Line graphs and scatter plots — basic interpretation: Exercises

Before we start, one shared picture. Everything below refers to the two graph shapes you are reading:

Figure — Line graphs and scatter plots — basic interpretation

Figure s01. Left panel — a line graph: seven yellow dots (Days 1–7 on the x-axis, Temperature in °C on the y-axis from about 20 to 30) joined by a blue line in time order, so the line tells a story of change. Right panel — a scatter plot: seven loose green dots (Hours studied on the x-axis $0$ – $5$ , Test score $%$ on the y-axis $40$ – $85$ ), each dot a single observation of two things measured together; the red arrow flags the meaning of one dot. Keep this picture in mind — the exercises keep pointing back to it.

Level 1 — Recognition

Goal: just read what the graph literally says. No arithmetic tricks.

L1.1 — Read one value off a line graph

A line graph of daily maximum temperature (°C) has these points: Mon $22$ , Tue $25$ , Wed $28$ , Thu $24$ , Fri $30$ , Sat $27$ , Sun $23$ .

(a) What was the maximum temperature on Thursday? (b) On which day was it hottest?

Recall Solution

(a) Find "Thu" on the x-axis, slide straight up to the line, read across to the y-axis. The Thursday point is $(Thu, 24)$ , so the answer is $== 24° C ==$ . (b) The hottest day is the highest point on the graph. Scanning the numbers, the biggest is $30$ on Friday.

L1.2 — Read a point off a scatter plot

A scatter plot of (hours studied, test score $%$ ) contains the point $(3, 70)$ . What does this single dot mean in words?

Recall Solution

Each dot is one student with two measurements: x = hours, y = score. So $(3, 70)$ means "a student who studied $3$ hours scored $70%$ ." That is the whole meaning of one dot — no calculation, just translation.

L1.3 — Name the trend (all four shapes)

A trend is the general direction a line graph moves as you read left to right. There are four basic shapes:

Increasing — the line only rises (y goes up as x goes up).
Decreasing — the line only falls (y goes down as x goes up).
Constant — the line is flat (y never changes).
Fluctuating — the line goes up and down repeatedly, with no single overall direction.

The figure below shows all four so you can compare them directly:

(a) Match each of graphs A, B, C, D in Figure s04 to one of the four trend words. (b) In your own words, how is a fluctuating trend different from random noise on top of a straight line?

Recall Solution

(a) Reading Figure s04: A = increasing (blue line climbing steadily), B = decreasing (red line falling), C = constant (yellow flat line), D = fluctuating (green line zig-zagging up and down). (b) A fluctuating trend genuinely has no persistent direction — it repeatedly reverses (peaks and valleys), like the stock price in L4.3. Noise is different: it is small random wobble sitting on top of a clear overall direction (e.g. temperatures that generally rise all week but jiggle a degree or two day to day). Rule of thumb: if you can still draw one honest arrow for the whole graph, it is a trend with noise; if no single arrow fits, it is fluctuating.

Level 2 — Application

Goal: do arithmetic with the numbers you read — change, range, rate.

L2.1 — Range of a line graph

Using the same temperature week (Mon $22$ , Tue $25$ , Wed $28$ , Thu $24$ , Fri $30$ , Sat $27$ , Sun $23$ ), find the range.

Recall Solution

Range measures total spread: $Range = Max - Min$ Max $= 30$ (Fri), Min $= 22$ (Mon). $Range = 30 - 22 === 8° C ==$

L2.2 — Amount of change

A shop's sales (in ₹ thousands — that is, thousands of rupees) are: Month 1 = $12$ , Month 2 = $15$ , Month 3 = $21$ . By how much did sales increase from Month 1 to Month 3?

Recall Solution

WHY subtraction measures change: "How much did it increase?" asks for the gap between where you ended and where you started. On the y-axis, that gap is the vertical distance between the two heights — and the distance between two numbers on a number line is exactly the larger minus the smaller. So subtracting final − start literally measures the jump: $Month 3 - Month 1 = 21 - 12 === 9 == (₹ thousand)$ If we had added them ( $21 + 12$ ) we'd get a meaningless total that grows even when nothing changed — only the difference captures movement.

L2.3 — Rate of change (slope of a segment)

A plant's height is: Week 1 = $2$ cm, Week 4 = $11$ cm. Find the average growth rate in cm per week.

Recall Solution

WHY a ratio and not a subtraction? Subtraction gives total growth ( $11 - 2 = 9$ cm), but that hides how spread out in time it was. To compare speeds we need "per week", so we divide the vertical change by the horizontal change. First, the notation. The symbol $Δ$ (Greek capital "delta") means "the change in". We attach it to whichever quantity changed:

$Δ y$ = change in the y-value = (end y) − (start y) = $11 - 2$ (the rise in height).
$Δ x$ = change in the x-value = (end x) − (start x) = $4 - 1$ (the run in weeks).

