A high-frequency Problem-Solving type (4.3% of the bank), and an interpretive one rather than a computational one. Covers mean, median, mode and range, reading a frequency table so each row counts as often as its frequency says, why the mean chases outliers and the median ignores them, comparing standard deviations without ever calculating one, and reading histograms and box plots — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 12 of 19
4.3% of the question bank — 72 of 1675 questions
Objectives
Mean, median, range and standard deviation, read off a list, a frequency table, a histogram or a box plot. What makes this type different from the rest of the section is that it is rarely asking you to compute anything. It is asking what happens to a statistic when the data changes, or which of two data sets is more spread out — and both of those are answered by reasoning, not arithmetic.
One sentence answers most of this type: the mean chases outliers, and the median ignores them.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 72 tagged questions of this type in the site's bank
Section
Section 1
Concept
Mean, median, range and standard deviation, read off a list, a frequency table, a histogram or a box plot. What makes this type different from the rest of the section is that it is rarely asking you to compute anything. It is asking what happens to a statistic when the data changes, or which of two data sets is more spread out — and both of those are answered by reasoning, not arithmetic.
You will see it phrased in these ways:
That last phrasing is the one worth preparing for. You are never asked to calculate a standard deviation on the SAT — only to say which of two is bigger, which is a question about width.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): One-variable data: centre and spread: the tell, the move, and the trap
The red panel saves more time than it saves marks. Students who try to compute a standard deviation lose two minutes and still cannot finish, because the formula is not provided.
Prediction
The four measures answer different questions.
Predict first
Which measure tells you how SPREAD OUT a data set is?
Correct: Standard deviation
Why: Standard deviation measures typical distance from the mean, so it describes width. Mean and median both describe the CENTRE — where the data sits — and say nothing about how tightly it clusters. The mode is simply the most frequent value. Range also measures spread, but crudely, since it uses only the two extreme values and ignores everything in between.
Concept
Four shapes of question, and only the first involves much arithmetic.
| variant | what it wants | the move |
|---|---|---|
| Compute a statistic | a mean, median, mode or range | order the data first, then apply the definition |
| What happens if | the effect of a change on the mean and median | test the outlier; compare against the current centre |
| Compare two sets | which has the larger spread | look at the width, not the centre |
| Read a display | a value from a histogram or box plot | read the axis and the frequencies carefully |
The middle two are the most common and involve essentially no computation. That is the surprise of this type: the reasoning questions outnumber the arithmetic ones.
Definition probe
Sorting the four measures is most of the vocabulary.
Sort into buckets
Does each measure describe the centre or the spread?
Discrimination
A data set has median 50. Six changes are made, one at a time.
Sort into buckets
What happens to the MEAN?
Warm-up
Try it before the rules.
Discussion prompt
A data set is 4, 5, 6, 7, 8. Now the 8 is changed to 800. What happens to the mean and to the median?
Hint: Compute both before and after.
Answer:
Before: the mean is 30 over 5, which is 6, and the median is the middle value, 6.
After: the data is 4, 5, 6, 7, 800. The mean is 822 over 5, which is 164.4 — it has moved enormously.
The median is still 6. The middle value of the ordered list did not change at all, because only the largest value moved and it was already largest.
That is the whole idea of the type: the mean is a balance point, so a distant value drags it; the median only counts positions, so it barely notices.
This single comparison answers most what-happens questions on the test.
Pattern
Three steps, and for the most common variants you never compute anything.
Decide whether the question is about CENTRE, SPREAD, or a CHANGE.
Why: Centre means mean or median, spread means range or standard deviation, and change means a what-happens question. The three need different tools.
If a statistic is wanted, order the data first, and expand any frequency table.
Why: The median is defined by position, so an unordered list gives the wrong answer, and a frequency of 5 means that value appears five times.
If a comparison or a change is wanted, reason about position rather than computing.
Why: Ask whether the new value is above or below the current centre, and whether the data is getting wider or narrower. Both are answerable by inspection.
