Mean, Median, and Standard Deviation

A data set is a list of numbers, and a statistic is one number that summarizes the list. The mean and the median both describe where the center of the data sits, but they answer that question in different ways: the mean balances the values, the median splits them in half. The range and the standard deviation describe how spread out the values are. Almost every question in this topic is a question about which of these numbers moves, and by how much, when the data change.

On the digital SAT these questions live in the Problem-Solving and Data Analysis domain and usually come with a short list, a frequency table, or a pair of dot plots. You will be asked to find a median from an even-length list, to find the score needed on one more test to reach a target average, or to say which of two distributions has the greater standard deviation. The ACT asks the same things with a little more arithmetic, including means from frequency tables and missing values in a list with a known average.

The standard deviation deserves one reassurance up front: you will never be asked to compute it. The tests only ever ask you to compare two spreads, and that is a judgment you can make by looking at how far the values sit from the mean. The computation that does matter is the one behind every mean question, namely that a mean times a count is a sum. Once you can move between means and totals without hesitation, the rest of the topic is bookkeeping.

The mean of a data set is its sum divided by its count. Think of it as a balance point: if you laid the values out on a number line as equal weights, the mean is where the line would balance, which is why every value pulls on it in proportion to how far away it sits. The mean carries one algebraic fact that matters more than any other on these tests: mean times count equals sum. Any question that gives you a mean and a count is really handing you the total, and any question that asks for a mean is really asking for a total divided by a count. Rewriting the problem in terms of the sum is the first move in almost every mean question.

The median is the middle value once the data are written in order from least to greatest, and ordering is not optional. If the count is odd there is a single middle value; with 7 values it is the 4th. If the count is even there are two middle values and the median is their average; with 8 values it is the average of the 4th and 5th, and it may be a number that does not appear in the list at all. In general the median sits at position (n + 1) / 2, so for n = 6 that is position 3.5, meaning halfway between the 3rd and 4th values. Two more summary numbers travel with these. The mode is the value that occurs most often, and a set can have no mode, one mode, or several. The range is the largest value minus the smallest, so it depends on only the two extreme values and ignores everything in between.

The mean and the median respond differently to an outlier, a value far from the rest. Because the mean is a balance point, an extreme value drags it toward that extreme; because the median depends only on position in the ordered list, an extreme value counts as one more value on one side and nothing more. In the data set 12, 14, 15, 15, 16, 18 both the mean and the median are 15. Replace the 18 with 60 and the mean jumps to 22 while the median stays at 15, because 60 is still simply the largest value. That is the whole story behind a rule the tests check constantly: a high outlier pushes the mean above the median, a low outlier pulls the mean below the median, and in a roughly symmetric data set the two agree. When a question asks which statistic would change if the largest value were doubled, the answer is the mean and the range; the median and usually the mode do not move.

The standard deviation measures spread. Informally, it is the typical distance of a value from the mean, computed from every value in the set. A set whose values cluster tightly around the mean has a small standard deviation, a set whose values scatter far from it has a large one, and a set whose values are all identical has a standard deviation of exactly zero. Neither the SAT nor the ACT asks you to compute it. They ask you to compare two data sets, usually shown as dot plots, histograms, or short lists, and to say which has the larger standard deviation. To do that, locate the mean of each set by eye and ask how far the values sit from it. Values piled up at the mean lower the standard deviation; values piled up at the extremes raise it. The size of the mean itself is irrelevant to the comparison, and so is the count. Two sets with the same range can have different standard deviations if one keeps most of its values near the center while the other pushes most of them to the ends.

Changing the data changes the statistics in predictable ways, and questions test whether you know which ones move. Adding the same constant to every value slides the whole set along the number line: the mean, median, and mode all increase by that constant, while the range and the standard deviation do not change at all, because every value is exactly as far from every other value as it was before. Multiplying every value by a positive constant k multiplies the mean, the median, the range, and the standard deviation all by k. Adding a new value above the current mean raises the mean, adding one below it lowers the mean, and adding a value equal to the mean leaves the mean alone while slightly shrinking the standard deviation, since the set now has one more value at distance zero. Removing an outlier moves the mean toward the remaining values, shrinks the range and the standard deviation sharply, and usually leaves the median close to where it was.

Working backwards from a mean is the most common calculation in this topic. If four test scores of 82, 90, 77, and 85 need to average 85 after a fifth test, the fifth score x satisfies $\frac{82 + 90 + 77 + 85 + x}{5} = 85$. Do not try to average the averages. Multiply the target mean by the final count to get the required total, 85 times 5 = 425, subtract the total you already have, 334, and the difference, 91, is the needed score. The same total-first thinking handles a missing value in any list and handles combined groups: the mean of two groups combined is the combined sum divided by the combined count, which equals the average of the two means only when the groups are the same size or the two means happen to be equal.

Finding the median without sorting. The median is the middle of the ordered list, not the middle of the list as written. A list such as 5, 12, 8, 3, 12, 14 has a median of 10, not 8 or 12. Rewrite the values in order every time, and count them so you know whether you need one middle value or the average of two.

Averaging averages when working backwards. If three quizzes average 78 and the target over four quizzes is 80, the fourth quiz does not need an 82, and it does not need an 80. The earlier quizzes are each 2 points short, so the shortfall is 6, and the fourth score must be 86. Convert every mean into a total with sum = mean times count, then subtract totals. This also protects you when two groups of different sizes are combined, since the combined mean is not the average of the two means.

Assuming the same range means the same standard deviation. The range depends only on the two extreme values, while the standard deviation depends on how far every value sits from the mean. Two lists can share a mean and a range while one keeps most of its values near the center and the other keeps most of them at the ends; the second has the larger standard deviation. When comparing spread, look at where the values pile up, not at the endpoints.

Expecting the median to move whenever the data change. Doubling the largest value, replacing it with an outlier, or removing it changes the mean and the range, but the median depends only on the middle position of the ordered list and usually stays exactly where it was. The reverse mistake also appears: assuming the mean is unaffected by one extreme value. The mean is a balance point, and one far-off value can move it a long way.

Moving the spread when you shift the data. Adding the same constant to every value raises the mean and the median by that constant and leaves the range and the standard deviation unchanged, because the values are all the same distances apart as before. Only multiplying every value by a constant scales the spread. On a which-must-be-true question, check each statistic separately and ask whether the change moved the values relative to each other or only relative to the number line.