1.3.3 · D5Basic Data & Probability
Question bank — Line graphs and scatter plots — basic interpretation
Before we start, three words we lean on constantly:
- A trend is the overall direction the data moves (up, down, flat, wobbly).
- A correlation means two measured quantities tend to move together — it lives only on scatter plots.
- An outlier is a point that sits far off the pattern everyone else follows.
True or false — justify
A line graph and a scatter plot can show the exact same data equally well.
False — line graphs need an ordered x-axis (usually time) so joining dots is meaningful; scatter plots show unordered pairs where joining dots would invent a false path.
If a line graph slopes upward, the quantity is definitely increasing.
True for that stretch — an upward segment means y rises as x rises, but the line can turn downward later, so "definitely increasing" only holds where the slope stays positive.
A steeper line always means a larger value.
False — steepness measures the rate of change (how fast), not the value itself; a gently rising line can still sit at a much higher height than a steep one.
On a scatter plot, "positive correlation" means all points lie exactly on a line.
False — positive correlation only means points tend upward; "exactly on a line" would be perfect (strongest possible) correlation, which real data rarely shows.
No correlation means the two variables have nothing to do with each other.
Careful — it means no straight-line relationship shows; a curved or U-shaped link (e.g. mood vs. hour of day) can exist while the linear correlation reads as zero.
A larger range always means the data is more spread out around its middle.
False — range only uses the max and min, so a single wild outlier can inflate it while every other point is tightly clustered.
Two graphs of the same data must look the same.
False — changing the y-axis scale or starting point stretches or squashes the visual slope, so identical numbers can look calm or dramatic. See Common Error #2 on scale.
Spot the error
"Ice-cream sales and drowning both rise together, so ice cream causes drowning."
Error: correlation is not causation. A hidden third variable — hot weather — drives both (more swimming and more ice cream). Always hunt for the lurking cause.
"The y-axis goes from 90 to 100, and the line jumps from 95 to 97 — that's a massive increase!"
Error: ignoring the scale. The real change is only +2; the truncated axis exaggerates a tiny rise into a visual cliff.
"This plant grew 5 cm/week for 8 weeks, so in 52 weeks it'll be 260 cm."
Error: extrapolating beyond the data. Growth trends flatten (plants stop growing); a pattern seen inside the data range need not continue outside it.
"There's one point far from the rest, so my whole scatter plot is wrong."
Error: an outlier is not automatically a mistake. It may be a real special case; investigate it before deleting — never erase data just because it's inconvenient.
"I studied 3 hours and scored 70, so the point (70, 3) goes on the graph."
Error: axis order matters. If x is hours and y is score, the point is (3, 70); swapping coordinates plots a completely different student.
"The line graph dips on Wednesday, so the value went negative that day."
Error: a dip means a decrease relative to Tuesday, not that the value is below zero. Read the actual y-height, not just the direction.
"Points are scattered randomly, but I'll still draw a line of best fit to predict."
Error: with no correlation, a best-fit line carries no predictive meaning — its slope is essentially noise, so predictions from it are worthless.
"The scatter shows negative correlation, so as x grows y must always drop for every single point."
Error: negative correlation is a tendency, not a guarantee for each pair; individual points can rise against the trend while the overall drift stays downward.
Why questions
Why do we join the dots on a line graph but leave them separate on a scatter plot?
Because a line graph's x-axis is continuous and ordered, so the space between points has meaning; a scatter plot's points are independent observations, and a connecting line would falsely suggest a journey between them.
Why is the slope the natural measure of "how fast something changes"?
Because it compares the rise (, the change in the measured quantity) to the run (, the change in time or input); dividing gives change per unit of x, which is exactly what "rate" means.
Why does a line of best fit put roughly equal points above and below it?
Because it aims to sit at the average relationship; balancing the misses above and below keeps it from being dragged toward one side, so it best represents the typical trend.
Why can a correlation be "strong" or "weak" rather than just present or absent?
Because strength describes how tightly points hug the trend line — tight clustering (strong) makes predictions reliable, wide scatter (weak) makes the same trend far less trustworthy. This is quantified later in 4.1.01-Correlation-coefficient.
Why must we always read the axis labels and units before interpreting anything?
Because the same shape can mean growth in ₹, decline in temperature, or anything else; without labels and units the graph is a picture with no meaning attached to its numbers (see 1.3.02-Reading-and-interpreting-tables for the same discipline with tables).
Edge cases
What does a perfectly horizontal line graph tell you?
The quantity is constant — no change over time; the rate of change is exactly everywhere.
What is the range of a data set where the maximum equals the minimum?
Zero — , meaning every value is identical and there is no spread at all.
Two students studied the same hours but scored differently — how does that appear on a scatter plot?
As two points with the same x but different y, stacked vertically; this is perfectly allowed and shows that x alone does not fully determine y.
Can a scatter plot show a clear pattern yet have a straight-line correlation of nearly zero?
Yes — a symmetric U-shape or curve rises then falls, so the linear trend cancels out even though a strong (non-linear) relationship exists.
What happens to the "trend" when a line graph has only one data point?
There is no trend — a trend needs at least two points to show a direction; a single point tells you a value but nothing about change.
If every point in a scatter plot lies exactly on one straight line, what is the correlation?
Perfect correlation (positive if the line rises, negative if it falls) — the strongest possible, letting you predict y from x with no error along that line.
Recall Quick self-test
Correlation proving causation? ::: Never — always suspect a hidden third variable. Truncated y-axis effect? ::: Exaggerates small changes into big-looking jumps. Range with an outlier? ::: Can be huge even if most points are tightly packed. Joining scatter-plot dots? ::: Wrong — invents a path between independent observations.