Step 3 is where the time is saved. Roughly half the questions on this type are answered by looking rather than calculating.
Section
Section 2
Concept
Mean is the balance point, median is the middle in order, mode is the most frequent, and range is largest minus smallest.
| measure | how to find it | what it tells you |
|---|---|---|
| Mean | add all values, divide by how many | the balance point |
| Median | order the data, take the middle | the middle value |
| Mode | the value appearing most often | the most common outcome |
| Range | largest minus smallest | the total width |
The median requires the data to be ordered first. Taking the middle of an unordered list is the most common arithmetic error in the type.
Concept
An extreme value pulls the mean strongly toward it and moves the median barely or not at all.
This one sentence answers the majority of what-happens questions, and it is worth being able to state and justify.
Prediction
One value moves a long way.
Predict first
The set 10, 12, 14, 16, 18 has its largest value changed from 18 to 180. What happens to the median?
Correct: it stays at 14
Why: The ordered list becomes 10, 12, 14, 16, 180, and the middle value is still 14. The median depends only on position, and the value that moved was already the largest, so it kept its position. The mean, by contrast, jumps from 14 to 46.4 — which is exactly the contrast this type is built on.
Concept
A value with frequency 7 appears seven times in the data, not once.
| value | frequency | contributes |
|---|---|---|
| 3 | 4 | 3 four times, so 12 to the total |
| 5 | 2 | 5 twice, so 10 to the total |
| 9 | 1 | 9 once |
| total | 7 values | sum 31, mean 31 over 7 |
Dividing by the number of rows rather than the total frequency is the standard frequency-table error, and it is invisible in the working.
Prediction
Count the data points, not the rows.
Predict first
A table shows value 2 with frequency 3, value 5 with frequency 1, and value 8 with frequency 4. How many data points are there?
Correct: 8
Why: The number of data points is the sum of the frequencies: 3 plus 1 plus 4, which is 8. There are only 3 rows, and answering 3 counts the rows instead — the standard frequency-table error. The 15 is the sum of the values, and 12 is one of the products, neither of which counts anything.
Concept
More spread means a larger standard deviation, and the SAT only ever asks you to compare.
That last point is worth holding: shifting data leaves its spread alone, while stretching data changes it.
Concept
A new value above the mean raises it, below the mean lowers it, and equal to it changes nothing.
You can answer almost every added-value question by comparing the new value to the current mean, with no arithmetic at all.
Prediction
What a constant does to spread.
Predict first
Every value in a data set is increased by 7. What happens to the standard deviation?
Correct: it is unchanged
Why: Adding a constant shifts every value and shifts the mean by the same amount, so every distance from the mean is exactly as it was. Standard deviation measures those distances, so it does not change. The mean does increase by 7 — the centre moves and the spread does not, which is precisely the distinction between the two kinds of measure.
Concept
The horizontal axis gives the values and the height of each bar gives how many data points fall in that interval.
That skew relationship is a compact way to answer which-is-larger questions: in a right-skewed set the mean exceeds the median.
Concept
The box spans the middle half of the data and the line inside it is the median.
The line in the box is the median, never the mean. Reading it as a mean is a reliable way to answer a comparison question wrongly.
Prediction
What the line inside the box is.
Predict first
On a box plot, what does the line inside the box represent?
Correct: the median
Why: The line inside the box is always the median, which is why the box is split at that point into two halves each containing 25 per cent of the data. A box plot shows no mean at all, so it cannot be used to compare means. The range is the full width from whisker tip to whisker tip, and the mode does not appear on a box plot in any form.
Two truths and a lie
Three of these statements about data are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: Mean and standard deviation are independent: the mean says where the data sits and the standard deviation says how wide it is. The sets 49, 50, 51 and 0, 50, 100 both have a mean of 50, and the second is vastly more spread out. This is the single most important idea in the type, because comparison questions ask about spread and offer the centres as distractors.
Check
Reason about direction before computing.
