PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Problem-Solving and Data Analysis Syllabus

A price rises from $50 to $65. What is the percent increase?

((65 - 50)/50) * 100 = (15/50) * 100 = 30%.

How do you compute a value after a percent increase or decrease using a single multiplier?

For a p% increase multiply by (1 + p/100); for a p% decrease multiply by (1 - p/100). Example: a 20% increase means multiplying by 1.20.

An item costs $80 and is discounted 25%. What is the sale price using a multiplier?

80 * (1 - 0.25) = 80 * 0.75 = $60.

Why is a 20% increase followed by a 20% decrease NOT a return to the original value?

The two percents apply to different bases. Multiplying by 1.20 then 0.80 gives 0.96 of the original, a net 4% decrease.

How do you reverse a percent change to find the original price (e.g., a price after a 25% increase is $100)?

Divide by the multiplier, not subtract. Original = 100 / 1.25 = $80, because the new price equals original * 1.25.

How is simple interest calculated, and what do the variables represent?

I = P * r * t, where P is principal, r is the annual interest rate as a decimal, and t is time in years. The total amount is P + I.

How do you apply sales tax of 8% to a $50 purchase?

Multiply by 1.08: 50 * 1.08 = $54, where the 0.08 portion ($4) is the tax.

What is the mean (average) of a data set and how is it calculated?

The mean is the sum of all values divided by the number of values. It represents the balance point of the data.

What is the median, and how do you find it for an even number of data points?

The median is the middle value when data are ordered. For an even count, it is the average of the two middle values.

What is the mode of a data set?

The value (or values) that appears most frequently. A data set can have one mode, multiple modes, or no mode.

What is the range of a data set?

Range = maximum value - minimum value. It measures the total spread of the data.

How does an outlier affect the mean versus the median?

The mean is strongly pulled toward an outlier, while the median is resistant (robust) and changes little. The median is the better center for skewed data.

What does standard deviation measure?

It measures how spread out the values are around the mean. A larger standard deviation means greater variability; a smaller one means data cluster tightly near the mean.

On the PSAT, how do you compare the standard deviations of two data sets without calculating them?

Judge how spread out the values are from their mean: the set whose values are more dispersed (farther from the mean) has the larger standard deviation.

In a histogram, what do the horizontal axis and the height of each bar represent?

The horizontal axis shows intervals (bins) of a quantitative variable, and each bar's height shows the frequency (count) of data values in that interval.

How do you read the value of a quantity from a line graph at a given point?

Locate the point on the horizontal axis, move up to the line, then read straight across to the vertical axis to find the corresponding value.

In a scatterplot, what does a line of best fit (trend line) represent?

It models the overall linear relationship between two variables and is used to estimate or predict y-values; its slope gives the rate of change.

What is the difference between interpolation and extrapolation when using a model?

Interpolation estimates within the range of observed data (more reliable); extrapolation predicts beyond the observed range (less reliable).

What does the shape of a distribution being 'skewed right' tell you about the mean and median?

A right (positive) skew has a long right tail, pulling the mean greater than the median. (Left skew: mean less than median.)

When comparing two distributions, which features should you describe?

Center (mean/median), spread (range/standard deviation), shape (symmetric/skewed), and any outliers or gaps.

If two data sets have the same mean but different ranges, what does that indicate?

They have the same center but different variability; the set with the larger range (and likely larger standard deviation) is more spread out.

What is the basic formula for the probability of a single event?

P(event) = (number of favorable outcomes) / (total number of equally likely outcomes). It is a value between 0 and 1.

What is the probability of rolling a number greater than 4 on a standard six-sided die?

Favorable outcomes are 5 and 6, so P = 2/6 = 1/3.

What is the relationship between the probability of an event and its complement?

P(event) + P(not event) = 1, so P(not event) = 1 - P(event).

How do you find the probability of two independent events both occurring?

Multiply their individual probabilities: P(A and B) = P(A) * P(B).

In a two-way frequency table, what does each cell in the body of the table represent?

The count (frequency) of individuals belonging to a specific combination of the two categorical variables (one row category and one column category).

In a two-way table, what are the 'marginal' totals?

The row totals and column totals located in the margins; they give the overall count for each single category, and the grand total is in the bottom-right corner.

Using a two-way table, how do you find the conditional probability of a column category given a specific row?

Divide the cell count by that row's total (not the grand total), restricting the sample to only members of that row.

What makes a sample random, and why does random sampling matter for valid conclusions?

In a random sample every member of the population has an equal chance of selection. Randomness reduces bias so results can be generalized to the whole population.

Under what conditions can the results of a study be generalized to a population, and when can a cause-and-effect conclusion be drawn?

Results generalize to the population only when subjects are randomly selected from it; cause-and-effect can be concluded only when subjects are randomly assigned to treatment groups in a controlled experiment.