without 36: 90 ÷ 6 = 15; middle of 11, 14, 15, 15, 17, 18 is 15. The mean changed more: 36 was an outlier (p.10).
7
85–115 · 70–130 · about 340 100 ± 15 and 100 ± 30. 0.68 × 500 = 340.
8
1.5 · −1 · the 34 (49 − 40) ÷ 6 = 1.5. (34 − 40) ÷ 6 = −1, about the 16th percentile (50% − 34%).
9
16% · 36% · r = −0.6 0.4² = 0.16; (−0.6)² = 0.36. The negative one is stronger: size matters, then sign (p.13).
10
128 flagged · about 23% 75% of 40 = 30. 5% of 1,960 = 98. 30 + 98 = 128; 30 ÷ 128 ≈ 23%. Most flags are false alarms (p.15).
Level Up
11
r = 44 (r − 10) ÷ 40 = 0.85 → r − 10 = 34 → r = 44.
12
50 responses · $140 90 + n = 15 + 2.5n → 75 = 1.5n → n = 50. 90 + 50 = 140.
13
z = 2.2 · raw 41 10z = 22 → z = 2.2. 30 + 2.2 × 5 = 41.
14
12 people n² − n − 132 = 0 → (n − 12)(n + 11) = 0 → n = 12.
Make your own: flip reversed items before adding, sort before finding a median, and remember that every score here describes made-up answers, not real people.
Profiles Practice · Psychological Profiles Path · UnizonAnswer key