Check your understanding
The set 5, 6, 7, 8, 9 has mean 7 and median 7. The value 9 is changed to 45. Which statement is true?
Answer: A
Why: The ordered set becomes 5, 6, 7, 8, 45, and the middle value is still 7, so the median is unchanged. The sum rises from 35 to 71, so the mean rises from 7 to 14.2. The value that moved was already the largest, so it kept its position and the median never noticed.
Notice that the direction of both answers was available without any arithmetic. Only the exact new mean required computing, and the question did not ask for it.
Section
Section 3
Worked example
Find all four measures for the data set 7, 3, 9, 3, 8, 12.
Figure (svg): A number line marking the mode at three, the mean at seven and the median at seven point five
Order the data first: 3, 3, 7, 8, 9, 12.
Why: The median is defined by position, so ordering must happen before anything else.
Mean: the sum is 42, and there are 6 values, so 42 over 6 is 7.
Why: The mean adds every value and divides by the count.
Median: six values, so average the two middle ones, 7 and 8, giving 7.5. Mode is 3, and range is 12 minus 3, which is 9.
Why: An even count means the median falls between two values.
Note that the three measures of centre give three different numbers. That is normal, and questions exploit it by asking which is largest.
Verify: check the median lies between the two middle values and the mode appears most often.
Why: 7.5 sits between 7 and 8, and 3 is the only repeated value, so both are consistent.
Answer: mean 7, median 7.5, mode 3, range 9
Worked example
A table shows: value 2 with frequency 5, value 4 with frequency 3, value 9 with frequency 2. Find the mean and the median.
Figure (svg): A frequency table with the products and running counts worked out
Count the data points: the frequencies sum to 5 plus 3 plus 2, which is 10.
Why: The number of values is the total frequency, not the number of rows.
Mean: multiply each value by its frequency and add, giving 10 plus 12 plus 18, which is 40. Then 40 over 10 is 4.
Why: Each value contributes as many times as its frequency says.
Median: with 10 values the middle is the average of the 5th and 6th. The 5th is 2 and the 6th is 4, so the median is 3.
Why: Counting through the running totals locates the middle positions.
Dividing 40 by 3 rows instead of 10 values would give about 13.3 — a number nowhere near the data, which a glance at the values would flag.
Verify: check the median is below the mean, consistent with a tail of high values.
Why: The two 9s pull the mean up while leaving the median low, which matches a right skew.
Answer: mean 4, median 3
Worked example
Set A is 48, 49, 50, 51, 52. Set B is 10, 30, 50, 70, 90. Both have mean 50. Which has the larger standard deviation?
Figure (svg): Two dot plots with the same mean, one tightly clustered and one widely spread
Note that both means are 50, so the centres are identical and tell you nothing.
Why: Standard deviation is about spread, and the centre is irrelevant to it.
Look at the distances from the mean: Set A's values are within 2, and Set B's reach 40 away.
Why: Standard deviation is the typical distance from the mean.
So Set B has the much larger standard deviation.
Why: Values further from the mean give a larger typical distance.
No formula was used and none was needed. Comparison questions are answered by looking at how far the values sit from the centre.
Verify: check the ranges: Set A has range 4 and Set B has range 80.
Why: Range is a cruder measure of the same thing and points the same way, confirming the comparison.
Answer: Set B, by a wide margin
Step zero
Before computing anything.
Discussion prompt
A question shows two dot plots and asks which data set has the greater standard deviation. What do you do, and what do you deliberately not do?
Hint: One of these is a two-minute mistake.
Answer:
Do: look at how far the values sit from their centres. The set whose points reach further from the middle has the larger standard deviation.
Do not: attempt to calculate it. The formula is not provided, the arithmetic is long, and the question never required it.
Ignore the centres entirely. Two sets can have identical means and completely different spreads, so comparing the means answers a different question.
A quick proxy: compare the ranges, or how tightly the points cluster. Both point the same way as the standard deviation in almost every SAT case.
