SAT Math — mean, median and spread, and what moves them

The mean and the median are both called averages, and they are built out of completely different things. The mean adds up every value, so the SIZE of each one matters to it. The median only asks which value sits in the middle, so only the ORDER matters. Almost everything the test asks about one-variable data follows from that difference, and this set is arranged to make it concrete rather than memorized.

Three questions change a data set and ask what happens. Adding one value ten times the size of the others moves the mean by fifteen and leaves the median exactly where it was. Adding a value equal to the mean moves nothing, because that value contributes precisely its fair share of the new total. Adding a value at the median leaves the middle where it was. Each explanation gives both figures before and after, so the difference is demonstrated rather than asserted, and each says why the structure of the two measures produces it.

Two questions read a median or a mean out of a distribution written as a sentence — how many households had each number of pets, how many people read each number of books. The trap in both is counting the CATEGORIES instead of the members, and both wrong options built from that mistake are named in the explanations. A median is a middle household, not a middle label, and a mean is weighted by how many people are in each group.

Two work backwards from an average, which is the most useful habit in this topic: a mean is a total shared out, so multiplying it by the count recovers the total and everything else is subtraction. One of them uses two different means and two different counts, which is harder and is exactly how the test likes to ask it.

Two concern spread, and neither requires calculating a standard deviation. One set has every value identical, so its spread is zero — the smallest any set can have. Another pair shares a mean of thirty while one set ranges from ten to fifty and the other from twenty-eight to thirty-two. The point of both is that a mean tells you nothing about spread, which is precisely why a second figure is needed.

One asks what a range of fifty guarantees, and the answer is: only that the largest value exceeds the smallest by fifty. It fixes no value, so a set from zero to fifty and a set from a thousand to a thousand and fifty share it, and a set running from minus thirty to twenty has that range with every value below fifty.

Every student, household and survey is invented, and nothing is drawn from any official publication.

• Explain why an extreme value moves the mean a long way and the median hardly at all
• Sort a list before reading a median, and use the right rule for an odd and an even count
• Read a median and a mean out of a distribution of counts without mistaking categories for members
• Work backwards from a mean to a total, including when two different means and counts are involved
• Compare the spread of two sets from their values alone, without calculating a standard deviation
• State exactly what a range does and does not tell you about a data set

Written for this catalog, with no source document. The content is the one-variable data material named in the College Board's own public description of the Problem-Solving and Data Analysis domain — measures of center and spread, and the effect of changing a data set; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test often presents this material as a histogram, a dot plot or a table on screen, which a text-only bank cannot reproduce: every distribution here is written out in a sentence instead. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an test center.