1 · Mean, median, mode, and range
Mean: add and divide by how many. Median: the middle value once the list is in order; with an even count, the mean of the two middle values. Mode: the most common value. Range: largest minus smallest.
15, 3, 9, 21, 6, 12 in order is 3, 6, 9, 12, 15, 21, so the median is (9 + 12) ÷ 2 = 10.5. Taking the middle of the list before sorting is the usual slip.
2 · Work with totals
A mean tells you the total: 5 quizzes with a mean of 80 total 400. If the mean of 6 quizzes is 82, they total 492, so the sixth score was 492 − 400 = 92.
3 · Weighted means
When groups are different sizes, do not average the averages. 20 students averaging 80 and 30 averaging 90 total 1,600 + 2,700 = 4,300 points, and 4,300 ÷ 50 = 86, not 85.
4 · Outliers
One very large or very small value pulls the mean toward it but barely moves the median. 3, 5, 6, 6, 10 has mean 6 and median 6. Add 36: the mean jumps to 66 ÷ 6 = 11, but the median is still 6.
5 · Box plots and quartiles
A box plot shows five numbers: the smallest, the first quartile (Q1), the median, the third quartile (Q3), and the largest. Each of the four parts holds about 25% of the data.
The box covers the middle half. Its length, Q3 − Q1, is the interquartile range. The range runs from whisker end to whisker end.
6 · Probability and complements
Probability = favorable outcomes ÷ equally likely outcomes. Rolling a number greater than 4 on a number cube: 2 of 6, or 1/3.
"At least one" is often easiest from the other side: P(at least one head in 3 tosses) = 1 − P(no heads) = 1 − 1/8 = 7/8.
7 · Independent and dependent events
For "this and then that," multiply. Two coin tosses both heads: ½ × ½ = 1/4.
Without putting the first item back, the second probability changes. A bag of 4 red and 6 blue: both red is 4/10 × 3/9 = 2/15, not 4/10 × 4/10.
8 · "Or" events
Add the two counts, then subtract anything counted twice. From 1 to 30, multiples of 3 or 5: 10 + 6 − 2 (15 and 30) = 14, so the probability is 14/30 = 7/15.
9 · Counting with order
Multiply the number of choices at each step. A president, vice president, and treasurer from 7 people: 7 × 6 × 5 = 210 ways, because the jobs are different.
Codes work the same way. If repeats are allowed, every place has all the choices; if not, each place has one fewer.
10 · Counting without order
When order does not matter (a team, a committee, a pair), count in order and divide by the number of ways to arrange the group. 3 of 7 students: 7 × 6 × 5 ÷ (3 × 2 × 1) = 35.
Handshakes among n people: n(n − 1) ÷ 2. Each of 10 people shakes 9 hands, and each handshake is counted twice: 45.
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