This is the one place on the Math section where the correct method is explicitly to not compute.
Worked example
The set 20, 22, 24, 26, 28 has mean 24 and median 24. A value of 90 is added. What happens to each?
Figure (svg): Bars showing the mean jumping to thirty-five while the median moves only to twenty-five
The new value 90 is far above the mean of 24, so the mean rises.
Why: Adding a value above the mean pulls the balance point toward it.
Compute it: the sum was 120, now 210, over 6 values, giving 35.
Why: The mean has moved 11 units on the strength of one value.
The median: the ordered set is 20, 22, 24, 26, 28, 90, so the median is the average of 24 and 26, which is 25.
Why: With six values the median falls between the third and fourth.
Notice you could have answered the direction of both without any arithmetic. Only the exact values needed computing.
Verify: compare the movements: the mean moved 11 and the median moved 1.
Why: The eleven-fold difference is exactly the outlier effect this type tests.
Answer: the mean rises from 24 to 35; the median rises only from 24 to 25
Worked example
A box plot has whiskers at 12 and 68, box edges at 28 and 52, and a line at 40. Give the median, the range and the interquartile range.
Figure (svg): A table naming each feature of a box plot and its position
The line inside the box is the median, so the median is 40.
Why: A box plot always marks the median inside the box, never the mean.
Range is maximum minus minimum: 68 minus 12, which is 56.
Why: The whiskers reach the extreme values.
Interquartile range is the box width: 52 minus 28, which is 24.
Why: The box spans from the first to the third quartile, holding the middle 50 per cent.
The mean cannot be determined from a box plot at all. A question asking to compare means from box plots is asking something the display does not show.
Verify: check the median lies inside the box and the box inside the whiskers.
Why: 40 is between 28 and 52, and the box is within 12 to 68, so the plot is consistent.
Answer: median 40, range 56, interquartile range 24
Faded example
From memory. Four lines that carry the type.
Fill in the blanks
The mean is dragged by an outlier, while the median barely moves. Standard deviation measures spread, not centre. And adding a constant to every value leaves the standard deviation unchanged.
Why: The last line is the one that separates a shift from a stretch. Adding a constant moves every value and the mean together, so the distances between them are untouched. Multiplying every value by a constant does change the spread, because the distances scale too.
Worked example
A histogram of household incomes has a long tail stretching to the right. Which is larger, the mean or the median?
Figure (svg): A dot plot with most values clustered low and a few stretching far to the right
A long right tail means a few very large values.
Why: Skewed right describes where the tail points, not where the bulk of the data sits.
Those large values pull the mean upward, while the median stays with the bulk of the data.
Why: The mean uses each value's size and the median only its position.
So the mean is larger than the median.
Why: The tail drags one measure and not the other.
This is why incomes and house prices are reported as medians. A handful of very large values would make the mean unrepresentative of a typical household.
Verify: check with the plotted values: the mean is 20.5 and the median is 15.5.
Why: The mean sits above the median, as the right skew predicts.
Answer: the mean is larger
Fill the middle
Each feature has one meaning.
Fill in the blanks
The line inside the box is the median. The two edges of the box are the quartiles. The distance across the box is the interquartile range. And the whisker tips give the minimum and maximum.
Why: The most consequential of these is the first: the line is always the median, and a box plot shows no mean at all. Questions that ask you to compare means from box plots are asking for something the display cannot provide, and no is not among the answer choices — so the real answer is usually about medians instead.
Estimation
The mean must land inside the data.
Predict first
A data set is 41, 44, 46, 47, 52. Roughly what is the mean?
Correct: about 46
Why: The mean of any data set lies between its smallest and largest values, so it must be between 41 and 52 — which eliminates three choices immediately. The values cluster near the middle forties, and the exact mean is 46. This bound is worth carrying as a check: a mean outside the range of the data is always an arithmetic error, most often dividing by the wrong count.
