Digital SAT · Math · Unit 3 · Problem Solving & Data

Lesson 3.4 · Mean, Median, Mode, Range & Spread

On the test: 2–3 questions a test — often “which statement compares…” with no arithmetic at all

Big idea. Two kinds of numbers describe a data set. CENTER — where the data sits: mean (the balance point), median (the middle value), mode (the most common). SPREAD — how scattered it is: range, interquartile range, standard deviation. Most SAT questions ask how one change (an outlier, a shift) moves each of them.

🎛 Move it · the live dot plot

Add and remove dots. Watch which measures move and which hold still.

▲

▲ mean 3.50│ median 3.5mode 3, 4range 5SD ≈ 1.38

Try: add a dot at 12. The mean (▲) jumps toward it; the median (green line) moves at most half a step. Pile dots in the middle: the SD bar shrinks.

🧑‍🏫 Rule 1 · Mean = sum ÷ countstep 1 of 4

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📋 The question · SAT · May 2024 · Math 800 Book Q7
Masses (g) of 5 lemurs: 3,910; 3,910; 3,130; 3,950; 3,150. Mean mass?
(type the answer)
the question
$3910, 3910, 3130, 3950, 3150$
May 2024: mean mass of 5 lemurs.

🧑‍🏫 Rule 2 · Mean from a frequency tablestep 1 of 3

1
the idea
$\overset{x}{ˉ} = \frac{\sum ( value \times frequency )}{\sum frequency}$
Frequency 4 means the value appears 4 times.

🧑‍🏫 Rule 3 · Total = mean × count (combined groups)step 1 of 5

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📋 The question · SAT · June 2024 · Math 800 Book Q30
175 compounds have mean boiling point 167°C; 50 of them have mean 327°C. Mean of the other 125?
(type the answer)
the question
$175 at 167, 50 at 327$
June 2024: mean of the other 125?

🧑‍🏫 Rule 4 · Median: sort, then take the middlestep 1 of 5

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📋 The question · SAT · March US 2023 · Math 800 Book Q5
Toronto snowfall (cm), 2002–2010: 114.9, 129.6, 134.9, 162.6, 32.4, 114.1, 216.5, 89.0, 45.6. Median?
1. A)32.4
2. B)114.9
3. C)115.5
4. D)184.1
sort
$32.4, 45.6, 89, 114.1, 114.9, 129.6, \dots$
March US 2023 snowfall: the years are NOT in order of size — sort first.

🧑‍🏫 Rule 5 · Mode and rangestep 1 of 4

1
mode
$most frequent value$
The tallest column of dots or the biggest frequency. There can be more than one.

🎛 Move it · the Box Plot Lab

Press “Next move” to build the box plot one move at a time. Then change the dots, switch to an even count, and add the outlier.

add an outlier (40)

move 1 of 8 · The data, sorted

9 values on the number line: 3, 4, 5, 6, 7, 9, 11, 13, 16. A box plot is built from them in six moves.

min

Q₁

4.5

median

Q₃

max

IQR

7.5

Not on a box plot: the mean (here 8.22), how many values there are, and the exact values inside each section.

🧑‍🏫 Rule 6 · Build a box plot (odd count)step 1 of 7

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📋 The question · Gen Z Study · build it (odd count)
The data set is 3, 16, 5, 9, 4, 13, 7, 11, 6. What is the first quartile, $Q_{1}$ ?
(type the answer)
what a box plot is
$min, Q_{1}, median, Q_{3}, max$
A box plot draws FIVE numbers of a data set — nothing else. We build them from these 9 values.

🧑‍🏫 Rule 7 · Build a box plot (even count)step 1 of 5

1
📋 The question · Gen Z Study · build it (even count)
The data set is 2, 3, 5, 6, 8, 9, 11, 13, 15, 18. What is the third quartile, $Q_{3}$ ?
(type the answer)
sort
$2, 3, 5, 6, 8, ∣, 9, 11, 13, 15, 18$
10 values — an even count.

