1 · Mean, median, mode, range
For 4, 7, 7, 9, 13: the mean is the total divided by the count, 40 ÷ 5 = 8. The median is the middle value in order, 7. The mode is the most common value, 7. The range is largest minus smallest, 9.
With an even number of values, the median is the mean of the two middle values: 3, 5, 8, 10 has median 6.5.
2 · Working with the total
Most mean problems are really total problems. Five tests averaging 84 add to 5 × 84 = 420.
"What score does Ana need on the 6th test to average 86?" Six tests at 86 add to 516, so she needs 516 − 420 = 96.
3 · Outliers
An outlier is a value far from the rest. Adding 95 to 10, 12, 13, 15 moves the mean from 12.5 to 29, but the median only from 12.5 to 13.
That is why the median is a better "typical value" when the data has an outlier.
4 · Stem-and-leaf and box plots
In a stem-and-leaf plot, the stem is the tens digit and each leaf is one value's ones digit: stem 4 with leaves 2 5 8 means 42, 45, 48.
A box plot shows five numbers: the lowest, the first quartile (Q1), the median, the third quartile (Q3), and the highest. The box runs from Q1 to Q3, so it holds the middle half of the data. Q3 − Q1 is the interquartile range.
5 · Graphs
Read the scale before you read a bar. If the grid lines go up by 20, a bar halfway between 60 and 80 is 70.
A circle graph shows percents of a whole. If 25% of 400 students walk to school, that is 100 students. The percent alone is not a count.
6 · Probability
Probability = favorable outcomes ÷ all equally likely outcomes. A bag with 3 red, 5 blue, and 2 green marbles gives P(blue) = 5/10 = 1/2.
The complement is everything else: P(not blue) = 1 − 1/2 = 1/2. "At least one" problems are often easier as 1 − P(none).
7 · Two events
For independent events (a coin and a number cube), multiply: P(heads and a 6) = 1/2 × 1/6 = 1/12.
Without replacement, the second pick changes. Two red marbles from 3 red and 5 blue: 3/8 × 2/7 = 6/56 = 3/28. Using 3/8 × 3/8 would be right only if the first marble went back in.
8 · Expected counts
If a spinner lands on red with probability 1/5, then in 200 spins you expect about 1/5 × 200 = 40 reds.
9 · Counting
Multiplication principle: 4 shirts, 3 pants, and 2 pairs of shoes make 4 × 3 × 2 = 24 outfits.
Orders: 5 runners can finish in 5 × 4 × 3 × 2 × 1 = 120 ways. Pairs: if 8 people each shake hands with everyone once, that is 8 × 7 ÷ 2 = 28 handshakes, because each handshake was counted twice.
10 · On the ISEE
A probability is always between 0 and 1. If you get more than 1, you added when you should have multiplied, or the other way round.
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