Check
Count the data points before dividing.
| value | 3 | 5 | 8 | 10 |
|---|---|---|---|---|
| frequency | 6 | 4 | 2 | 8 |
Check your understanding
What is the mean of the data in the table?
Answer: A
Why: Multiply each value by its frequency: 3 times 6 is 18, 5 times 4 is 20, 8 times 2 is 16, and 10 times 8 is 80. Those sum to 134. The frequencies sum to 20, so there are 20 data points, and the mean is 134 over 20, which is 6.7. The answer lies between the smallest value 3 and the largest 10, as any mean must.
The two distractors here are the two standard frequency-table errors: ignoring the frequencies in the numerator, and counting rows in the denominator.
Section
Section 4
Trap
The trap. A question asks which of two data sets has the larger standard deviation, and you start computing means, deviations and squares.
Two minutes later you have half a calculation and no answer, and the formula was never provided in the first place.
The SAT does not ask you to calculate standard deviation. It only ever asks you to compare.
The fix. Treat it as a question about width and answer by inspection.
The set whose values sit further from the centre has the larger standard deviation.
Trap
The trap. The data is given as 7, 3, 9, 3, 8, and you take the middle of the list as written, answering 9.
The median is defined by position in the ORDERED data. Sorted, the set is 3, 3, 7, 8, 9, so the median is 7.
The error is invisible because taking the middle item genuinely is the method — on sorted data.
The fix. Write the data in order as your first action, before reading the question again.
Then count to the middle position.
Error analysis
A student computing a mean from a frequency table. One divisor is wrong.
Annotate
On: \( \text{values } 2,4,9 \text{ with frequencies } 5,3,2 \;\Rightarrow\; \frac{2 + 4 + 9}{3} = 5 \)
A frequency table is a compressed list. The first thing to do with one is decide how long the list it represents actually is.
Trap
The trap. A frequency table has 4 rows, and you divide the total by 4 to get the mean.
The number of data points is the SUM of the frequencies, which might be 30.
The resulting mean is far too large and usually sits outside the range of the data entirely.
The fix. Add the frequency column first and write the total down before doing anything else.
That total is your divisor, and it is also the count you use to locate the median.
Elimination
Set P is 30, 31, 32, 33, 34. Set Q is 2, 17, 32, 47, 62. Both have mean 32.
Eliminate the wrong options
Which statement is true? Three can be ruled out by inspection.
Survives elimination: b
Why: Set Q's values reach 30 units from the mean while Set P's reach only 2, so Q has by far the larger standard deviation. Choice C is the important distractor: equal means say nothing whatever about spread, and these two sets exist precisely to demonstrate that. No arithmetic was needed for any of the four eliminations.
Trap
The trap. Two box plots are shown and you compare the lines inside the boxes, calling them means.
Those lines are medians. A box plot does not display the mean anywhere.
The comparison you made may still be true of the medians, but it is not the statement you think you made.
The fix. Learn the five features of a box plot and what each is: minimum, first quartile, median, third quartile, maximum.
If a question asks about means and only box plots are given, the answer is usually that it cannot be determined — or the question is really about medians.
Counterexample
A claim that feels like it should follow.
Discussion prompt
A student says: if two data sets have the same mean and the same median, they must have the same spread. Give a counterexample.
Hint: Build two symmetric sets of different widths.
Answer:
Counterexample: 49, 50, 51 and 0, 50, 100. Both have mean 50 and median 50, and their spreads could hardly be more different.
The first has range 2 and the second has range 100.
The mechanism: mean and median both describe WHERE the data sits. Neither carries any information about how far the values stray from that point.
A second, sharper version: 50, 50, 50 and negative 1000, 50, 1050 also share both measures of centre, and one has zero spread.
Why it matters on the test: comparison questions are about spread, and the answer choices routinely offer statements about the centres. Recognising that they are independent is the whole skill.
Edge cases
The mean chases outliers and the median ignores them. Always?
Discussion prompt
When does an added value move the median noticeably, and when does the mean barely move at all?
