Data Analysis and Statistics

Pentru elevii români: SAT-ul pune accentul pe interpretarea datelor (grafice, tabele, eșantioane, „margin of error”) mai mult decât pe calcul, iar termeni ca „median”, „range”, „standard deviation” și „line of best fit” trebuie recunoscuți rapid în engleză. Tabelele cu două intrări (two-way tables) și concluziile despre eșantioane sunt mult mai frecvente decât în programa de la școală.

What the SAT tests here

This lesson covers most of the data side of the Problem-Solving and Data Analysis domain:

• one-variable data: mean, median, mode, range, standard deviation, and the effect of outliers;
• two-variable data: scatterplots, lines of best fit, linear versus exponential trends;
• probability and conditional probability, usually from a two-way table;
• inference from sample statistics and margin of error;
• evaluating statistical claims: what a study design allows you to conclude.

Key rules and formulas

Measures of center

• Mean $= \frac{\text{sum of values}}{\text{number of values}}$
• Median: the middle value of the ordered data (the average of the two middle values when there is an even count)
• Outliers pull the mean toward them; the median barely moves.

Measures of spread

• Range $=$ maximum $-$ minimum.
• Standard deviation measures how far values typically are from the mean. You will not need to compute it by hand on the SAT; you need to compare: more spread out means larger standard deviation.

Probability

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

For conditional probability ("given that..."), the denominator is only the group named after "given" or "selected from."

Lines of best fit. In $y = mx + b$, $m$ is the predicted change in $y$ per one-unit increase in $x$; $b$ is the predicted value when $x = 0$.

Sampling and studies

• A random sample lets you generalize to the population it was drawn from, and only that population.
• Random assignment to treatment groups allows conclusions about cause and effect. Without it, you can only claim an association.
• A margin of error gives a range of plausible values: estimate $\pm$ margin. Larger samples generally produce smaller margins of error.

Worked examples

Example 1 (outliers). The data set 12, 15, 15, 18, 20, 22, 95 has a sum of 197, so the mean is $\frac{197}{7} \approx 28.1$, and the median is 18. If the outlier 95 is removed, the mean becomes $\frac{102}{6} = 17$ and the median becomes $\frac{15 + 18}{2} = 16.5$. The mean dropped about 11 points; the median dropped only 1.5.

Example 2 (two-way table). A school surveyed 200 students.

(a) If a Grade 12 student is chosen at random, the probability that the student plays an instrument is $\frac{44}{110} = 0.4$. (b) If a student who plays an instrument is chosen at random, the probability that the student is in Grade 12 is $\frac{44}{80} = 0.55$. Same cell (44), different denominators: the group you choose from is the denominator.

Example 3 (margin of error). A random sample of 500 voters in a city shows 54% support for a new park, with a margin of error of 3 percentage points. Appropriate conclusion: it is plausible that between 51% and 57% of all voters in that city support the park. Inappropriate conclusions: "exactly 54% support it," or anything about voters in other cities.

Test-taking strategy

• Find the denominator first in every probability question. Underline the phrase that names the group ("of the students in Grade 12...").
• Desmos statistics. Enter a list such as L=[12,15,15,18,20,22,95], then type mean(L), median(L), or stdev(L) on new lines. This avoids arithmetic errors on long lists.
• Desmos regression. If a table of data is given, enter it in a Desmos table (columns $x_1$, $y_1$) and type y1 ~ m x1 + b to get a line of best fit, or y1 ~ a b^x1 for an exponential model.
• Compare spread by eye. For standard deviation questions, look for which data set has values clustered near the center (smaller) or pushed to the extremes (larger). Calculation is rarely necessary.
• Read claims word by word. "Causes," "all," and "proves" are red flags unless the study used random assignment or the claim stays within the sampled population.

Common traps

• Swapping the conditional. $P(\text{Grade 12} \mid \text{instrument})$ is not $P(\text{instrument} \mid \text{Grade 12})$.
• Overgeneralizing. A sample from one school, city, or species says nothing reliable about others.
• Confusing association with causation in observational studies.
• Treating a prediction as exact. A line of best fit gives predicted or estimated values; individual data points lie above or below it.
• Misreading graph scales. Check axis labels, units, and whether an axis starts at zero before comparing heights of bars or points.

Summary

Know how outliers affect the mean and median, read standard deviation as spread, and always choose the correct denominator in probability questions. Interpret the slope of a line of best fit as a predicted rate of change. Limit conclusions to the population that was randomly sampled, claim causation only with random assignment, and use the margin of error to describe a plausible range.