SAT (Scholastic Assessment Test) Math: Problem-Solving and Data Analysis Syllabus

How do you find the original amount when you know the amount after a percent increase? (e.g., after a 25% increase the price is $50)

Divide by the growth factor: original = 50 / 1.25 = $40.

How do you express the statement '12 is what percent of 48'?

Divide part by whole and multiply by 100: (12/48) × 100 = 25%.

How do you find the whole when a part and its percent are known? (e.g., 18 is 30% of what number?)

Divide the part by the decimal percent: 18 / 0.30 = 60.

What is the formula for the arithmetic mean (average)?

Mean = (sum of all values) / (number of values).

How do you find the median of a data set?

Order the values; the median is the middle value if there is an odd count, or the average of the two middle values if there is an even count.

What is the mode of a data set?

The mode is the value that occurs most frequently; a set can have no mode, one mode, or multiple modes.

What is the range of a data set and what does it measure?

Range = maximum value − minimum value. It is a simple measure of spread (total variability).

What does standard deviation measure, and what does a larger value indicate?

Standard deviation measures how spread out values are around the mean. A larger standard deviation means the data are more spread out (greater variability).

What is the difference between a measure of center and a measure of spread?

A measure of center (mean, median, mode) describes a typical value; a measure of spread (range, standard deviation, IQR) describes how dispersed the values are.

What is an outlier?

An outlier is a data value that is much larger or much smaller than the other values in the set.

Why is the median often preferred over the mean for skewed data or data with outliers?

The median is resistant to outliers and skew, so it better represents the typical value, while the mean is pulled toward extreme values.

How does adding a single very large outlier to a data set affect the mean and the range?

It increases (pulls up) the mean and increases the range, since both depend on extreme values.

If the same constant is added to every value in a data set, how do the mean and the standard deviation change?

The mean increases by that constant; the standard deviation (spread) is unchanged.

In a histogram, what do the horizontal axis and the bar heights represent?

The horizontal axis shows intervals (bins) of a numerical variable, and bar heights show the frequency (count) of values in each interval.

What does a scatterplot show, and what does its line of best fit represent?

A scatterplot shows the relationship between two quantitative variables. The line of best fit models the trend, predicting y from x; its slope is the predicted change in y per unit change in x.

In a box plot, what five values are displayed?

The five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.

What is the interquartile range (IQR) and how is it found?

IQR = Q3 − Q1. It is the spread of the middle 50% of the data and is resistant to outliers.

When comparing two data sets, which statistics compare center versus spread?

Compare centers using mean or median; compare spread (consistency) using range, IQR, or standard deviation.

If two data sets have the same mean but different standard deviations, what does that tell you?

They have the same typical value, but the set with the larger standard deviation is more spread out (less consistent).

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

P(event) = (number of favorable outcomes) / (total number of equally likely outcomes).

What is the range of possible probability values, and what do 0 and 1 mean?

Probabilities range from 0 to 1. P = 0 means impossible; P = 1 means certain.

How do you find the probability that an event does NOT happen (complement)?

P(not A) = 1 − P(A).

For independent events A and B, how do you find P(A and B)?

Multiply the probabilities: P(A and B) = P(A) × P(B).

For mutually exclusive events A and B, how do you find P(A or B)?

Add the probabilities: P(A or B) = P(A) + P(B), since they cannot occur together.

What is the general addition rule for P(A or B) when events can overlap?

P(A or B) = P(A) + P(B) − P(A and B), subtracting the overlap counted twice.

What is a sample statistic versus a population parameter?

A statistic is a numerical summary computed from a sample (e.g., sample mean); a parameter is the corresponding true value for the whole population.

What is required for a sample to allow valid generalization to a population?

The sample must be randomly selected (representative) from the population; random selection allows results to be generalized.

What does a margin of error indicate in a sample estimate?

It gives the range above and below the sample estimate within which the true population value is likely to fall; a larger sample generally yields a smaller margin of error.

When is it valid to conclude a cause-and-effect relationship from a study?

Only when subjects are randomly assigned to treatment groups in a controlled experiment; observational studies show association, not causation.

What is the difference between random selection and random assignment, and what does each allow?

Random selection (sampling) of subjects allows generalizing results to the population; random assignment to groups allows concluding cause and effect.