Demonstration 1 of 4
Read the word among
In the same 100-parcel audit, why do 80% and 60% both come out right?
All three shares use the same four cells. Choosing a group in the menu outlines the row or column that becomes the whole; the bold cell is the part being counted.
\[P(D\mid A)\]
D means the parcel is damaged and A means the detector alerted. P(D | A) is read "the probability of damage given an alert": the group after the bar supplies the denominator.
Predict first. If 20 sound parcels set off alerts instead of 8, does the share of damaged parcels that alert change? Does the share of alerted parcels that are damaged change?
Choose an example
Constructed example: the chapter's 100-parcel audit, with one alternative false-alert count.
Calculated values
- Reference group
- damaged parcels (15)
- Alert share
- 12/15 = 80.0%
- Damaged with alerts
- 12
- Sound with alerts
- 8
- Total parcels
- 100
Among damaged parcels, the alert share is 12/(12 + 3) = 12/15 = 80.0%. The outlined cells are the denominator: the word among picked them. The bold count is the numerator. Check the whole table: 12 + 3 + 8 + 77 = 100.
Use the idea
When someone quotes a percentage, ask "among which group?" before you compare it with any other percentage.
Where the conclusion applies
A completed audit where every parcel's damage was checked independently of the alert. The sound group stays at 85 parcels when the false alerts change. If damage were judged by the alert itself, the table would have no missed cases and would fail.
Check your understanding: In an audit of 15 damaged parcels with 12 alerts, and 85 sound parcels with 28 alerts, what share of alerted parcels are damaged?
Chapter 22 source: section "Worked example: read the word “among”". Demonstration C22-D01.
Demonstration 2 of 4
Split the population first
Before anyone is tested, how lopsided are the two starting groups?
The base rate cuts the population into two rows. Every later rate acts inside one of these rows, so their sizes decide how many people each rate can touch.
\[10{,}000 - 100 = 9{,}900\]
The base rate is the share of a named population with the condition before any result is known. Each square stands for 100 of the 10,000 fictional people.
Predict first. At a 10% base rate, how many squares are hatched, and how many people stand in the plain squares?
Choose an example
Constructed example: the chapter's 10,000-person model, with other base rates.
Calculated values
- Base rate
- 1%
- With the condition
- 100
- Without the condition
- 9,900
- Without : with
- 99 to 1
1% of 10,000 = 100 people with the condition, and 10,000 - 100 = 9,900 without it. Before any test is applied, the group without the condition is 99 times as large (9,900 / 100).
Use the idea
Before reading any test or alert result, write down the two starting counts. They are the first line of the table.
Where the conclusion applies
A fictional population of 10,000 and a stated base rate. It describes no real condition or group. The split fails if the base rate belongs to a different population than the one being tested.
Check your understanding: With a 5% base rate in 10,000 people, how many people are without the condition?
Chapter 22 source: section "Begin with the population split". Demonstration C22-D02.
Demonstration 3 of 4
Three rates, three denominators
The same 90 true positives sit in two fractions. What makes the fractions different?
Each rate outlines a different set of cells as its whole. Doubling the population doubles both numerator and denominator, so every fraction stays the same.
\[\text{sensitivity} = 90/(90 + 10)\]
\[\text{positive predictive value} = 90/(90 + 495)\]
Sensitivity looks across the condition row. The false-positive rate looks across the no-condition row. Positive predictive value looks down the positive column.
Predict first. Double the fictional population to 20,000. Will the positive predictive value change?
Choose an example
Constructed example: the chapter's 10,000-person table and its 20,000-person scaling.
Calculated values
- Rate
- Positive predictive value
- Denominator group
- positive results (585)
- Value
- 2/13 = 15.4%
- Fictional population
- 10,000
Positive predictive value = 90/(90 + 495) = 90/585 = 2/13, about 15.4%. The outlined cells are the positive results.
Use the idea
Name the denominator group out loud whenever you read a rate: across a row, or down a column.
Where the conclusion applies
The chapter's fictional table of 90, 10, 495 and 9,405. Scaling works only when every cell is multiplied by the same factor; changing the base rate is a different change.
Check your understanding: In the 20,000-person table, what are the true and false positives, and what fraction do they give?
Chapter 22 source: section "Three rates, three denominators". Demonstration C22-D03.
Demonstration 4 of 4
The positive column is a mixture
With a test that catches 90% of cases, why are most positive results false here?
The positive results come from both rows. A small rate applied to the large condition-free row can outnumber a large rate applied to the small condition row.
\[P(\text{condition} \mid \text{positive}) = 90/585\]
Sensitivity is held at 90%. The false-positive rate is the share of people without the condition who still test positive. P(condition | positive) is the share of positive results that are true positives.
Predict first. Keep the false-positive rate at 5% and move the base rate from 1% to 2%. Does the true share among positives roughly double, or less than double?
Choose an example
Constructed example: the chapter's 1% and 2% models, plus other fictional rates.
Calculated values
- True positives
- 90
- False positives
- 495
- All positive results
- 585
- P(condition | positive)
- 90/585 = 2/13 = 15.4%
- Sensitivity (fixed)
- 90%
90% of 100 = 90 true positives; 5% of 9,900 = 495 false positives. Among the 90 + 495 = 585 positive results, 90/585 = 2/13, about 15.4%, have the condition. Sensitivity stayed at 90% the whole time.
Use the idea
When a result is positive, count the true positives and false positives in a stated population before deciding what the result means.
Where the conclusion applies
Fictional rates applied to a fictional 10,000 people, chosen so every count is whole. Sensitivity and the false-positive rate stay fixed when the base rate moves. No real test, diagnosis or risk is described.
Check your understanding: With a 1% base rate, 90% sensitivity and a 1% false-positive rate, what share of positive results are true positives?
Chapter 22 source: section "Work the fictional 10,000-person model". Demonstration C22-D04.