Hint: Think about the number of data points.
Answer:
The median can move noticeably when the data set is small. Adding one value to a set of three shifts which value is in the middle, and with an even count the median becomes an average of two values that may not have been adjacent before.
The mean barely moves when the data set is large. Adding one extreme value to a set of a thousand changes the mean only slightly, because it is diluted by everything else.
So the rule is really about relative influence: the mean is always sensitive to size and the median to position, but how much either moves depends on how many values are already there.
On the SAT the data sets are small, so the classic contrast holds clearly, and the phrasing usually makes the outlier extreme enough that the direction is obvious.
The safest habit: state the DIRECTION of the change from the rule, and only compute if the question asks for the exact new value.
Check
Look, do not calculate.
Check your understanding
Set X is 20, 20, 20, 20, 20. Set Y is 18, 19, 20, 21, 22. Which statement is true?
Answer: A
Why: Every value in X equals the mean, so every distance from the mean is zero and the standard deviation of X is exactly zero — the smallest it can ever be. Set Y has values spread from 18 to 22, so its typical distance from the mean is positive. Y is therefore more spread out.
A data set of identical values is the extreme case worth remembering: standard deviation zero, range zero, and mean equal to every value.
Section
Section 5
Matching
Six statements, six measures.
Match the pairs
Why: The bottom three are all measures of spread, and they differ in how much of the data they use: range uses two values, interquartile range uses the middle half, and standard deviation uses every value. That is why range is the crudest and standard deviation the most informative.
Sorting
Each change is made to a whole data set.
Sort into buckets
What happens to the standard deviation?
The pattern worth carrying: shifting leaves spread alone, stretching changes it. That single distinction answers most standard-deviation questions on the test.
Comparison
Fill the blanks from memory.
Comparison matrix
| mean | median | |
|---|---|---|
| What it uses | every value's size | only position in the ordered list |
| Effect of an outlier | dragged strongly toward it | barely moves, often not at all |
| Needs the data ordered? | no | yes, always |
| In a right-skewed set | larger | smaller |
The bottom row is the compact way to answer skew questions. A long tail to the right pulls the mean above the median; a long tail to the left pulls it below.
Trade off
Fill in when each is the better summary.
Comparison matrix
| situation | better measure of centre | why |
|---|---|---|
| House prices in a city | median | a few very expensive houses distort the mean |
| Test scores in a class of 30 | mean | no extreme values, and it uses all the data |
| Most common shoe size sold | mode | the question is about frequency, not centre |
| Salaries including one executive | median | one enormous value pulls the mean above everyone |
The rule of thumb: use the median when the data has extreme values, and the mean when it does not. That is exactly why national statistics report median income rather than mean income.
Real world
One minute on why the mean-median distinction matters.
Discussion prompt
Why do governments report MEDIAN household income rather than mean household income, and how could someone mislead with the choice?
Answer:
Because income is strongly right-skewed. A small number of very high earners pull the mean well above what a typical household actually earns.
The median is the typical household — half earn more, half earn less — which is what the statistic is supposed to communicate.
How to mislead: quote the mean when you want the figure to sound high, and the median when you want it to sound low. Both are honest numbers describing the same data.
The same trick works with averages of anything skewed: house prices, wealth, city populations, response times.
The general lesson, and it is the SAT's too: a measure of centre is a choice, and which one you choose depends on whether the data has a tail. Knowing that both exist and behave differently is what stops you being misled.
Ranking
Order from smallest spread to largest. No calculation.
Put in order
Why: All four have a mean of 10. Set (a) has every value at the mean, so its standard deviation is exactly zero. Set (b) strays at most 1 from the mean, (c) strays up to 5, and (d) up to 9. The ordering is by how far the values reach from the centre, and it needs no arithmetic at all — which is exactly how the SAT expects these to be answered.
