SAT One-Variable Data: The Complete Guide

In a frequency table, multiply each value by its frequency before summing: $\bar{x}=\frac{\sum xf}{\sum f}$. The count is the sum of frequencies, not the number of rows. For the median, use cumulative frequencies to locate the middle observation or pair. Rewriting a small table as an ordered list is a reliable check.

In a dot plot, each dot is one observation at the horizontal value below it. Stacked dots show repetition, not larger measurement values. Count all dots, then count from the left to the middle position. Compare clusters, tails, and unusual points. A set can have the same median as another and still have a much larger range or standard deviation.

For example, a dot plot with two dots at 3, three at 4, and one at 9 represents the ordered list 3, 3, 4, 4, 4, 9. There are six observations, so the median averages positions three and four, giving 4. The mode is also 4, while the range is 6. Moving the single dot at 9 to 15 leaves the median and mode unchanged but increases the mean and range. This is a concrete example of why center and spread must be read separately.

For a histogram, read the interval boundaries and bar heights. With equal-width bins and a frequency axis, heights give counts in those intervals. A bin labeled 10 to less than 20 excludes 20. A histogram usually does not reveal exact observations, so an exact mean or median generally cannot be recovered. You can often locate the median's bin using cumulative counts.

A box plot marks the median inside a box from the first quartile $Q_1$ to the third quartile $Q_3$. The interquartile range is $Q_3-Q_1$, describing the middle half of the data. A longer segment means more numerical spread, not automatically more observations. Whiskers may show extrema or use an outlier convention, so read the problem's labels before treating their endpoints as minimum and maximum.

A box plot does not normally show the mean, modes, or sample size. If one box is wider than another, its middle half covers a wider interval; it does not follow that its mean is larger or that it contains more people. Use only the statistics actually marked in the display.

If six measurements have mean 12, their sum is 72. Adding a measurement of 19 produces a sum of 91 across seven observations, so the new mean is 13. Adding a value equal to the old mean leaves the mean unchanged. Adding a value above it increases the mean; adding a value below it decreases the mean.

Removing a value reverses the direction: removing a value above the mean lowers the mean, provided observations remain. But center and spread must be examined separately. Removing one extreme value can sharply reduce range, while removing a nonextreme value can leave range unchanged. For a median claim, test the ordered positions after the change rather than borrowing the mean rule.

An extreme observation can affect mean and range much more than median or interquartile range. But avoid absolute claims such as an outlier never changes the median. Replacing an existing extreme value farther outward may leave central positions unchanged; adding a new observation changes the count and may change which positions are central.

A mean of means requires weights. If one class has ten students and another has thirty, the combined mean is based on the sum of both classes' totals divided by forty students. Averaging the two class means equally would give a ten-student group the same influence as a thirty-student group.

Suppose nine measurements have mean 18. Removing one measurement leaves eight with mean 17. The original total was $9(18)=162$, and the remaining total is $8(17)=136$. The removed value is $162-136=26$. A difference of means alone, 1, is not the removed measurement because the counts differ.

For a box-plot comparison, read numerical features directly. If one set has quartiles 12 and 20 and another has quartiles 15 and 19, their interquartile ranges are 8 and 4. You can compare middle-half spread exactly. You cannot infer which mean is greater without more information, even if one median is marked farther right.

When a question asks what must be true after a change, construct a small counterexample to test broad claims. Adding a large value always increases the mean when it exceeds the old mean, but the median may remain the same if central values repeat. A single valid counterexample rules out a claimed guarantee. This is often faster than trying to derive every possible new distribution.

Plan about 25 minutes. First, use five minutes to read a frequency table in three ways: count observations, calculate a weighted total, and locate central positions. Then spend ten minutes on Solar one-variable-data questions. Alternate exact-value questions with conceptual spread comparisons so you do not treat every display as a request for arithmetic.

Use another five minutes to sketch a dot plot, histogram, and box plot of a small collection. Write one question each display answers and one it cannot answer exactly. The distinction matters: a histogram can identify an interval containing a median without identifying the median's exact value, while an exact dot plot preserves individual observations.

Finish by explaining one wrong answer from a missed question. On a later day, revisit additions and removals without reading the old solution. Time pressure should shorten repeated arithmetic, not erase the initial count or the check that the available data supports an exact answer.