SAT Math Review · Problem-Solving and Data Analysis
Spread: range, standard deviation and comparing distributions
Two data sets can share a mean, a median and a range and still look nothing alike. Spread is the word for the difference.
The dot plots show the 11 values in each of two data sets.
Which of the following statements about the standard deviations of data sets A and B is true?
Standard deviation measures how far the values sit from the mean. Both sets have a mean of 22 and both run from 20 to 24, but A piles nine of its eleven values on 21, 22 and 23, while B pushes eight of its eleven all the way out to 20 and 24. B's values are typically farther from 22, so B has the greater standard deviation. A reverses it, reading the tall middle stack of A as “more spread”; a tall stack is the opposite of spread. B and D are the two traps this lesson is built around: an equal mean says nothing about spread, and an equal range only fixes the two end values — it says nothing about the nine values in between.
1
The range is the crude measure: largest minus smallest.
Here are the commute times, in minutes, of 11 employees, already in order. Everything in this lesson comes back to this list.
1216202224262933364048
range = maximum − minimum
the definition, and the whole method
= 48 − 12 = 36 minutes
one subtraction, no ordering needed beyond the ends
The range looks at two values and ignores the other nine
Change the 48 to 148 and the range leaps from 36 to 136 while every other commute is untouched. That is why the opening problem's two data sets have the same range and nothing else in common: the range fixes the two ends of the picture and leaves the middle free.
2
Quartiles cut the sorted list into four equal pieces.
Find the median first. Q1 is the median of the values below it, Q3 the median of the values above it, and IQR = Q3 − Q1.
n = 11, so the median is the 6th value
five below, five above
median = 26
also written Q2
lower half: 12, 16, 20, 22, 24 → Q1 = 20
the middle of the five below
upper half: 29, 33, 36, 40, 48 → Q3 = 36
the middle of the five above
IQR = 36 − 20 = 16 minutes
the width of the middle half
range 36
the two extremes decide it
IQR 16
the extremes are thrown away
With an odd count, the median joins neither half
Eleven values split into 5 below, the median itself, and 5 above. With an even count there is no middle value to set aside, so the list splits cleanly down the middle and each half gets n2 values. Getting this wrong shifts Q1 and Q3 by one position and every answer after that.
3
A box plot is those five numbers, drawn to scale.
Minimum, Q1, median, Q3, maximum — the five-number summary. Nothing else is in the picture.
Part of the plot
Runs from
Holds
left whisker
12 to 20
the bottom 25%
the box
20 to 36
the middle 50%
right whisker
36 to 48
the top 25%
line inside the box
at 26
the median
A long whisker means spread, not more data
Every quarter of the plot holds the same number of values. The right whisker here is 12 minutes wide and the left one only 8, but each covers the same two or three employees. And a box plot never shows the mean, so a question asking you to compare means from box plots alone has no answer — that choice is there to be eliminated.
4
Standard deviation is the typical distance from the mean.
The SAT never asks you to compute a standard deviation. It asks you to look at two data sets and say which one is more spread out. Read that sentence twice — it is most of this lesson.
Here are five hourly customer counts: 4, 7, 8, 11 and 15. Their mean is 45 ÷ 5 = 9, and the picture below shows how far each value sits from it.
distances: 5, 2, 1, 2, 6
how far each value is from 9
average distance = 16 ÷ 5 = 3.2
a fair description of “typical”
standard deviation ≈ 3.7
the official version, which counts the far values more heavily
The real definition squares each distance, averages the squares and takes the square root, which is why the 5 and the 6 pull the answer above 3.2. You will not be asked for that number. What you will be asked is whether one set's distances are typically bigger than another's — and the picture answers that without a single calculation.
values huddled near the mean → small SDvalues scattered far → large SDevery value identical → SD = 0SD is never negative
5
Comparing two distributions: look at the clustering, not the centre.
Go back to the two dot plots from the opening problem. Both have a mean of 22 and both have a range of 4, and their standard deviations are not close.
Distance from 22
Set A: how many values
Set B: how many values
0
5
1
1
4
2
2
2
8
total distance
0 + 4 + 4 = 8
0 + 2 + 16 = 18
Eight of B's eleven values sit as far from the mean as the data allows, against two of A's. More than twice the total distance, on the same range — so B's standard deviation is the larger one.
Three facts settle almost every comparison item on the test:
If two data sets…
then their standard deviations…
have the same mean
may be anything — the mean says nothing about spread
have the same range
may still differ, as A and B do
differ only by a shift (every value +10)
are exactly equal
Sliding a data set does not stretch it
Add 10 to every commute time and the whole picture slides 10 minutes to the right: the mean rises from 27.8 to 37.8, and the median, Q1 and Q3 all rise by 10 too. But every value is still exactly as far from the mean as it was, so the range stays 36, the IQR stays 16 and the standard deviation does not move at all. “The mean changed, so the spread changed” is the wrong instinct here.
6
The four shapes this question takes.
Spread items on the Digital SAT are short. Recognise which of these four you are holding and the work is nearly done.
What the stem shows
What it wants
What to do
one dot plot or list
the range
maximum − minimum, once
one box plot
the IQR or the median
read the box edges, subtract
two dot plots or histograms
which SD is greater
the tighter cluster has the smaller SD
a table of means and SDs
which sentence is true
larger SD = more spread out from the mean, nothing more
On that last one, be strict about what the table supports. A larger standard deviation does not mean a larger mean, a larger median, a taller maximum or a single unusual value. It means the values are typically farther from their own mean.
Where the calculator earns its keep
Desmos will do the arithmetic if you want a check: type L = [12, 16, 20, 22, 24, 26, 29, 33, 36, 40, 48] and then stdev(L), quartile(L, 1) or median(L) on the next line. One thing to know if you check a number and it disagrees with ours: stdev is the sample standard deviation, and stdevp is the population one. They give different answers on the same list, and the SAT never asks you to compute either — only to compare two spreads. It is a safety net, not the method: the comparison items are built so that the eye beats the keyboard, and typing eleven numbers costs more time than the question is worth.
✓
Check: is each spread a sensible size?
IQR vs range16 ≤ 36the middle half can never be wider ✓
SD vs range3.7 well under 11an SD is always smaller than the range ✓
the comparisonB's dots sit at the endsfarther from the mean means the greater SD ✓
Remember the flow
Range = max − min · IQR = Q3 − Q1 · SD = typical distance from the mean
Compare spreads by eye. You are never asked to compute a standard deviation.
Now you try
Practice problem
Which statement is true?
Data set S consists of 20 values with a mean of 50 and a standard deviation of 6. Data set T is created by increasing each of the 20 values in S by 10. Which of the following describes the mean and the standard deviation of data set T?
Adding 10 to every value slides the whole data set 10 to the right. The mean slides with it, 50 + 10 = 60, but no value has moved relative to any other, so every distance from the mean is unchanged and the standard deviation is still 6. Test it on the two-value set 47, 53: mean 50, distances 3 and 3; after adding 10 it is 57, 63 — mean 60, distances still 3 and 3. B adds 10 to the spread as well, which would require the values to move apart, not along. C shifts the spread and leaves the mean, exactly backwards. D multiplies the standard deviation by 10 instead of leaving it alone; a standard deviation as large as the mean would need values scattered from about 0 to 120.
Hint: write down any tiny data set with mean 50 — say 44, 50, 56 — add 10 to each value and look at what happened to the gaps between them. Spread is about gaps, and a shift leaves every gap where it was.