Concept
This type is 4.3 per cent of the section and roughly half its questions need no calculation, so the gains come from reasoning habits rather than arithmetic practice.
| session | what you do | why |
|---|---|---|
| 1 | Fifteen computations of mean, median, mode and range, always ordering the data first. | Makes the basic definitions automatic and kills the unordered-median error. |
| 2 | Ten frequency tables, writing the total frequency down before anything else. | The divisor error is the type's most common arithmetic failure. |
| 3 | Fifteen what-happens questions, stating the direction before computing anything. | Trains the outlier reasoning, which answers these in seconds. |
| 4 | Ten comparison questions answered by inspection, with a stopwatch. | Proves to yourself that standard deviation questions need no formula. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 72 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, explain why an outlier moves the mean but not the median, and say what happens to the standard deviation when a constant is added to every value.
Hint: One measure uses sizes and the other uses positions.
Answer:
The mean uses every value's SIZE, so a very large value contributes a very large amount and drags the balance point toward it.
The median uses only POSITION in the ordered list, so moving the largest value further out does not change which value sits in the middle.
Adding a constant to every value leaves the standard deviation unchanged, because the mean shifts by the same amount and every distance from it is preserved.
Multiplying every value by a constant does change it, because all those distances scale too.
If you gave the shift-versus-stretch distinction as well, you have the fact that answers most standard-deviation questions on the test.
Explain it
Two minutes, out loud.
Discussion prompt
A friend starts calculating standard deviations whenever a question mentions them, and runs out of time. What do you tell them?
Answer:
Tell them the formula is not provided, which is the strongest possible hint that it is not needed.
Show them what the questions actually ask: which set has the larger standard deviation, or what happens to it when the data changes. Both are comparisons.
Give the reading: standard deviation is how far the values sit from their centre. Wider means bigger. That is enough for every SAT question of this kind.
Show the two-set example: 49, 50, 51 against 0, 50, 100. Same mean, and anyone can see which is more spread out in a second.
Then give the discipline: if a question mentions standard deviation, spend ten seconds looking at widths and move on. Any longer means you have misread the question.
Commit first
Commit before you check.
Predict first
A distribution has a long tail stretching to the left. Which is true?
Correct: The mean is less than the median
Why: A long left tail means a few unusually SMALL values, and those drag the mean downward while the median stays with the bulk of the data. So the mean falls below the median. The memory hook is that the mean follows the tail: a right tail pulls it up, a left tail pulls it down. In a symmetric distribution the two are equal.
Connect it up
Blank paper.
Draw it
Draw a map of one-variable data. Split the page into CENTRE and SPREAD. Under centre, put mean and median side by side, and write beside each what it uses — sizes for the mean, positions for the median — and what an outlier does to it. Under spread, put range, interquartile range and standard deviation, ordered by how much of the data each uses. Draw a right-skewed distribution with the mean marked to the right of the median, and a left-skewed one with them reversed. In a box, write the two change rules: adding a constant leaves spread unchanged, multiplying changes it. At the bottom, write in large letters: NEVER CALCULATE A STANDARD DEVIATION.
Exit ticket
One question before you close the deck.
Predict first
A question shows two histograms with the same mean and asks which has the greater standard deviation. What do you do?
Correct: Compare how far the data reaches from the centre in each
Why: Standard deviation is a measure of width, so the set whose values spread further from the centre has the larger one — visible at a glance. Computing it is never required and the formula is not provided. The means are stated to be equal and in any case say nothing about spread. And it certainly can be determined: comparison by inspection is exactly what the question expects.
Recap
One type, one sentence: the mean chases outliers and the median ignores them.
| never do this | do this instead |
|---|---|
| Calculate a standard deviation | Compare how far each set reaches from its centre |
| Take the middle of an unordered list | Sort the data first, then count to the middle |
| Divide by the number of rows in a frequency table | Divide by the sum of the frequencies |
| Assume equal means mean equal spread | They are independent — check the widths |
| Read the box plot line as a mean | It is the median; a box plot shows no mean |
| Compute an exact value for a what-happens question | State the direction from the outlier rule |
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