Medium: Problem Solving and Data Analysis

At the Medium level the data questions stop being one-step. A percent is applied and then another percent; a rate must be converted before it is used; a scatterplot comes with a line of best fit and asks about the gap between a point and the line; a two-way table asks a probability about a subgroup, not the whole; two distributions must be compared by spread, not center; a survey result comes with a margin of error and asks what it means.

Each of these has a standard move. Percent changes are multipliers you can chain. Rates convert by cancelling units. A line of best fit is read like any line, and the distance from a data point to it is actual minus predicted. Conditional probability shrinks the denominator to the given group. Standard deviation is spread around the mean, larger when values sit farther from it. A margin of error turns one sample statistic into a plausible range for the population.

The Advanced lesson pushes each of these further: weighted means, the effect of changing one value on mean and median, exponential versus linear fits, and the difference between what an experiment and an observational study can prove.

A percent change is a multiplication. Increasing a quantity by r percent multiplies it by 1 + r/100; decreasing it multiplies by 1 - r/100. Because they are multiplications, successive changes combine by multiplying: a 20 percent increase followed by a 25 percent decrease is 1.20 x 0.75 = 0.90, a net decrease of 10 percent, regardless of the starting value. The reverse question, finding the original from the changed value, is a division by the multiplier: a jacket that costs 68 dollars after a 15 percent discount cost 68 / 0.85 = 80 dollars. The percent change between two values is the difference divided by the original value: from 40,000 to 46,000 is an increase of 6,000 / 40,000 = 0.15 = 15 percent. A percent of a percent is common in context: if 40 percent of a school's students play a sport and 25 percent of those play soccer, then 0.25 x 0.40 = 0.10, so 10 percent of the school plays soccer.

Rates and unit conversions at this level take more than one step. Speed in kilometers per hour becomes meters per second by converting each unit in turn: 72 km/h = 72,000 m / 3,600 s = 20 m/s. Density is mass per volume, so mass is density times volume and volume is mass over density. A question that gives a rate and asks for a total over a time, or gives two rates and asks which is faster, is solved by writing each rate with units and cancelling. The units are the method: if the unit you want is left after cancelling, the arithmetic was set up correctly.

A scatterplot shows paired data, one dot per observation. The association is positive if the dots rise left to right, negative if they fall, and its strength is how tightly they cluster around a line or curve. A line of best fit is the line that summarizes a linear trend; the SAT gives it as a drawn line or an equation such as y = 3.2x + 15. Its slope is the predicted change in y per unit of x, in the units of the axes; its y-intercept is the predicted value at x = 0, which may or may not be meaningful. For a particular data point, the predicted value is the line's y at that x and the actual value is the dot's y; the difference actual minus predicted is the residual, positive for points above the line. Predictions inside the range of the data are interpolation and are usually reasonable; predictions far outside it are extrapolation and are not to be trusted, a fact the test sometimes asks about directly.

A two-way table sorts a group by two categorical variables. The probability that a randomly chosen member has property A is the count with A divided by the grand total. A conditional probability changes the group: the probability of A given B is the count with both A and B divided by the count with B. In a table of 150 students where 24 of 60 juniors and 45 of 90 seniors have jobs, the probability a random student is a junior with a job is 24/150, but the probability a randomly chosen student who has a job is a junior is 24/69, because 69 students have jobs. Relative frequencies are these same fractions expressed as decimals or percents, and a table may give them instead of counts.

Measures of spread describe how far the data extend. The range is the largest minus the smallest value and is sensitive to a single outlier. The standard deviation measures the typical distance of values from the mean: data bunched near the mean have a small standard deviation, data spread far from it have a large one. Adding the same constant to every value shifts the mean and median but leaves the range and standard deviation unchanged; multiplying every value by a constant scales all of them. Comparing two dot plots or histograms, the one whose values are more concentrated near its center has the smaller standard deviation, whatever its mean. An outlier pulls the mean toward it and inflates the range and standard deviation while barely moving the median, so a distribution with a long tail has its mean pulled toward the tail.

Statistics from a sample are used to estimate a population. The estimate is trustworthy only if the sample was selected at random from the population in question; a survey of people leaving a gym says something about gym-goers, not about all residents of the city. A margin of error gives the uncertainty: if a random sample of 500 residents finds 62 percent support for a proposal with a margin of error of 4 percentage points, the plausible values for the whole population's support run from 58 to 66 percent. The margin of error shrinks as the sample grows and depends on the sample, not on the population size. It says nothing about the individuals sampled, only about the estimate for the population, and it does not mean the true value is certainly inside the interval.

Adding or subtracting successive percents instead of multiplying their multipliers: up 20 then down 25 is a 10 percent decrease, not a 5 percent one.

Dividing a change by the new value instead of the original when computing percent change.

Reading the slope of a line of best fit as a value of y rather than a rate of change of y per unit of x.

Computing predicted minus actual when the question asks how much the actual exceeds the predicted, and getting the sign backwards.

Using the grand total as the denominator of a conditional probability. "Of those who" and "given that" name the denominator.

Judging standard deviation by the number of data points or by the mean. It is about distance from the mean.

Applying a margin of error to the sample instead of the population, or believing a larger sample gives a larger margin of error.

Generalizing from a sample to a population the sample was not drawn from.