Data, Counting, and Probability: Chapter Review

Vocabulary and notation

Outlier

A value far from the rest. It drags the mean toward itself but barely moves the median, which is why the two disagree.

Frequency table

Pairs each category (what was counted) with its frequency (how many). The frequencies add to the total number of data values.

Key (pictograph)

The note saying how many one symbol stands for. Without it a row of symbols has no value.

Double bar graph

Two bars per category, one per group, with a legend naming which is which. Pick the category, then read that group's bar.

Experiment, outcome, sample space

An experiment has an uncertain result, each possible result is an outcome, and the sample space lists them all: a die gives $\{1, 2, 3, 4, 5, 6\}$.

Event

Any collection of outcomes from the sample space. "Roll an even number" is the event $\{2, 4, 6\}$.

Equally likely

Every outcome has the same chance. Promised by fair, equal sectors, or at random, and it must be checked, never assumed.

Law of large numbers

As independent trials of the same experiment pile up, the experimental probability settles toward the theoretical value. It promises nothing about one short run.

Formulas and theorems

• Mean

  $$\text{mean} = \frac{\text{sum}}{n}, \qquad \text{sum} = n \times \text{mean}$$

  Use when Numeric data, $n \ge 1$. The second form answers a missing-value question; as a check, deviations from the mean always add to $0$.

• Median

  $$\begin{gathered} n \text{ odd: middle value} \\ n \text{ even: mean of the middle two} \end{gathered}$$

  Use when SORT least to greatest first, then count in to the middle. With an even $n$ the median can fall between data points and need not be in the list.

  e.g. $2, 4, 7, 10$: median $\tfrac{4 + 7}{2} = 5.5$.

• Mode

  $$\text{mode} = \text{the most frequent value}$$

  Use when Counting only, no arithmetic. SEVERAL modes when values tie, NONE when every value occurs equally often. Also the only summary here that works on non-numeric data.

  e.g. $1, 1, 5, 8, 8$ has two modes, $1$ and $8$; $3, 6, 9$ has none.

• Range

  $$\text{range} = \text{maximum} - \text{minimum}$$

  Use when Measures SPREAD, not center, so it says nothing about where the data sits. Never divided by the count, never negative.

  e.g. $-3, 5, -8, 2$: range $5 - (-8) = 13$.

• Pictograph value

  $$\text{value} = \text{symbols} \times \text{key}$$

  Use when A symbol is never worth $1$ by default, and a partial symbol is that fraction of the key.

• Circle graph: count and angle

  $$\begin{gathered} \text{count} = \frac{\text{percent}}{100} \times \text{total} \\ \text{angle} = \frac{\text{count}}{\text{total}} \times 360^\circ \end{gathered}$$

  Text description

  A circle graph whose shaded wedge is the 15 percent slice, worth 36 of 240, drawn with a 54 degree angle at the center.

  Use when Slices must be parts of ONE whole, and their percents add to $100\%$, which finds a missing slice. A count needs the total stated.

  e.g. A $15\%$ slice of $240$: $36$ students, drawn as $54^\circ$.

• Multiplication principle

  $$\begin{gathered} (\text{stage 1 choices}) \\ \times (\text{stage 2 choices}) \times \cdots \end{gathered}$$

  Text description

  A tree for choosing an outfit. Its single start splits into 3 stage one branches, one for each shirt, and each of those splits into 2 stage two branches, one for each hat, ending in 6 highlighted end dots: the 6 possible outfits.

  Use when Each stage must offer the same NUMBER of choices next, no matter what came before; the choices themselves may differ. If an earlier pick changes how many choices remain, adjust that count before multiplying.

  e.g. $3$ flavors, $2$ sauces, $4$ toppings: $3 \times 2 \times 4 = 24$ sundaes.

• Addition principle

  $$(\text{group 1 size}) + (\text{group 2 size}) + \cdots$$

  Use when Exactly ONE item is chosen, and the groups must not overlap, or a shared item is counted twice. "Or" adds, "and then" multiplies.

• Arrangements

  $$\begin{gathered} n \times (n - 1) \times (n - 2) \times \cdots \\ (r \text{ factors for } r \text{ positions}) \end{gathered}$$

  Use when Distinct items, no reuse, and ORDER matters, so ABC and CBA are different. Stop after one factor per position; filling all $n$ runs the product down to $1$.

