One-variable data: distributions, centre and spread

This skill point covers one-variable data: computing and comparing measures of centre (mean, median, mode) and spread (range, standard deviation), reading distributions off dot plots, frequency tables, histograms and box plots, and predicting how those statistics respond when the data changes. It does not cover relationships between two variables — scatterplots, lines of best fit and two-way tables are separate skills.

College Board publishes the domain weight, not the skill weight: Problem-Solving & Data Analysis is about 15% of the scored Math questions, roughly 5 to 7 of them. How those split across the six skills inside the domain is not published. The commonly repeated prep-industry figure of two or three one-variable-data items per form is an inference from released material, not an official number — treat it as a rough prior rather than something to plan a schedule around.

The test asks it in five recognisable shapes. Compute-from-a-display: a dot plot or frequency table and a request for the median or the mean. Compare-two-sets: two displays and a question about which has the larger standard deviation, or how the means compare. Predict-the-change: a value is added, removed or altered, and you must say which statistics move and in which direction. Read-the-shape: a histogram and a question about skew, or about which measure of centre best represents the data. And read-the-box: a box plot and a question about quartiles, interquartile range, or what fraction of the data lies in some interval.

One structural fact worth carrying into every item on this page: the arithmetic is genuinely easy, and the test knows it. So the difficulty is loaded into the reading — which statistic was asked for, which display can answer it, and whether the answer is a count or a percent. That is why the careless errors on this topic are not slips in computation. They are fluent, confident computations of the wrong quantity.

If you can look at a frequency table and produce the median without hesitating over whether to sort the rows, skip to the next block; nothing here will be new. If you cannot, this is the block that matters most on the page, and it starts from nothing.

The four basic measures, on the data set 4, 7, 7, 9, 13. The MEAN is the total divided by how many values there are: (4 + 7 + 7 + 9 + 13)/5 = 40/5 = 8. The MEDIAN is the value in the middle position once the values are sorted from smallest to largest: five values, so the third one, which is 7. The MODE is the value that appears most often: 7, which appears twice while everything else appears once. The RANGE is the largest value minus the smallest: 13 − 4 = 9.

Two rules about the median that decide more SAT items than anything else in this block. First, sort before you do anything — the middle of the list as printed is not the median unless the list happens to arrive sorted, and on the SAT it usually does not. Second, when there is an even number of values there is no single middle, so you average the two middle ones: for 2, 5, 9, 10 the median is (5 + 9)/2 = 7, and note that 7 is not one of the values in the set. That is allowed and it is a favourite answer choice.

The position formula, for when the list is long. Sort the n values. If n is odd, the median sits at position (n + 1)/2 — for 15 values, the 8th. If n is even, average the values at positions n/2 and n/2 + 1 — for 20 values, the 10th and the 11th.

A FREQUENCY TABLE and a DOT PLOT are the same object drawn two ways: each one tells you a value and how many times that value occurs. To get the mean, multiply each value by its frequency, add those products, and divide by the total frequency — divide by how many data points there are, never by how many rows the table has. To get the median, do not sort the rows: they are already in value order. Instead find the position you need from the total frequency, then walk down the cumulative count until you reach it.

A HISTOGRAM is a bar chart of frequencies over intervals rather than over single values: a bar spanning 10–14 with height 6 means six data points landed somewhere in 10 to 14, and it does not tell you where inside that interval any of them landed. So you generally cannot compute an exact mean or median from a histogram — what you can do is determine which interval the median falls in, by adding bar heights until you pass the middle position. Answer choices for those items are intervals, not numbers, which is the tell.

A BOX PLOT draws the five-number summary: the minimum, the first quartile Q1, the median, the third quartile Q3, and the maximum. The box runs from Q1 to Q3 with a line at the median, and the whiskers run out to the extremes. The four sections each contain about a quarter of the data — about 25% below Q1, about 25% between Q1 and the median, and so on. The INTERQUARTILE RANGE is Q3 − Q1, the width of the box, and it is a measure of spread that ignores the outer quarters entirely.

SPREAD, in two flavours. The RANGE is maximum minus minimum: one subtraction, and it uses exactly two of the values, so a single extreme point sets it. The STANDARD DEVIATION uses every value — it is, in plain language, the typical distance of a value from the mean of its own set. Larger standard deviation means the values sit farther from their mean on average; a standard deviation of 0 means every value is identical, which is the only way to get one. The SAT never asks you to compute a standard deviation. It asks you to compare two, or to say what happens to one.

SHAPE, in three words. A distribution is SYMMETRIC when the two halves roughly mirror each other. It is SKEWED RIGHT (or positively skewed) when there is a long thin tail stretching toward the high values, and SKEWED LEFT when the tail stretches toward the low values. The name comes from where the tail is, not from where most of the data sits — a right-skewed distribution has most of its data on the left, which is exactly why this gets called backwards under time pressure.

Outliers. An unusually extreme value moves the MEAN, because the mean is a total and the total absorbs the value at full size. It moves the RANGE, because if the outlier is the new maximum or minimum then it is one of the two numbers the range is made of. It moves the STANDARD DEVIATION, because that value now sits a long way from the mean. It generally does not move the MEDIAN or the MODE, because those are decided by position and by frequency, and one extreme value occupies exactly one position and appears once.

