Problem-Solving and Data Analysis

Fewer questions than Algebra or Advanced Math, and mostly arithmetic — but with wording that's easy to misread under time pressure.

#Percentages

The three sentences that cover nearly all of it:

• X% of Y → (X/100) × Y
• Percent change → (new − old) / old × 100. Divided by the old value, always.
• Increase by X% → multiply by (1 + X/100). Decrease → (1 − X/100).

Successive changes multiply; they don't add. Up 20% then down 20% is 1.20 × 0.80 = 0.96 — a 4% net loss, not zero. This exact question appears constantly.

To reverse a percentage — "after a 15% increase the price is $92, what was it before?" — divide: 92 / 1.15. Don't subtract 15%.

#Ratios, rates, and units

Set up proportions with units written in, and cancel them like algebra. Most rate errors are unit errors, and units are self-checking:

120 miles     1 hour       120
--------- × ---------- = ------- gallons⁻¹ ...
1           30 miles/gal

If the units don't cancel to what the question wants, the setup is wrong — before you've done any arithmetic.

Unit conversion: multiply by fractions equal to 1, arranged so the unwanted unit cancels.

#Statistics

• Mean — sum ÷ count. If you know the mean and the count, you know the sum, and "the mean of 5 numbers is 12, four of them are..." questions are asking for exactly that.
• Median — middle value when ordered. Sort first, every time.
• Mode — most frequent.
• Range — max − min.
• Standard deviation — spread. You will never compute it; you'll compare two data sets and say which is more spread out.

Skew: in a right-skewed (long tail to the right) distribution, mean > median. Left-skewed, mean < median. Outliers drag the mean and leave the median alone — which is why the median is the better measure for skewed data, and the test asks that.

#Study design and inference

Two rules, and they carry every question in this area:

1. Random selection permits generalisation. If subjects were randomly selected from a population, results generalise to that population — and no further. A study on 500 randomly chosen students at one university says nothing about students elsewhere.

2. Random assignment permits causal claims. Only if subjects were randomly assigned to groups can you say the treatment caused the difference. Observational data gives association, never cause.

The wrong answers are always the overreach: cause from an observational study, or generalising past the sampled population. Check which of the two randomisations the stem mentions — that alone decides most of these.

Margin of error widens with more variability and narrows with a larger sample. A plausible-value range is the estimate ± the margin.