Statistics: Data, Models & Chance

Use dot plots, histograms, and box plots. Match the statistic to the shape: mean and SD for roughly symmetric data; median and IQR when outliers or skew dominate.

How to do it:

1. Plot first.

2. Choose center and spread that fit the shape.

3. Interpret outliers in context, do not delete them silently.

Worked example: Incomes are right-skewed; median income tells a typical household better than the mean. Open the Statistics lab with a pasted list.

Check: a leftover Grade 6 habit of only reporting the mean is not enough here.

How to do it:

1. Fit a line to a scatter plot (or read a given fit).

2. Interpret slope as the predicted change in $y$ per unit $x\text{,}$ with units.

3. Do not jump from a strong correlation to "x causes y." A lurking variable can drive both.

Worked example: Ice cream sales and drowning both rise in summer. Heat is the lurking variable. Grade 8 scatter plots started this; here we say it explicitly (HSS-ID.C.9).

Check: a strong $r$ still does not justify "x causes y" without a randomized experiment.

How to do it:

1. A survey samples a population and asks.

2. An observational study records what already happened, with no assigned treatment.

3. An experiment assigns treatments. Random assignment is what makes a causal claim possible.

Worked example: "Students who chose honors chemistry scored higher" is observational. To test a new lab curriculum you must randomly assign classes to old vs new.

Check: a large sample does not turn an observational study into an experiment. Randomization of treatment is the mark.

How to do it:

1. Independent events: $P(A\text{ and }B)=P(A)P(B)\text{.}$

2. Conditional: $P(A|B)=P(A\text{ and }B)/P(B)$ when $P(B)>0\text{.}$

3. Addition: $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\text{.}$

Worked example: Fair die, $P(\text{even or }<3)=P(\text{even})+P(1\text{ or }2)-P(2)=1/2+1/3-1/6=2/3\text{.}$

Check: do not add overlapping events without subtracting the intersection. Grade 7 probability is the base.

How to do it:

1. Find the mean.

2. Square each deviation from the mean, average the squares (divide by $n$ for a whole population, $n-1$ for a sample), and take the square root.

3. For a roughly bell-shaped distribution, about $68\%$ of values lie within $1$ SD of the mean, $95\%$ within $2\text{,}$ and $99.7\%$ within $3\text{.}$

Worked example: the data $2, 4, 4, 4, 5, 5, 7, 9$ has mean $5\text{.}$ The squared deviations $9, 1, 1, 1, 0, 0, 4, 16$ sum to $32\text{;}$ $32/8=4\text{;}$ the SD is $\sqrt{4}=2\text{.}$ Adult heights with mean $170$ cm and SD $8$ cm: about $68\%$ are between $162$ and $178$ cm, and about $95\%$ between $154$ and $186$ cm.

Check: SD is in the same units as the data and is never negative. A value more than $2$ SD from the mean is unusual, roughly $1$ in $20$ by chance.

How to do it:

1. Put both groups on the same scale, with parallel box plots or dot plots.

2. Compare center, then spread, then shape, in context.

3. A value is an outlier if it lies more than $1.5\times\text{IQR}$ below $Q_1$ or above $Q_3\text{.}$

Worked example: Class A quiz scores have $Q_1=20$ and $Q_3=32\text{,}$ so $\text{IQR}=12$ and the fences are $20-18=2$ and $32+18=50\text{.}$ A score of $55$ is an outlier; $48$ is not. Class B has median $30$ against A's median $26$ with a similar IQR: B typically scored about $4$ points higher with about the same consistency.

Check: name the group and the statistic: "B's median is $4$ points higher" beats "B did better." An outlier gets flagged and investigated, not deleted.

How to do it:

1. Residual $=$ actual $-$ predicted. Positive means the point sits above the line.

2. A residual plot with no pattern says a line fits; a curve in the residuals says it does not.

3. $r$ runs from $-1$ to $1\text{:}$ the sign gives the direction and the size gives the strength of the linear association.

Worked example: the fit is $\hat{y}=2x+3\text{.}$ At $x=4$ the actual $y$ is $12$ and the prediction is $11\text{,}$ so the residual is $+1\text{.}$ An $r$ of $-0.9$ is a strong negative association (as $x$ rises, $y$ falls, tightly); $r=0.2$ is weak. $r=0.9$ and $r=-0.9$ are equally strong.

Check: $r$ has no units and does not change when you swap $x$ and $y$ or change units. $r=0$ means no linear relationship; the data could still follow a curve.

How to do it:

1. $P(A\mid B)$ means "given $B$": restrict to the row or column for $B\text{,}$ then divide.

2. Compare $P(A\mid B)$ with $P(A)\text{.}$ If they are equal, $A$ and $B$ are independent.

3. Read "given" carefully: $P(\text{pass}\mid\text{athlete})$ and $P(\text{athlete}\mid\text{pass})$ use different denominators.

Worked example: $100$ students: $60$ athletes, of whom $45$ passed; $40$ non-athletes, of whom $25$ passed. $P(\text{pass}\mid\text{athlete})=45/60=0.75\text{,}$ while $P(\text{pass})=70/100=0.70\text{.}$ They differ, so passing and being an athlete are not independent here. $P(\text{athlete}\mid\text{pass})=45/70\approx 0.64$ answers a different question.

Check: the product test agrees: $P(\text{pass and athlete})=0.45\text{,}$ but $P(\text{pass})\cdot P(\text{athlete})=0.70\cdot 0.60=0.42\text{.}$ Totals must add up before you divide.

How to do it:

1. Different random samples give different results. That spread is sampling variability, not a mistake.

2. For a proportion from a random sample of size $n\text{,}$ a rough margin of error is $\tfrac{1}{\sqrt{n}}\text{,}$ about two standard deviations of the sample proportion.

3. Report an interval, and say what would narrow it (a larger sample) or break it (a biased sample).

Worked example: a poll of $400$ voters finds $52\%$ in favor. The margin is about $\tfrac{1}{\sqrt{400}}=0.05\text{,}$ so the interval is $47\%$ to $57\%\text{.}$ It contains $50\%\text{,}$ so the poll cannot say the measure is ahead. Halving the margin to $2.5$ points takes four times the sample: $1{,}600$ voters.

Check: a simulation makes it concrete: $60$ or more heads in $100$ fair flips happens about $3\%$ of the time, so $60$ heads is suspicious but not proof of a biased coin. A bigger sample never fixes a biased one.