Inequalities, intervals and formulas with cases

Formulas aren't only about "equals". Inequalities say which side of a boundary a number falls on, and the two kinds differ by exactly one value, the boundary itself.

• <, "less than", excludes the boundary: p < 0.05 does not include 0.05.
• ≤, "at most", includes it: p ≤ 0.05 does.
• > and ≥ work the same way upwards; ≥ reads "at least".

Most numbers never land exactly on a boundary, so the sign seems pedantic until one does: a rating of exactly 4.0, a p-value of exactly 0.05, sales of exactly 100,000 UAH. Then the sign alone decides the outcome, and a rule that doesn't say which side the boundary is on starts an argument.

When you write a rule, make every boundary belong to exactly one side: "from 100,000 up to, but not including, 300,000" is 100,000 ≤ s < 300,000. When you read one, check the sign before the number.

±, "plus or minus", gives two numbers at once: a ± b means everything from a − b to a + b. A lift of 1.3 pp ± 0.9 pp is the range from 0.4 to 2.2 pp.

Written as an interval: [0.4; 2.2]. Square brackets include the ends, round ones exclude them: (0.4; 2.2) is the same range without its endpoints. A mix such as [0; 5) includes 0 and excludes 5.

The middle of a ± interval is the estimate, and b is its margin of error: how far the true value may plausibly sit on either side. So the interval is 2b wide, and halving b halves the uncertainty.

Two slips: reading 1.3 ± 0.9 as "from 0.9 to 1.3", and giving the margin in percent when the estimate is in points. Keep the units of the estimate: a lift in pp has a margin in pp.

For a test result, the key question about an interval is whether it contains 0, because 0 means "no effect".

• An interval entirely above 0, like [0.4; 2.2], rules out "no effect": every plausible value is a gain.
• An interval that contains 0, like [−0.3; 1.9], doesn't. The symbol ∈ reads "is in": 0 ∈ [−0.3; 1.9]. ∉ means "is not in".
• An interval entirely below 0 points to a loss.

An interval that only touches 0, like 0.8 ± 0.8 = [0; 1.6], still contains it, so it doesn't rule out "no effect".

The arrow ⇔, "if and only if", ties a decision to a condition both ways: "reject 'no effect' ⇔ p < α" means a small p leads to rejecting, and rejecting means p was small. There's no third case. Convert a ± into an interval first; then the 0 check takes one glance.

Some formulas branch: the answer depends on which case you're in. A curly bracket on the left lists the cases, each with its condition on the right. Read the condition first, then use the formula next to it.

The median is a classic. With n odd, it's the middle value after sorting, x₍₍ₙ₊₁₎/₂₎; with n even, there's no single middle, so it's the mean of the two middle ones, (x₍ₙ/₂₎ + x₍ₙ/₂₊₁₎)/2. For 4, 6, 7, 9 (n = 4): (6 + 7) ÷ 2 = 6.5. For 4, 6, 7, 9, 14 (n = 5): x₍₃₎ = 7.

Business rules are written the same way: a bonus that is 0 below one rating, one rate in a middle band and another above it. The cases must cover every possible value exactly once, with no gaps and no overlaps, and that's where the < and ≤ signs earn their keep.

Most errors come from skipping the condition and using the first formula in sight.

A common rule for flagging unusual values is a pair of inequalities, the fences. A value is flagged if

x < Q₁ − 1.5·IQR or x > Q₃ + 1.5·IQR,

where Q₁ and Q₃ are the lower and upper quartiles and IQR = Q₃ − Q₁ is the spread of the middle half of the data.

With Q₁ = 6 and Q₃ = 10 lessons a month, IQR = 4, the lower fence is 6 − 6 = 0 and the upper fence is 10 + 6 = 16. A learner with 18 lessons is flagged; one with exactly 16 is not, because the rule uses a strict >.

"Or" matters: a value is flagged if it breaks either fence. And a flag is a reason to look, not to delete: 18 lessons may be a real, very keen learner.

max picks the larger of its inputs and min the smaller. That makes them the shorthand for lower limits and caps.

• max(0, x − 50) never goes below 0: it's 0 until x passes 50, then grows by 1 for each unit above. An overage charge of 30 UAH per hour above 50 is 30 × max(0, h − 50): 0 for 42 hours, 30 × 14 = 420 UAH for 64 hours.
• min(cap, x) never goes above the cap: a referral reward of 10% of the first payment, capped at 500 UAH, is min(500, 0.1 × payment). A 3,000-UAH payment gives 300; a 7,000-UAH one gives 500.
• max(a, b, c) of several durations is how long a plan takes when its tracks run in parallel: 5, 8 and 6 weeks give 8.

To find where a cap starts to bind, set the inside equal to the cap and solve: 0.1 × payment = 500 gives a payment of 5,000 UAH.

Payback combines a running total, as in the lesson on Σ, with an inequality: it's the first month k for which the running total of margin per learner reaches the acquisition cost, Σ from i = 1 to k of mᵢ ≥ CAC.

Both "first" and "≥" matter. Margins of 140, 120 and 110 UAH give running totals 140, 260 and 370. Against a CAC of 300, month 2 gives 260 < 300 and month 3 gives 370 ≥ 300, so payback is month 3, even though the total keeps growing afterwards.

With ≥, a running total exactly equal to the CAC counts as paid back; with a strict > it wouldn't, and the answer could move by a month. Read the sign in the definition you're given.

Two slips: reporting the month after the crossing, and looking for a month whose single margin exceeds the CAC instead of the running total.

A tiered rate applies each rate only to its own slice. Klasna pays sales partners 10% on the first 100,000 UAH of monthly sales, 15% on the part from 100,000 to 300,000 and 20% on everything above 300,000.

For 380,000: 10% × 100,000 = 10,000; 15% × 200,000 = 30,000; 20% × 80,000 = 16,000. Total 56,000 UAH. The effective rate is total ÷ base, 56,000 ÷ 380,000 ≈ 14.74%; the marginal rate, on the next hryvnia, is 20%. Being "in the 20% bracket" doesn't mean 20% of everything: that would be 76,000, overpaying by 20,000.

When rates rise slice by slice, the effective rate is always below the marginal one.

How it fits together: inequalities and intervals say which side of a line a number falls on; cases and max/min turn those sides into rules; tiers apply a rule slice by slice. Next module: averages and shares of shares.