The direction is not always up. Students taught "outliers pull the mean up" get half the items right. A low outlier pulls the mean DOWN, and removing a low outlier pulls it UP — a data set of exam scores where one student scored 12 out of 100 has a mean below its median, and deleting that score raises the mean while leaving the median almost where it was. Read which end the outlier is on before deciding which way anything moves.

Skew and the two centres. When the tail runs right, the mean sits above the median. When the tail runs left, the mean sits below it. When the distribution is symmetric they are approximately equal. The line to carry is that the mean chases the tail — the mean is the one that can be dragged, so it ends up on the side the tail is on.

Which centre to report. When a distribution is roughly symmetric with no extreme values, the mean and the median say the same thing and the mean is the conventional choice. When there are outliers or a long tail, the MEDIAN is the better summary of a typical value, precisely because it is the one the tail cannot move. This is why published figures for house prices and salaries are medians and not means, and the SAT asks it in exactly that form: "which measure best represents the data?"

Adding a constant versus multiplying by one. If every value in a set has 4 added to it, the mean rises by 4 and the median rises by 4, but the RANGE and the STANDARD DEVIATION do not change at all — the whole set slid along the number line without changing its internal distances. If every value is multiplied by 3, then the mean, the median, the range and the standard deviation are all multiplied by 3, because every distance between values was tripled too. Shifting moves the centre; scaling moves the centre and the spread.

In Bluebook, the built-in Desmos calculator is available on every Math question, which is College Board's own published statement rather than a prep-industry claim. Typed as a list, it will return mean, median, total and standard deviation directly. Meridian's recommendation, not College Board's: treat it as a verification tool and a tie-breaker rather than a default, because typing a twenty-value data set out of a dot plot costs more time than the arithmetic saves — and it offers no protection whatsoever against computing the wrong statistic, which is where the points on this topic are actually lost. If you plan to lean on it, confirm it behaves the way you expect inside a Bluebook practice test, not on test day.

At 1500 none of the definitions are the problem. The points are going to edge cases in the invariance rules, to displays being asked for information they do not carry, and to fluent computation of a quantity nobody asked for. Start with the invariance case that surprises almost everyone: add one new value exactly equal to the current mean, and the mean does not move — but the standard deviation goes DOWN. The new point contributes a deviation of zero to the total squared deviation while increasing the number of values that total is divided by, so the average squared deviation falls. This holds whether the standard deviation is computed with n or with n − 1 in the denominator, so the distinction cannot rescue a wrong answer here. The companion fact: adding a value equal to the current median may or may not leave the median where it is, because adding one value changes n and therefore changes which position the median occupies.

Averaging averages. Class A has 20 students with a mean score of 80; class B has 30 students with a mean of 90; the mean for all 50 students is not 85. Recover the totals and then divide: (20 × 80) + (30 × 90) = 1,600 + 2,700 = 4,300 over 50 students, giving 86. The naive average of the two means is correct only when the two groups are the same size, and the SAT writes this item specifically with unequal groups. The general habit: a mean is a total divided by a count, so to combine means you must return to totals first. Medians do not combine at all — there is no arithmetic that recovers the median of a merged set from the two medians.

What a display cannot tell you. A box plot shows no mean, no mode, no sample size, and nothing about how the data is distributed inside any of its four sections — two wildly different data sets can share a five-number summary exactly. A histogram gives you the interval the median falls in, not the median. A dot plot does show every value, which makes it the only one of the three you can compute a mean from directly. When an answer choice asserts something a display physically cannot support, that is the answer being wrong, not you missing a step — and "cannot be determined from the information given" is a real credited answer on this topic.

Honesty about the skew rule. "Right tail means mean above median" is a reliable heuristic and it is true of every distribution the SAT uses, but it is not a theorem — statisticians have constructed right-skewed distributions where the mean sits below the median. The converse fails more accessibly: mean equal to median does not prove a distribution is symmetric, since two opposing skews can cancel in the total. Use the rule with confidence on the test; do not carry it out of the room as a proof.

The sample-versus-population question is a non-issue and it is worth saying so once. Statistics distinguishes dividing by n from dividing by n − 1, and the two give different standard deviations for the same data. College Board's framework describes standard deviation as a quantity you interpret and compare rather than compute, and released material bears that out — no item requires the arithmetic, so no item can turn on which denominator you used. Time spent memorising the formula is time not spent on the reading, which is where the item is actually decided.

Finally, the execution errors, which cost more at the top of the scale than any misconception does. Asked for the median and reporting the mean, or asked for the interquartile range and reporting Q3. Asked how many values exceed some threshold and reporting the percentage, or the reverse — box plot items give you percentages and need the sample size from the question stem to become counts. Reading a frequency axis marked in steps of 2 or 5 as though it were marked in steps of 1. Converting a dot plot to a list and writing each value once instead of once per dot. Assuming an added value changes the median without checking whether n went from odd to even. Removing a low outlier and expecting the mean to fall. None of these are gaps in knowledge, which is precisely why they survive to test day — they are caught by naming the statistic and its unit out loud before selecting, and by nothing else.