SAT Math Review · Problem-Solving and Data Analysis
Reading histograms, dot plots and box plots
Each display throws something away to show you something else. Knowing what is missing is half the question.
The histogram shows the distribution of the times, in minutes, that each of 45 callers spent on hold. The first bar represents times of at least 0 minutes but less than 4 minutes, and so on.
Which of the following can be determined from the histogram?
A histogram reports how many values landed in each interval and nothing else. “Less than 8 minutes” is exactly the first two intervals, so the count is 6 + 14 = 20 callers — a question the display answers precisely. A asks for one value inside an interval: 12 callers fall in the 8-to-12 interval, but the histogram never says how many of them were at exactly 10. B asks for the maximum: the last non-empty bar tells you the longest wait was somewhere between 16 and 20 minutes, not what it was. D needs every individual value to add up, and those were destroyed when the values were sorted into bars.
1
Three displays, three different bargains.
Every display trades detail for shape. Ask what it kept and what it threw away, and most questions answer themselves.
Display
Shows
Hides
dot plot
every individual value, one dot each
nothing — but it is unreadable past about 40 values
histogram
how many values fell in each interval
the individual values inside a bar
box plot
the five-number summary and the spread
the mean, the counts, and the shape within each quarter
dot plot → every value surviveshistogram → only the interval survivesbox plot → only five numbers survive
Read that middle column from the top down and you are losing information at every step. That is not a flaw: a dot plot of 2,000 salaries is a smear, while a box plot of the same 2,000 tells you in one glance where the middle half sits.
2
A histogram bar is a count, and the count is all you get.
The height of a bar, read off the vertical axis, is how many values landed in that interval. Here is the opening histogram with the five counts written on.
6 + 14 + 12 + 8 + 5 = 45
the bar heights add to the whole group
less than 8 minutes → 6 + 14 = 20 callers
two whole bars, so the count is exact
at least 12 minutes → 8 + 5 = 13 callers
the last two bars, again exact
less than 6 minutes → cannot be found
6 cuts a bar in half, and the bar does not say where its 14 values sit
The boundary belongs to the bar on its right
“At least 4 but less than 8” is the standard SAT wording, so a caller who waited exactly 8 minutes is counted in the third bar, not the second. That is why “less than 8” is a clean 20 and “8 or less” is not answerable. Whenever a cut-off falls on a boundary between two bars you can count; whenever it falls inside a bar you cannot.
3
A dot plot keeps every value, so you can count anything.
One dot is one member of the data set. Nothing has been grouped, so no question about the values is out of reach.
n = 1 + 3 + 5 + 4 + 3 + 1 + 1 = 18
count the dots, never the tick marks
at least 4 → 3 + 1 + 1 = 5 students
the value 4 itself is included
fewer than 2 → 1 + 3 = 4 students
“fewer than 2” stops before the 2s
median: 9th and 10th of 18 → (2 + 3) ÷ 2 = 2.5
a histogram could never give you this
“At least”, “more than”, “fewer than”, “at most”
These four words decide whether the dots sitting on the cut-off are in or out, and an SAT choice list almost always contains the answer you get by reading the wrong one. “At least 4” and “at most 4” both include the three dots sitting at 4; “more than 4” and “fewer than 4” both exclude them. Underline the phrase before you start counting.
4
A box plot is five numbers, and each section holds a quarter of the data.
Minimum, Q1, median, Q3, maximum. The box runs from Q1 to Q3, so the box holds the middle 50% of the values.
range = 68 − 16 = 52 years
the two whisker ends
IQR = 44 − 25 = 19 years
the two box edges — the width of the middle half
24 members ÷ 4 = 6 per section
about 6 under 25, about 12 inside the box
A long whisker is spread, not a crowd
The right whisker here covers 24 years and the left one only 9, but both hold about six members. The long whisker means those six are strung out from 44 to 68, not that there are more of them. And notice what is absent: the plot never shows the mean, never shows how many members there are unless the stem says so, and never shows whether the ages inside the box are bunched at 26 or at 43.
5
Running totals find the median interval; nothing finds the mean.
The hardest histogram items ask where the middle value sits. Add the bars left to right until you pass the middle position.
Interval (minutes)
Callers
Running total
0 to 4
6
6
4 to 8
14
20
8 to 12
12
32
12 to 16
8
40
16 to 20
5
45
n = 45, so the median is the 23rd value
(45 + 1) ÷ 2 = 23
the running total reaches 20, then 32
position 23 lands in the third bar
median is between 8 and 12 minutes
the interval is the answer; the exact value is gone
The mean cannot be pinned down at all, but it can be trapped. Push every value to the bottom of its interval, then to the top, and the true mean must lie between the two results.
This is the difference the hard items live on. “The median is less than 12” must be true, because position 23 is fixed by counting. “The mean is less than 10” only could be true, since the mean is anywhere from about 7.3 to 11.3. When a statement about a histogram depends on where values sit inside a bar, it can never be forced.
6
Pick the display that answers the question you were asked.
Some items hand you the question and ask which display would settle it. Match the question to the thing the display kept.
The question
Dot plot
Histogram
Box plot
How many values equal 7?
yes
no
no
How many values are between 10 and 20?
yes
yes, if 10 and 20 are bar edges
no
What is the exact mean?
yes
no
no
What is the median?
yes, exactly
the interval only
yes, exactly
What is the IQR?
yes, with work
no
yes, at a glance
Which of two groups is more spread out?
yes
yes
yes, and fastest
Two rows are worth memorising. Only the dot plot gives an exact mean. Only the box plot gives an IQR without counting.
Where the calculator earns its keep
Desmos will build the display for you if a stem hands you a raw list. Paste the values into a Desmos table, then plot histogram(L, 4), dotplot(L) or boxplot(L) to see the same data three ways. It is a good way to build the instinct for what each one hides — on the test itself the display is already drawn, and the work is reading it.
✓
Check: does the display actually contain the answer?
the total6 + 14 + 12 + 8 + 5 = 45bar heights match the stated group size ✓
the cut-off“less than 8” sits on a bar edgeso the count is exact ✓
the dot plot18 dots, median 2.5every value survived, so anything is countable ✓
the box plot16 ≤ 25 ≤ 32 ≤ 44 ≤ 68five numbers in order, four equal quarters ✓
Remember the flow
What did the display keep? · Count whole bars or whole stacks · Quarters, not crowds
If the question needs a value the display threw away, the answer is “cannot be determined”.
Now you try
Practice problem
The box plot summarizes the weights, in pounds, of the 40 dogs at an animal shelter.
Based on the box plot, which of the following statements is true?
The box runs from Q1 = 20 to Q3 = 50, and the box always holds the middle 50% of the data, so about 20 of the 40 dogs weigh between 20 and 50 pounds. A is the long-whisker trap: the right whisker covers 30 pounds and the left one only 10, but each whisker holds a quarter of the dogs — about 10 either way. C reads the median line as a mean; a box plot never shows the mean, and with a long right tail the mean here is probably above 30. D reads it as a mode; the median is the value that splits the list in half, which says nothing about which weight occurs most often.
Hint: go through the four choices asking one question of each — is this something a box plot keeps? Three of them ask for information that was thrown away when the 40 weights were reduced to five numbers.