• Repeated choices

  $$\underbrace{c \times c \times \cdots \times c}_{n \text{ stages}} = c^{\,n}$$

  Use when Repetition allowed, so every stage keeps all $c$ choices. The base counts the choices at one stage, the exponent the stages.

• Theoretical probability

  $$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total equally likely outcomes}}$$

  Use when Valid only when the outcomes are EQUALLY LIKELY (fair, equal sectors, at random). "Favorable" means counted, not good. Simplify the fraction.

• The probability scale

  $$0 \le P(\text{event}) \le 1$$

  Use when A THEORETICAL $P = 0$ is impossible, $P = 1$ certain; zero successes in a finite experiment does not itself prove impossibility. Every probability here is a ratio of whole numbers, so it can be written as a fraction, decimal, or percent. A result off the scale means the counts are wrong.

  e.g. $\tfrac{1}{4} = 0.25 = 25\%$; rolling a $7$ on a standard die has $P = 0$.

• Complement rule

  $$P(\text{not } A) = 1 - P(A)$$

  Text description

  One bar of total 1 split into a short piece for the probability of A and a shaded remainder for the probability of not A.

  Use when Holds for any event in any sample space. All the outcome probabilities add to $1$: SUBTRACT from $1$, do not flip the fraction.

• Experimental probability

  $$P_{\text{exp}}(\text{event}) = \frac{\text{times the event happened}}{\text{trials}}$$

  Use when Comes from recorded data, so it needs no equally-likely assumption. A short run is not guaranteed to match the theoretical value exactly, though it sometimes will by chance.

  e.g. $27$ heads in $50$ flips: $\tfrac{27}{50} = 0.54$, against a theoretical $0.5$.

Problem types, step by step

Exam traps

• Trap Reading the median off the list as handed to you, so $5, 9, 1$ appears to have median $9$.

  Fix Sort first: $1, 5, 9$ has median $5$. "Middle" always means middle in ORDER.

• Trap Adding when the problem is a sequence of choices, or multiplying when it is one single choice.

  Fix $5$ books and $3$ games give $5 \times 3 = 15$ book-and-game pairs but only $5 + 3 = 8$ single prizes.

• Trap Treating any two outcomes as $\tfrac{1}{2}$, so "rain or no rain" becomes a $50\%$ chance of rain.

  Fix Counting gives a probability only when the outcomes are equally likely; otherwise you need data.

• Trap Shrinking the choices where repeats are allowed, so a $4$-digit code counts as $10 \times 9 \times 8 \times 7 = 5040$.

  Fix A digit may repeat, so every position keeps all $10$: $10^4 = 10{,}000$. Shrink only when an item is used up.

• Trap Swapping the base and the exponent, writing $5$ coin flips as $5^2$.

  Fix The base counts the choices at one stage, the exponent the stages: $2^5 = 32$, not $5^2 = 25$.

• Trap Reporting a circle graph's $40\%$ slice as $40$ people.

  Fix A slice is a share: $\tfrac{40}{100} \times 300 = 120$. Percent and count agree only when the total is $100$.

• Trap Comparing bar heights on a value axis that does not start at $0$.

  Fix Read the numbers off the scale: bars at $52$ and $48$ differ by $4$, not by a factor of two. A truncated axis breaks the length-to-value proportion.

• Trap Taking a complement by flipping the fraction, turning $P(A) = \tfrac{1}{6}$ into $\tfrac{6}{1}$ or $\tfrac{1}{5}$.

  Fix Subtract from $1$: $1 - \tfrac{1}{6} = \tfrac{5}{6}$, matching the $5$ faces that are not the one.

• Trap Counting pictograph symbols as the value, or rounding a half symbol up.

  Fix Multiply by the key, a partial symbol getting that fraction of it: with a key of $10$, four and a half symbols are $45$, not $4.5$ or $50$.

• Trap Reading a cause, or a value past the last plotted point, out of a graph.

  Fix A graph reports only what was measured: ice cream sales and sunburns rising together shows no causation, and a line says nothing past its last point.