doing math

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Base Rates & Conditional Rates

lesson · about 3 minutes

Always ask 'out of which group?' before you divide.

In your head, jot if needed · no calculator why?

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the idea

Every rate is a count divided by a group, and the group decides the answer. Before you divide, ask "out of which group?"

"The conversion rate among referrals" divides by referrals. "The share of buyers who were referrals" divides by buyers. Same counts, different questions, different answers.

The biggest trap is a warning sign for something rare. If 2% of orders are fraud, many flags can land on honest orders, because there are so many honest ones to be wrong about.

techniques

Name the group, then divide

Any rate or share: among a group, or out of all buyers.

  1. Find the group after "among" or "of all". It goes on the bottom.
  2. On top, count only members of that group with the outcome.
  3. Say it back in words before you divide.
worked example

Example: Referrals: 30 buy, 20 don't. Non-referrals: 10 buy, 40 don't. What share of all buyers were referrals?

  1. The group is buyers: 30 + 10 = 40.
  2. 30 ÷ 40 = 0.75 = 75%.
  3. A different question: among referrals, 30 ÷ 50 = 60% buy.

Answer: 75%

Turn rates into counts

A filter flags something rare, and you want the chance a flag is real.

  1. Split the population into the rare group and the rest.
  2. Apply each flag rate to its own group.
  3. Real share of flags: flagged rare ÷ all flagged.
worked example

Example: Of 1,000 orders, 20 are fraudulent. A filter flags 90% of fraudulent orders and 5% of honest orders. If an order is flagged, what is the chance it's fraudulent? Round to the nearest whole percent.

  1. Flagged fraud: 90% of 20 = 18.
  2. Flagged honest: 5% of 980 = 49.
  3. 18 ÷ 67 ≈ 26.9%, about 27%.

Answer: 27%

watch out for

practice

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Rate within a group

worked example

There are 260 leads. 40 are referrals, and 7 of the referral leads buy. What is the conversion rate among referrals?

Answer: 17.5%

  1. "Among referrals" means divide by referrals: 7 ÷ 40 = 17.5%.
  2. Dividing by all 260 leads would answer a different question.

Share of buyers from a group

worked example

Leads from trade shows: 105 buy, 105 don't. Leads from the website: 35 buy, 115 don't. What share of all buyers came from trade shows? Round to one decimal place if needed.

Answer: 75%

  1. Now the group is "buyers": 105 ÷ (105 + 35) = 105 ÷ 140 = 75%.
  2. That's a different question from "what share of trade-show leads buy" (105 ÷ 210 = 50%).

Flagged: how likely is it real?

worked example

Of 10,000 orders, 200 are fraudulent. A filter flags 90% of fraudulent orders and 2% of honest orders. If an order is flagged, what is the chance it's fraudulent? Round to the nearest whole percent.

Answer: 48%

  1. Flagged fraud = 90% of 200 = 180; flagged honest = 2% of 9,800 = 196.
  2. Among the 376 flagged orders: 180 ÷ 376 ≈ 47.9%, about 48%. When the bad group is small, false alarms pile up.

Which denominator?

worked example

You want the no-show rate among online bookings. What goes in the denominator?

  1. All bookings
  2. All online bookings
  3. All no-shows
  4. Online bookings that were no-shows

Answer: All online bookings

  1. The group after "among" is the whole you divide by: all online bookings.
  2. Only the ones that were no-shows go on top.

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