Math 30 min

Statistics & Probability

Mean, median, spread, data displays, counting and probability.

Stats and probability questions reward careful reading more than hard math. Figure out exactly which group you’re counting, and you’re most of the way there.

What it tests

This skill covers describing data and predicting chance: mean, median, mode, range and spread; reading tables, box plots and scatterplots; counting arrangements; and finding probabilities, including from two-way tables.

How it shows up

The Math test is 45 questions in 50 minutes — a little over a minute each. Every question has four answer choices, a calculator is allowed on the whole section, there is no penalty for guessing, and no formula sheet is given.
Statistics & Probability is a smaller Math category, but the questions are often quick.
Look for averages (“what score does she need on the next test?”), data tables, “what is the probability that…”, and “how many different ways…”.

Key rules & concepts

The content you actually need to know.

Center: mean, median, mode

Mean = sum ÷ count. The flip side is the key trick: sum = mean × count.
Median = middle value after SORTING. With an even count, average the two middle values.
Mode = most frequent value. Range = max − min.

Spread and outliers

An outlier pulls the mean toward it a lot; the median barely moves.
Adding the same number to every value shifts the mean and median by that number but doesn’t change the range or standard deviation.
Multiplying every value by a positive number k multiplies the mean, median, range and standard deviation by k.
Standard deviation measures spread: data bunched close to the mean → small SD. You compare SDs; you almost never compute one.

Data displays

Box plot: min, Q1, median, Q3, max. Interquartile range IQR = Q3 − Q1.
Scatterplot: positive association (up to the right), negative (down to the right). The slope of a best-fit line is the predicted change in y per 1 unit of x.
Two-way table: read row and column totals carefully.

Counting

Fundamental counting principle: independent choices multiply. 3 shirts × 4 pants = 12 outfits.
Order matters (president, VP, secretary) → permutation: n × (n − 1) × … for r spots, or n!/(n − r)!.
Order doesn’t matter (a group of 3) → combination: n! / [r!(n − r)!].

Probability

P(event) = (favorable outcomes) / (total outcomes), between 0 and 1.
P(not A) = 1 − P(A).
Independent events: P(A and B) = P(A) · P(B). Without replacement, the second draw’s numbers change.
P(A or B) = P(A) + P(B) − P(A and B).
Conditional (“given that…” or “of those who…”): shrink the total to that group only.
Expected value = Σ (value × probability).

Strategy & traps

Step by step

1Identify exactly what’s being asked: a mean, a median, a count, or a probability.
2For average problems, convert averages into sums.
3For probability, write “what I want / what’s possible,” and check whether the question limits the group.
4For counting, ask “does order matter?” before choosing multiply, permutation or combination.

Unsorted median

Always put the data in order before picking the middle.

Wrong total in conditional probability

“Of the students who chose robotics…” means the denominator is the robotics total, not the whole table.

Replacement confusion

Without replacement, both the numerator and denominator drop by 1 on the second draw.

Board Buddy tip: Averages hide sums. The moment you see “average,” multiply to get the total.

Worked examples

Try each one first, then reveal the solution.

Example 1

Keisha’s average score on 4 tests is 85. What score does she need on the 5th test to raise her average to 87?

Example 2

Club choices of 80 students

	Robotics	Art	Total
Grade 10	18	12	30
Grade 11	22	28	50
Total	40	40	80

One of the students who chose Robotics is selected at random. What is the probability that the student is in Grade 11?

A11/40
B11/25
C1/2
D11/20

Example 3

A club of 8 members will choose a president, a vice president and a secretary. No one can hold two offices. How many different ways can the offices be filled?

A24
B56
C336
D512

Practice

Pick an answer for instant feedback. Answers lock once chosen.

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Question 1

What is the median of this data set: 14, 9, 21, 9, 17, 12?

Question 2

A data set is 10, 12, 13, 15, 20. If the value 80 is added to the set, how do the mean and median change?

Question 3

A bag holds 5 red marbles and 3 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both are red?

Question 4

A pizza shop offers 7 toppings. How many different pizzas with exactly 3 different toppings are possible?

Question 5

In a carnival game, a player wins $10 with probability 0.2, wins $2 with probability 0.5, and wins nothing with probability 0.3. What is the expected value of the winnings for one play?