Worked examples — Range — definition and calculation

This page is a drill through every kind of range problem you can meet. We start from zero: the parent note told you that

Everything below is that one rule, pushed into every corner where a beginner slips.

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The scenario matrix

Before working anything, let us list every kind of case a range question can throw at you. Think of this as a checklist — by the end of the page every row has a worked example.

Cell	Case class	What makes it tricky
A	All positive numbers	The plain, warm-up case
B	Repeated values	"Does repetition change the extremes?"
C	Mixed positive & negative	Subtracting a negative flips a sign
D	All negative numbers	"Which of two negatives is bigger?"
E	Degenerate: all values equal	Range collapses to zero
F	Single data point	The smallest possible list
G	Outlier present	Formula is right but picture is misleading
H	Grouped / class-interval data	Extremes come from class boundaries
I	Word problem (real world)	You must extract the numbers first
J	Exam twist: work backwards	Range given, find a missing value

The figure below is a map of the whole matrix — a number line showing how each case sits.

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[!example] Example 1 — Cell A: all positive

Statement: Heights (cm) of 6 plants: $\{12,20,8,15,22,11\}$. Find the range.

Forecast: Guess the answer before reading on. What are the two dots at the ends?

1. Sort the numbers in your head, left to right on the line. $8,11,12,15,20,22$. Why this step? Sorting makes the smallest and biggest dots jump out — no formula juggling.
2. Read off the extremes. $x_{\min }=8$, $x_{\max }=22$. Why this step? Range only ever needs these two; the middle four never enter the arithmetic.
3. Subtract. $\text{Range}=22-8=14$ cm. Why this step? The rule says max minus min, and that difference is the length of the gap.

Verify: $14$ is positive ✓ (must be, since $22\ge 8$). Units are cm, same as the data ✓. Sanity: the tallest plant is $14$ cm taller than the shortest — plausible ✓.

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[!example] Example 2 — Cell B: repeated values

Statement: Hourly wages (₹): $\{150,150,175,200,150,225\}$. Find the range.

Forecast: Three of the values are $150$. Do the repeats push the minimum around? Guess.

1. List the distinct extremes. The smallest value present is $150$ (it appears three times, but it is still just "$150$"). The largest is $225$. Why this step? Range cares only which value is smallest/largest, not how many times it occurs. A repeated minimum is still the minimum.
2. Subtract. $\text{Range}=225-150=75$ ₹. Why this step? Same rule as always; repetition never enters it.

Verify: $75>0$ ✓, units ₹ ✓. Cross-check: if I delete two of the three $150$s the list is $\{150,175,200,225\}$ — extremes unchanged, range still $75$ ✓. This proves repeats are irrelevant.

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[!example] Example 3 — Cell C: mixed positive & negative

Statement: Winter temperatures (°C): $\{-5,-2,0,3,7\}$. Find the range.

Forecast: The minimum is a negative number. Do you add or subtract it? Guess first.

1. Identify extremes on the line. Leftmost dot is $-5$ (further left = smaller). Rightmost dot is $7$. So $x_{\min }=-5$, $x_{\max }=7$. Why this step? On the number line "smaller" literally means "further left", and $-5$ is left of everything else.
2. Substitute into the rule. $\text{Range}=7-(-5)$. Why this step? We never change the rule; we just plug the negative min in honestly.
3. Resolve the double sign. Subtracting a negative is the same as adding: $7-(-5)=7+5=12$ °C. Why this step? $a-(-b)=a+b$. Geometrically, the gap from $-5$ to $7$ is $5$ units to reach $0$ plus $7$ more — that is $12$ total.

The figure shows exactly this: the gap crosses zero, so its length is the two pieces added.

Verify: $12>0$ ✓. Count units on the line: $-5\to 0$ is $5$, $0\to 7$ is $7$, total $12$ ✓. Units °C ✓.

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[!example] Example 4 — Cell D: all negative

Statement: Freezer readings (°C): $\{-3,-11,-7,-1,-9\}$. Find the range.

Forecast: Everything is negative. Which one is the maximum? The trap is picking the "biggest-looking" digit.

1. Remember the rule for negatives: closer to zero = larger. So among $\{-3,-11,-7,-1,-9\}$ the largest is $-1$ (nearest zero), the smallest is $-11$ (furthest left). Why this step? $-1$ sits to the right of $-11$ on the line; right means bigger. The size of the digit lies to you here.
2. Substitute. $\text{Range}=x_{\max }-x_{\min }=(-1)-(-11)$. Why this step? Rule unchanged — max minus min, whatever their signs.
3. Simplify. $(-1)-(-11)=-1+11=10$ °C. Why this step? Subtracting the negative $-11$ adds $11$.

Verify: $10>0$ ✓ (as required). Gap on line from $-11$ to $-1$ is $10$ steps ✓. If I had wrongly done $-11-(-1)=-10$, the negative answer would flag the mistake instantly.

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[!example] Example 5 — Cell E & F: degenerate cases

Statement (E): $\{5,5,5,5\}$. Statement (F): a single reading $\{7\}$. Find each range.

Forecast: Can the range be zero? Can a one-number list even have a range?