🧑‍🏫 Rule 8 · Read a box plotstep 1 of 3

1
📋 The question · SAT · May 2024 · Math 800 Book Q14
$figure from the test$
A box plot of units sold per day for 28 days. Minimum units sold per day?
(type the answer)
the whiskers
$min = left end, max = right end$
May 2024: the left whisker starts at 85, so the minimum is 85.

🧑‍🏫 Rule 9 · A quarter of the data in each sectionstep 1 of 4

1
four sections
$whisker ∣ half-box ∣ half-box ∣ whisker$
Each holds about 25 % of the values — that is what “quartile” means.

🧑‍🏫 Rule 10 · IQR, range and outliersstep 1 of 4

1
IQR
$I QR = Q_{3} - Q_{1} = 31 - 18 = 13$
The width of the box: how spread out the MIDDLE half is.

🧑‍🏫 Rule 11 · Compare two box plotsstep 1 of 4

1
📋 The question · Gen Z Study · compare two
Class A
Class B
The box plots summarize the test scores of class A and class B. Which statement must be true?
1. A)Class A has the greater median
2. B)Class B has the greater interquartile range
3. C)Class A has the greater range
4. D)Class B has the greater mean
center
$median_{A} = 30 > 28 = median_{B}$
Compare the LINES INSIDE the boxes. That statement must be true (A).

🧑‍🏫 Rule 12 · What a box plot cannot tell youstep 1 of 3

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📋 The question · Gen Z Study · what it cannot tell
From the box plot alone, which can be determined? I. the median II. the mean III. the number of values
1. A)I only
2. B)I and II
3. C)I and III
4. D)I, II and III
it shows
$min, Q_{1}, median, Q_{3}, max (\Rightarrow range, I QR)$
Five numbers and anything made from them.

Real question 1SAT · May 2024 · Math 800 Book Q14type the answer

$figure from the test$

A store recorded the number of units sold per day for 28 days (box plot). What is the minimum number of units sold per day?

🧑‍🏫 Rule 13 · SD measures distance from the meanstep 1 of 4

1
idea
$SD small ⟺ values bunched near the mean$
The SAT never makes you compute it — only compare.

🧑‍🏫 Rule 14 · Shift or scale the datastep 1 of 4

1
add a constant
$x + c$
Mean, median, mode all move by $c$ ; range and SD do NOT change (every distance stays the same).

🧑‍🏫 Rule 15 · Adding or removing an extreme valuestep 1 of 5

1
📋 The question · SAT · May US 2024 · Math 800 Book Q23
$figure from the test$
Orbital periods of 13 moons are shown. A moon with period 628 days is added (14 values). Which compares the means and medians?
1. A)Mean equal, median equal
2. B)Mean equal, median less
3. C)Mean less, median less
4. D)The mean of the new data set is less than the mean of the original, and the median of the new data set is equal to the median of the original.
the question
$+ 628 to periods around 720 - 732$
May US 2024: 13 moons, add one tiny value.

🧑‍🏫 Rule 16 · Skew: which is bigger, mean or median?step 1 of 4

1
right tail
$mean > median$
A few large values pull the mean up.

Real question 2SAT · March US 2023 · Math 800 Book Q3

$figure from the test$

The dot plot shows 9 values with mean $a$ and median $b$ . The value 46 is removed to make 8 values with mean $y$ and median $z$ . Which statement best compares them?

Take with you

▸Mean = sum ÷ count; total = mean × count (use totals to combine or split groups).
▸Median: SORT, middle value (odd) or average of the two middles (even); count down a frequency table.
▸Mode = most frequent; range = max − min; IQR = Q₃ − Q₁ (the box).
▸Box plot = five numbers. Q₁ and Q₃ are the medians of the lower and upper halves; each section holds a quarter of the data (a long section = spread out, not more values); no mean, no count.
▸SD = typical distance from the mean: bunched → small, spread to the ends → large.
▸Adding a constant shifts the centre only; multiplying scales everything.
▸An outlier pulls the mean, barely the median. Right tail: mean > median.

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