Line graphs
A city's temperature climbs through summer, a plant grows a little taller each week, a shop's sales rise and fall through the year — all of these are stories of a quantity changing over time. A line graph is the picture that tells such a story at a glance: plot each (time, quantity) pair as a point, join consecutive points with straight segments, and the resulting zig-zag instantly shows whether things are rising, falling, or holding steady, and exactly where the change is fastest. This article builds the skill from the ground up — reading a data table, choosing a sensible scale for the axes, plotting and joining the points correctly, and reading the finished picture for trends — using the same worked examples as the Class 8 chapter "Tales by Dots and Lines," and adds the one calculation every scale choice quietly relies on.
A line graph plots (time, quantity) pairs as points and joins them in order with straight segments, so the direction and steepness of each segment reveal whether a continuously changing quantity rose, fell, or stayed flat over that interval.
The core idea
What gets plotted, and on which axis. A line graph always has two axes: the horizontal axis carries the variable that changes steadily and continuously — almost always time, in days, months, or years — and the vertical axis carries the quantity being tracked, such as temperature, visitor count, or height. Each row of the data table becomes one point, and the points are joined in the order the data occurs, left to right.
Why joining is allowed. Joining two dots with a straight line makes an implicit claim: that the quantity could, in principle, be imagined to change smoothly between the two measured moments — the way temperature drifts minute by minute even though we recorded it only once an hour. That claim only makes sense when the horizontal axis is a genuinely continuous variable like time. If the axis instead lists unrelated categories — five different students, five different fruits — there is no "in-between" to speak of, and joining the dots is misleading; a bar graph is the honest choice there.
Reading the trend. Once the points are joined, the picture reads itself: a segment climbing left to right means the quantity increased over that interval, a falling segment means it decreased, and a flat segment means no change. The steepest segment marks the interval of fastest change — but "steepest" is only meaningful relative to the scale chosen for the axes, which is why scale choice is not a cosmetic detail but part of the mathematics.
Choosing the scale. Before plotting, you must decide how many units of the tracked quantity each grid square on the graph paper will represent. A good scale uses the available grid squares efficiently — neither so coarse that all the points crowd into one corner, nor so fine that the graph runs off the page. The calculation behind this choice is simple:
where: is the scale, in data-units per grid square; the range of the data is the largest value in the table minus the smallest; and the number of grid squares available is how many squares the graph paper offers along that axis. The raw value of is then rounded up to a nearby convenient number — 5, 10, 20, and so on — so that the axis labels are easy to read and mark. When the data values sit well above zero (temperatures in the high teens and twenties, say, rather than near zero), the vertical axis need not start at 0 at all; it can start just below the smallest value, which keeps the graph from being squashed into a thin strip at the top.
Why the scale calculation matters
The goal is to turn a raw data table into a working scale for the vertical axis before a single point is plotted, so that the finished graph uses the page well, keeps every value clearly separated, and does not mislead the eye by starting from an unnecessary zero — using nothing beyond subtraction, division, and a sensible rounding rule. We use the six-month temperature record already met, 18, 20, 25, 30, 36, 38 degrees Celsius.
Step 1. Find the range of the data. The largest value is and the smallest is , so the range is degrees. This is the spread the vertical axis must comfortably cover.
Step 2. Decide how many grid squares are available and divide. Suppose the graph paper offers about usable squares along the vertical axis. The raw scale is degrees per square — already a clean, easy-to-read number, so no rounding is needed here.
Step 3. Decide the starting value. Since the smallest reading, , is far from zero, starting the axis at would waste eight full squares before the data even begins and would flatten the differences between months. Instead, start just below the smallest value, at , so the first point sits close to the bottom of the graph rather than the top.
Step 4. Check that the largest value fits. With the axis starting at and a scale of per square, the top point, , sits squares up — so squares comfortably contain the whole data set. This confirms the scale " grid line represents degrees Celsius, starting from degrees" used for this exact temperature graph, and shows the calculation behind what might otherwise look like an arbitrary choice. Verified: range ; raw scale ; top point at squares, within a -square axis.
Worked example
The table below shows the number of visitors to a small museum on five consecutive days.
| Day | Mon | Tue | Wed | Thu | Fri |
|---|---|---|---|---|---|
| Visitors | 40 | 55 | 30 | 65 | 50 |
Step 1. Find the range: the largest count is and the smallest is , so the range is visitors.
Step 2. Choose the scale. With about usable vertical squares on the graph paper, the raw scale is , which rounds up to a convenient visitors per grid square — matching the scale used in the figure above.
Step 3. Decide the axes and plot the points. Put "Day" on the horizontal axis and "Number of visitors" on the vertical axis, starting from (just below the smallest value) rather than . The five points are , , , , .
Step 4. Join the points in order with straight segments: Mon to Tue rises by , Tue to Wed falls by , Wed to Thu rises by , Thu to Fri falls by .
Step 5. Read the trend. The steepest segment is Wed to Thu, a rise of visitors in a single day, so visitor numbers grew fastest between Wednesday and Thursday. Wednesday was the quietest day (30 visitors) and Thursday was the busiest (65 visitors) — a picture that communicates the week's pattern far faster than the raw table, and the scale calculation from Step 2 is exactly what made that picture readable rather than cramped.
Common traps
- Joining points across unrelated categories. If the horizontal axis lists five different students or five different fruits rather than a continuous variable like time, joining the dots falsely implies a smooth change between them; a bar graph is correct there.
- Skipping the scale calculation. Guessing a scale instead of computing often produces a graph that is either squashed into a corner or too wide to fit the page.
- Starting at zero out of habit. When values cluster far above , starting the vertical axis there squashes real differences into a small band near the top; starting just below the smallest value (as in both worked examples) is the better choice.
- Confusing steepness with total value. A steep segment shows a fast change, not necessarily a large one — always check the scale before comparing steepness across two different graphs.
Ask the tutor: Want to work through choosing a scale and plotting a line graph for your own data set, step by step?
Why can't I just always start my vertical axis at zero to be safe?
Starting at zero is not wrong, but when every value in your data sits well above zero, it wastes most of the graph paper on empty space and squeezes all the real differences into a thin band near the top, making the trend hard to read. Starting the axis just below the smallest value, as done for the temperature data here, uses the page efficiently and shows the ups and downs clearly. Just make sure you don't start above your smallest value, or that point will not fit on the graph at all.
How do I actually pick the number to use for the scale, like "1 square equals 10 visitors"?
Take the range of your data — the largest value minus the smallest — and divide it by the number of grid squares you have available along that axis. That gives a raw number like visitors per square. Round it up to the nearest easy-to-read number, usually , , , or depending on the size of your data, so every grid line lands on a value you can read at a glance rather than an awkward fraction.
How is a line graph different from a bar graph, and when should I use which?
A line graph joins points with straight segments and is used when the horizontal axis is a continuous variable, usually time, so that "in between" the recorded moments has meaning — a city's temperature between two hourly readings, for instance. A bar graph uses separate, unjoined bars and is used when the horizontal axis lists distinct categories with no continuous variable connecting them, such as different students, different fruits, or different cities compared at one moment. If joining two points would suggest a meaningless "in-between" value, use a bar graph instead.