Standard Deviation and Counting

• Interpret standard deviation as a measure of how spread out a data set is
• Compare two data sets and identify which has the larger standard deviation by inspection, and say what adding a constant to every value (mean shifts, spread stays) or multiplying every value (both scale) does
• Apply the fundamental counting principle: if event A has m outcomes and event B has n outcomes, the total is m × n
• Use nPr for ordered arrangements (permutations) and nCr for unordered selections (combinations)
• Distinguish problems that require permutations (order matters) from combinations (order does not matter)

Standard deviation (SD) measures how spread out a data set is around its mean. You will almost never compute an SD by hand on the ACT. The test gives you two data sets and asks which has the larger SD. The intuition: a data set with all values close to the mean has a small SD; a data set with values far from the mean has a large SD. The set {50, 50, 50, 50, 50} has SD = 0. The set {10, 30, 50, 70, 90} has a much larger SD even though both have mean 50. Extreme values (outliers) increase SD. Adding the same constant c to every value adds c to the mean and the median but leaves the SD unchanged, because every distance from the mean stays the same. Multiplying every value by k multiplies the mean and the SD by |k|: if scores have mean 70 and SD 8, adding 5 to each gives mean 75 and SD 8, while doubling each gives mean 140 and SD 16. Same range does not mean same SD: 1, 1, 9, 9 and 1, 2, 8, 9 both run from 1 to 9 with mean 5, but every value of the first sits 4 from the mean, so it has the larger SD. The fundamental counting principle: if a task has stages with m, n, p, … independent choices, the total number of outcomes is m × n × p. This works only when the number of choices at a stage is the same whatever was chosen earlier; if the available choices change, count each case separately. Example: 3 shirts, 4 pants, 2 shoes → 3 × 4 × 2 = 24 outfits. Permutations count ordered arrangements: nPr = n! / (n − r)! is the number of ways to arrange r items from n, where order matters. Example: 5 runners, medals for top 3 → 5 × 4 × 3 = 60 ways (gold-silver-bronze is a different outcome than bronze-silver-gold). Combinations count unordered selections: nCr = n! / (r!(n − r)!) is the number of ways to choose r items from n, where order does not matter. Example: pick 3 students from 10 for a committee → 10C3 = 120 ways. The key question: does swapping two picked items give you a new outcome? If yes, use permutations. If no, use combinations. Both nPr and nCr are built into most calculators (math → prb menu on a TI-84), and a graphing calculator can verify this kind of count.

• If all values in a data set are identical, SD = 0. If values are spread wide, SD is large. You can eyeball this.
• Adding or subtracting a constant to every value does not change SD. Multiplying changes it.
• Fundamental counting principle: multiply the options at each stage. Do not add.
• Ask yourself: 'Does order matter?' If yes → permutation (nPr). If no → combination (nCr).
• Use your calculator's nPr and nCr functions. Computing factorials by hand wastes time.