So the two subscripts just tell you which axis the change lives on. Now divide rise by run: $rate = \frac{Δ y}{Δ x} = \frac{11 - 2}{4 - 1} = \frac{9}{3} === 3 == cm/week$

Level 3 — Analysis

Goal: compare, interpret, and judge — not just compute.

L3.1 — Which period grew faster?

Plant height: Week 1 = $2$ , Week 4 = $11$ , Week 8 = $35$ (cm). In which interval — Weeks 1→4 or Weeks 4→8 — did the plant grow faster?

Figure s02. Two joined line segments on axes "Week" (x, from 1 to 8) and "Height (cm)" (y, from 2 to 35). The blue segment runs from $(1, 2)$ to $(4, 11)$ and is labelled "3 cm/week"; the red segment runs from $(4, 11)$ to $(8, 35)$ and is labelled "6 cm/week". Arrows point out that the blue segment is a gentle slope (slower) and the red segment a steep slope (faster).

Recall Solution

Compare the two rates (the two slopes in the figure): $Weeks 1-4: \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3 cm/week$ $Weeks 4-8: \frac{35 - 11}{8 - 4} = \frac{24}{4} = 6 cm/week$ $6 > 3$ , so growth was faster in Weeks 4–8. Notice in the figure the second segment is steeper — a steeper line is a bigger rate. Growth is accelerating.

L3.2 — Classify the correlation (with a strength scale)

Now we make the words positive/negative and strong/weak precise.

Direction: if y tends to rise as x rises → positive correlation; if y tends to fall as x rises → negative correlation; if there's no lean → no correlation.
Strength: how tightly the dots hug a single straight line. Use this eyeball scale (it previews the numerical measure in 4.1.01-Correlation-coefficient, where strength becomes a number $r$ between $- 1$ and $+ 1$ ):

What you see	Strength	Rough $r$
Dots almost exactly on one line	strong	$
Dots lean the same way but scattered around the line	moderate	$0.4 \le
Dots barely lean; a cloud with a hint of tilt	weak	$
No tilt at all — round blob	none	$r \approx 0$

A scatter plot of temperature vs. ice cream sales (sales in ₹, rupees) has points: $(20, 300), (22, 400), (25, 500), (26, 550), (28, 650), (30, 800), (31, 900)$ . Describe the correlation (direction and strength), using the scale.

Recall Solution

Sort mentally by x and watch y: as temperature climbs $20 \to 31$ , sales climb $300 \to 900$ — y rises with x, so this is positive correlation. Now strength: every single point moves upward with almost no dot sitting off the upward line — they nearly sit on one straight line. By the table that is the top row, so it is strong (eyeball $∣ r ∣ \geq 0.8$ ). Direction = positive, strength = strong.

L3.3 — Spot the outlier

Age vs. reaction time (seconds): $(20, 0.25), (25, 0.26), (30, 0.28), (35, 0.30), (40, 0.33), (45, 0.35), (50, 0.40), (30, 0.60)$ . Which point is the outlier, and why?

Recall Solution

The overall pattern: reaction time creeps slightly up with age (around $0.25$ – $0.40$ ). The point $== (30, 0.60) ==$ breaks this — a $30$ -year-old at $0.60$ s is far slower than the other $30$ -year-old ( $0.28$ s) and even slower than the $50$ -year-old. It sits far off the trend, so it is the outlier. Possible causes: fatigue, distraction, or a measurement error worth checking.

Level 4 — Synthesis

Goal: build a line of reasoning — predict, and mix ideas together.

L4.1 — Interpolate a missing value

Ice cream data includes $(25, 500)$ and $(28, 650)$ — temperature in °C, sales in ₹ (rupees). Estimate sales at $27° C$ by interpolation.

Recall Solution

Interpolation = estimating between two known points by assuming the line is straight there. Slope between them: $\frac{650 - 500}{28 - 25} = \frac{150}{3} = 50$ (₹ per °C). From $25°$ to $27°$ is $+ 2$ degrees, so add $2 \times 50 = 100$ : $500 + 100 === 600 == (₹)$ So about ₹ $600$ at $27° C$ .

L4.2 — Build a line-of-best-fit prediction

For "hours studied vs. score", a best-fit line is $y = 8 x + 40$ (score in $%$ ). (a) What score does it predict for $5$ hours? (b) What does the slope $8$ mean in plain words?

Recall Solution

The best-fit line has the form $y = m x + c$ , where $m$ = slope (rate) and $c$ = y-intercept (value at $x = 0$ ). (a) Substitute $x = 5$ : $y = 8 (5) + 40 = 40 + 40 === 80 == %$ . (b) The slope $m = 8$ means: each extra hour studied is associated with about $+ 8$ points on the test, on average. The intercept $c = 40$ is the predicted score for someone who studied $0$ hours.