1. Case E — all equal. Here $x_{\max }=5$ and $x_{\min }=5$ (same dot). Why this step? Every value is the same point on the line, so the leftmost and rightmost dots coincide.
2. Compute. $\text{Range}=5-5=0$. Why this step? No gap means no spread — the data has literally zero variability.
3. Case F — single point. With one value, $x_{\max }=x_{\min }=7$, so $\text{Range}=7-7=0$. Why this step? A lone dot cannot be spread from anything but itself; the gap is zero.

Verify: Both give $0$, which is the smallest allowed range (never negative) ✓. This is the "everyone is the same height" picture — no stretch at all. It matches Property 2 in the parent note.

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[!example] Example 6 — Cell G: outlier present

Statement: Daily visitors: $\{20,22,21,23,22,200\}$. Find the range and comment.

Forecast: The formula will run fine — but will the number describe the data honestly?

1. Apply the rule. $x_{\max }=200$, $x_{\min }=20$, so $\text{Range}=200-20=180$. Why this step? The rule is mechanical and gives $180$ correctly.
2. Now compute the range without the suspicious value $200$. Of $\{20,22,21,23,22\}$: $x_{\max }=23$, $x_{\min }=20$, range $=23-20=3$. Why this step? Comparing $180$ against $3$ exposes that one point is doing all the "stretching".
3. Report both. "Range $=180$, but this is dominated by a single outlier ($200$). Excluding it, the typical range is $3$." Why this step? Statistics is interpretation, not just computation. A correct number can still mislead.

Verify: $180>0$ ✓ and $3>0$ ✓. The huge gap between $180$ and $3$ confirms extreme outlier sensitivity — exactly why Interquartile Range and Standard Deviation exist as robust alternatives.

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[!example] Example 7 — Cell H: grouped data

Statement: A frequency table of marks:

Class interval	Frequency
$10\text{–}19$	4
$20\text{–}29$	9
$30\text{–}39$	6
$40\text{–}49$	2

Find the range.

Forecast: You don't have the raw numbers, only bins. Where do the extremes come from?

1. Take the boundaries of the outer classes. The data can be as low as the bottom of the lowest class ($10$) and as high as the top of the highest class ($49$). Why this step? With grouped data we no longer know individual values, so we use the outermost class limits as the best estimate of the extremes.
2. Subtract. $\text{Range}=49-10=39$ marks. Why this step? Same max-minus-min rule; the "max" and "min" are now the top and bottom class limits.

Verify: $39>0$ ✓, units marks ✓. Frequencies ($4,9,6,2$) never entered the arithmetic — correct, because range ignores how many fall in each bin, only how far the bins stretch. (Some textbooks use class boundaries $9.5$ and $49.5$, giving $40$; state which convention you use.)

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[!example] Example 8 — Cell I: real-world word problem

Statement: A shop recorded daily earnings over a week: Monday ₹1200, Tuesday ₹950, Wednesday ₹1450, Thursday ₹1100, Friday ₹1600, Saturday ₹2100, Sunday ₹800. On which two days does the range live, and what is it?

Forecast: The numbers are buried in words. Guess the best and worst day before computing.

1. Extract the dataset. $\{1200,950,1450,1100,1600,2100,800\}$. Why this step? You cannot find extremes until the numbers are pulled out of the sentences.
2. Find the extremes. Best day = Saturday $2100$, worst day = Sunday $800$. So $x_{\max }=2100$, $x_{\min }=800$. Why this step? Range questions in words always reduce to "highest minus lowest".
3. Subtract. $\text{Range}=2100-800=1300$ ₹. Why this step? The gap between the best and worst trading day is what "spread of earnings" means here.

Verify: $1300>0$ ✓, units ₹ ✓. The range "lives" between Saturday and Sunday — check both are genuinely the outer dots (nothing exceeds $2100$, nothing below $800$) ✓.

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[!example] Example 9 — Cell J: exam twist, work backwards

Statement: A dataset $\{14,x,9,23,17\}$ has range $30$, and $x$ is the largest value. Find $x$.

Forecast: You are given the range and must recover a missing number. Which equation do you invert?

1. Locate the current known extremes among the fixed numbers. Of $\{14,9,23,17\}$, the smallest is $9$. We are told $x$ is the largest, so $x_{\max }=x$ and $x_{\min }=9$. Why this step? To use the range rule we need to know which value plays "max" and which plays "min".
2. Write the rule as an equation. $\text{Range}=x_{\max }-x_{\min }$ becomes $30=x-9$. Why this step? The same rule now becomes a solvable equation with one unknown.
3. Solve for $x$. Add $9$ to both sides: $x=30+9=39$. Why this step? We invert the subtraction — the reverse of "max minus min" is "range plus min".

Verify: Put $x=39$ back: dataset $\{14,39,9,23,17\}$, extremes $39$ and $9$, range $=39-9=30$ ✓. And $39$ really is the largest, matching the condition ✓.

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[!recall]- Quick self-test on the matrix

Which cell does each match?

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[!mnemonic] One-line summary of the whole page

"Sort on the line, take the two ends, subtract — answer is always $\ge 0$." Every one of the ten cells above is that single sentence in a different disguise.

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Connections

• Range — definition and calculation — the parent rule these examples exercise
• Outliers — why Example 6 needs a caveat
• Interquartile Range · Standard Deviation · Variance — robust spread measures for when range misleads
• Measures of Central Tendency — pair spread with a centre for a full summary
• Data Visualization — range is the whisker span in a box plot