Fractions, ratios and rates

a = 30 / 200
b = 45 / 450
print(a, b)

A fraction is a division

A fraction says how much of a whole you have. 3/4 of a pizza means it was cut into 4 equal slices and you have 3. The line in the middle means divide: 3 ÷ 4 = 0.75. The same amount can be written three ways:

Fractions that look different can be the same share: 3/4, 6/8 and 75/100 are all 0.75. Dividing turns each one into a single number, which is the easiest way to compare them.

Percent means "out of a hundred", so to turn a decimal into a percent, multiply by 100. To go back, divide by 100:

part = 3
whole = 8
share = part / whole
print(share)              # as a decimal
print(share * 100, "%")   # as a percent
print(25 / 100)           # 25% back as a decimal

Rates: making groups comparable

Counts mislead whenever the groups differ in size. A newsletter that got 300 clicks sounds better than one that got 120, until you learn the first went to 10,000 people and the second to 2,000.

A rate is a count divided by how many chances there were. It answers "out of how many?":

big_clicks = 300
big_sent = 10000
small_clicks = 120
small_sent = 2000
print("Big list:", big_clicks / big_sent)
print("Small list:", small_clicks / small_sent)

The small list's rate, 0.06 (6%), is double the big list's 0.03 (3%). Rates are often quoted per 1,000 or per 100,000 to avoid tiny decimals: a rate of 0.003 is 3 per 1,000.

Choosing the whole matters as much as the part. A click rate usually divides by the times an ad was shown; a conversion rate divides by visits, or by visitors, which aren't the same, since one person can visit twice. Two teams that divide by different wholes will report different rates for the same campaign, so always say what you divided by.

north = 180 / 240
south = 95 / 100
print(north > south)

a_signups = 48
a_clicks = 1200
b_signups = 30
b_clicks = 600

rate_a = 0   # sign-ups out of clicks, as a percentage
rate_b = 0

print("A:", rate_a, "%")
print("B:", rate_b, "%")

Percent change versus percentage points

A checkout page converted 4% of visitors last month and 5% this month. Did conversion go up by 1% or by 25%? Both numbers are honest, which is why people trip over them:

• The difference between the rates is 1 percentage point: 5 − 4.
• The percent change compares that difference with where you started: 1 ÷ 4 = 0.25, a 25% increase.

The recipe for a percent change is (new − old) ÷ old × 100:

old_rate = 4.0
new_rate = 5.0
points = new_rate - old_rate
percent_change = (new_rate - old_rate) / old_rate * 100
print(points, "percentage points")
print(percent_change, "% change")

Headlines love whichever sounds bigger. When you report a change in a rate, say which one you mean.

One more trap: percent changes don't cancel out. A price that rises 50% and then falls 50% doesn't end where it began. $100 becomes $150, then $75, because the fall is half of the bigger, later amount.

old_signups = 250
old_visitors = 5000
new_signups = 330
new_visitors = 5500

old_rate = 0
new_rate = 0
points = 0
change = 0

print(old_rate, new_rate, points, change)

Averages of rates can lie

On Monday, 9 of 10 visitors to a new landing page signed up. On Tuesday, after an ad went out, 10 of 100 did. What was the page's sign-up rate over the two days?

It's tempting to average the two daily rates, 90% and 10%, and say 50%. But Tuesday had ten times the visitors, so it should count ten times as much. The honest overall rate adds up all the parts and all the wholes first:

monday = 9 / 10
tuesday = 10 / 100
print("Average of the rates:", (monday + tuesday) / 2)
print("Overall rate:", (9 + 10) / (10 + 100))

The overall rate is about 17%, nowhere near 50%. Whenever groups differ in size, combine the counts, then divide. It's the same reason an average over customers and an average over orders can disagree: they weigh things differently.

print(round((20 + 30) / (40 + 160), 2))

Ratios: scaling up and down

A ratio compares two amounts. A pancake recipe with 2 cups of flour to 1 cup of milk has a 2 : 1 ratio of flour to milk. Keep the ratio and you can make any amount. To scale a recipe, work out the scale factor, what you need divided by what the recipe makes, and multiply every amount by it. A ratio can also be read as a share of the total: in a 2 : 1 mix, flour is 2 of every 3 cups, or 2/3 of the batter.

serves = 4
guests = 10
factor = guests / serves      # 2.5 times the recipe
print(factor, 2 * factor, 1 * factor)

The same idea gives a unit price: divide the price by the amount to get the price of one unit. That's how you compare a 750 ml bottle with a 1,000 ml one, or a monthly plan with a yearly one. Bigger packs are often, but not always, cheaper per unit, which is exactly why it's worth dividing.

small = round(6.00 / 750, 4)
large = round(7.50 / 1000, 4)
print(small, large)

recipe_hikers = 6
nuts_per_recipe = 300
raisins_per_recipe = 150
hikers = 15

factor = 0
nuts = 0
raisins = 0

print(factor, nuts, raisins)

Fractions all through AI

You'll meet these ideas in every phase. An eval's accuracy is correct answers over all answers. A model's probabilities are shares of one whole, so they add up to 1. The share of a budget each campaign gets, a classifier's error rate, an API's cache hit rate: each is a part over a whole.

Fractions also hide traps you'll meet again. A model that's right 95% of the time sounds good, until you learn that 95% of the test answers were "no" and it always says no. You'll take that one apart in Phase 3.

Two questions catch most mistakes. Whenever you see a count, ask "out of how many?" Whenever you see a change in a rate, ask "points, or percent?"

a = 0.5
b = 0.3
c = 0.2
print(a + b + c)