L4.3 — Combine range + trend into a sentence

Stock price over 10 days peaks at ₹ $62$ (Day 7) with a low of ₹ $48$ (Day 3), ending at ₹ $55$ (Day 10). Give the range, and write one sentence describing the story.

Recall Solution

$Range = 62 - 48 === 14 == (₹)$ Story: "The stock swung within a ₹ $14$ band, dipped to its low on Day 3, rose to a Day-7 peak of ₹ $62$ , then partly pulled back to ₹ $55$ — a fluctuating graph, not a steady trend."

Level 5 — Mastery

Goal: the full expert move — combine reading, computing, and critical judgement.

L5.1 — Correlation is not causation

A town's records show that months with higher ice cream sales also have more drowning incidents — a clear positive correlation. A newspaper writes: "Ice cream causes drowning." Explain, using the idea of a hidden variable, why this conclusion is unjustified. Would removing ice cream reduce drownings?

Recall Solution

Correlation means the two quantities move together; it does not mean one causes the other. Here a third (hidden) variable — hot weather — drives both: hot days → more people swim (more drownings) and hot days → more ice cream bought. Ice cream and drowning are both effects of heat, so they rise and fall together without any link between them. Removing ice cream would not reduce drownings, because ice cream was never the cause. The correct question is always: "Could a hidden third factor explain both?"

L5.2 — Misleading scale

Graph A shows a company's revenue on a y-axis running $0$ to $100$ ; Graph B shows the same data on a y-axis running $90$ to $100$ . Both display a change from $95$ to $97$ . (a) What is the actual change? (b) Why does Graph B look far more dramatic, and what should a careful reader do?

Figure s03. Two panels plotting the identical two points $(1, 95)$ and $(2, 97)$ against "Year" (x = 1, 2) and "Revenue" (y). Left (Graph A) — y-axis runs $0$ to $100$ ; the green line is nearly flat (a "tiny bump", per the red arrow). Right (Graph B) — y-axis runs $90$ to $100$ ; the same two points now make a red line that looks steep ("looks huge! same +2", per the yellow arrow). The only difference is the y-axis range.

Recall Solution

(a) Actual change $= 97 - 95 === 2 ==$ units — a small rise either way. (b) In Graph B the y-axis only spans $90$ – $100$ (a height of $10$ units), so $2$ units fills a huge fraction of the picture and the slope looks steep. In Graph A the axis spans $0$ – $100$ , so the same $2$ units is a tiny bump. Our eyes read the visual slope, not the numbers. The fix: always read the y-axis range first before trusting how "big" a rise looks.

L5.3 — Full graph read-out

Ice cream data (temperature °C, sales ₹): $(20, 300), (22, 400), (25, 500), (26, 550), (28, 650), (30, 800), (31, 900)$ . Produce a complete interpretation: (i) type & strength of correlation, (ii) range of sales, (iii) an interpolated estimate for $24° C$ , (iv) one caution about the conclusion.

Recall Solution

(i) As temperature rises, sales rise and the dots hug an upward line tightly → strong positive correlation (top row of the L3.2 scale, $∣ r ∣ \geq 0.8$ ). (ii) Sales range $= 900 - 300 === 600 ==$ (₹). (iii) Interpolate between $(22, 400)$ and $(25, 500)$ . Slope $= \frac{500 - 400}{25 - 22} = \frac{100}{3} \approx 33.33$ ₹/°C. From $22°$ to $24°$ is $+ 2$ : add $2 \times \frac{100}{3} = \frac{200}{3} \approx 66.7$ . $400 + 66.7 \approx== 466.7 == (₹, about ₹467)$ (iv) Caution: this correlation shows association, not proof of cause — and any estimate outside $20$ – $31°$ C (extrapolation) is unreliable. Warm weather likely drives sales, but the graph alone can't prove ice cream causes anything.

Recall Quick self-test (cloze)

Range is defined as ::: Maximum − Minimum A dot far from the overall pattern is called ::: an outlier Estimating a value between known data points is called ::: interpolation Two variables moving together does NOT prove ::: causation (one causing the other) The slope $m$ of a best-fit line represents ::: the rate of change (change in y per unit x) A trend with no single overall direction (repeated ups and downs) is called ::: fluctuating

Where this connects

Computing mean, median, range of these datasets → 1.3.04-Mean-median-mode-range
The equation $y = m x + c$ of the best-fit line → 2.4.01-Linear-equations and 3.2.01-Functions-and-graphs
Measuring correlation strength with a number → 4.1.01-Correlation-coefficient
Other data displays → 1.3.01-Bar-charts-and-pie-charts, 1.3.02-Reading-and-interpreting-tables