Why Axis Choices Matter
A scatterplot places each observation according to its values on two variables: the horizontal axis shows the predictor and the vertical axis shows the response. Once the variables and plotting method are settled, axis choices determine how much of their observed variation is visible and how easily a reader can recover values from the graph. A plot can include the correct points and still give a misleading impression if its limits or tick marks are poorly chosen.
A very wide axis range can compress the observed points into a small part of the plotting area, making genuine variation hard to see. A very narrow range can make modest differences occupy most of the plot and appear dramatic. Neither a zero start nor a close-up view is automatically right for every scatterplot. The goal is to show the data needed for the question, while making the scale transparent and keeping all relevant observations visible.
Limits, Units, and Tick Spacing
Choose the horizontal and vertical scales separately, using the variable’s units and observed values. Limits should include the observations being displayed. A small amount of space between the outermost points and the plot boundary can make a graph easier to read, but it should not be so large that it needlessly compresses the data. If a legitimate value is far from the others, do not silently omit it to make the remaining points easier to see. A plot that cannot show all values legibly may need a clearly explained alternative display, not an unannounced exclusion.
Tick marks should be regular and labelled clearly enough that a reader can estimate positions. For instance, an axis measured in minutes might use ticks every 5 or 10 minutes, depending on its range. Very dense labels can overlap; very sparse or irregular marks make values difficult to judge. Choose values that are convenient to interpret and make sure the labels state units. Avoid relying on unlabeled gridlines to communicate scale.
When selecting limits, consider the question as well as the data. If a response has a meaningful zero and the question concerns its size relative to zero, including zero may help. In a scatterplot, however, values are located on two quantitative axes, and the useful comparison may be among observed values within a narrower interval. Showing only that interval can be reasonable if the limits are explicit and the reader is not led to confuse a small observed difference with a large difference on the full measurement scale.
Worked Example: Showing a Narrow Score Range
Suppose a hypothetical class has five quiz scores, in points: 72, 75, 77, 80, and 84. The graph is intended to show how scores vary within this class, not to compare scores with the full possible score range of 0 to 100.
With vertical limits from 0 to 100, the observed range uses only \(84-72=12\) points of a 100-point axis. The difference between the highest and lowest scores occupies \(12/100=0.12\), or 12% of the axis height. The scores will appear clustered in the upper part of the plot.
A close-up axis from 70 to 85 has a range of \(85-70=15\) points. The same 12-point observed spread now occupies \(12/15=0.80\), or 80% of the axis height. Thus, the spread takes up \(0.80/0.12=6.67\) times as much of the plotting height as it did on the 0-to-100 axis. The actual scores and their differences have not changed; only their displayed proportions have.
For the stated purpose, limits of 70 and 85 with ticks every 5 points show the class’s variation clearly. The axis must be labelled in points, and its nonzero lower limit must remain visible. If the purpose instead were to show how these scores compare with the full possible range, a 0-to-100 axis would be relevant. The scale choice should follow the question, and the limits should make the choice apparent.
How a Scale Changes Visible Distance
The effect of an axis range can be stated precisely. Let \(L\) be the lower limit and \(U\) the upper limit of an axis. A value \(v\) appears at a fraction \((v-L)/(U-L)\) of the way from the bottom of the axis to the top. Consequently, a difference of \(\Delta\) units occupies a fraction \(\Delta/(U-L)\) of the axis length, provided both values lie within the limits.
This relation explains why the same difference appears larger when the range is narrowed: the numerator stays fixed while the denominator decreases. It does not mean a close-up is always misleading. A close-up can be informative when it answers a specific question and its limits are plainly shown. The risk arises when a reader is not told how much of the measurement scale is displayed, or when the visible size of a difference is treated as if the axis began at zero.
Horizontal and vertical ranges also affect the apparent shape of a scatterplot. The angle of a cloud or trend on the screen depends not only on the data but also on the two axis ranges and the physical width and height of the plotting region. Therefore, the visual steepness of a pattern should not be interpreted without reading both axes and their units. A change of one hour horizontally and one point vertically does not have an inherent visual angle independent of how the axes are drawn.
Here \(W\) and \(H\) are the plot’s width and height, and \(L_x,U_x,L_y,U_y\) are the axis limits. The subtraction in the vertical expression reflects that larger vertical values appear higher on the screen. Changing either range or the plot’s shape changes the on-screen geometry, even when the plotted observations are identical.
Worked Example: Choosing Readable Ticks and Limits
Imagine a hypothetical scatterplot of daily outdoor temperature, \(x\), in degrees Celsius, and water use, \(y\), in litres per household. The recorded temperatures are 11, 14, 18, 22, and 26 degrees, and the corresponding water-use values are 118, 125, 139, 151, and 160 litres.
The observed temperature range is \(26-11=15\) degrees. A horizontal axis from 10 to 30 degrees has range \(30-10=20\), includes all five observations, and leaves 1 degree of space below the smallest value and 4 degrees above the largest. Ticks every 5 degrees give the labels 10, 15, 20, 25, and 30, which are easy to read.
The observed water-use range is \(160-118=42\) litres. A vertical axis from 110 to 170 litres has range \(170-110=60\), includes every value, and adds modest space at both ends. Ticks every 10 litres allow readers to estimate the plotted values without crowding the axis labels. Both axes should state their units.
These limits serve the purpose of comparing the displayed observations: they show all points and make their locations legible. They do not claim that temperature or water use can only take values in those intervals. If the graph is meant to compare the observations with a broader scientifically meaningful range, that broader range may be more suitable, but the resulting compression should be considered rather than ignored.
Keeping Extreme Observations in View
A value far from the others can force an axis to cover a wide range. That may leave most observations packed together, but removing the extreme point from the displayed range conceals information. First check that the value is a legitimate observation using the data-quality principles appropriate to the situation. An unusual value is not by itself proof of an error. If it is valid and part of the data being described, the display should account for it.
One option is to use limits that include the full range and accept that the other points occupy less of the plot. If that makes their detail hard to see, a second clearly labelled close-up can accompany the full-range plot. The close-up should identify which observations or range it displays and should not be presented as the complete picture. Any display that omits observations must say so and explain the scope; an axis limit should not quietly turn omitted points into invisible data.
Worked Example: A Valid Value Stretches the Axis
Suppose a hypothetical set of recorded study times, in hours per week, is 1, 2, 3, 4, 5, and 24. The question is how these values are distributed across the recorded students. The largest value is far above the others, but assume it has been checked and is a valid report.
An axis from 0 to 25 hours includes all observations. The first five values lie between 1 and 5 hours, while the value 24 appears near the right end. The display makes the separation and the full range visible, though the five smaller values are close together.
If a second view uses limits from 0 to 6 hours to show the first five values in more detail, it does not include the 24-hour observation. It must therefore be labelled as a close-up of values from 0 to 6 hours, and it should be paired with or explicitly connected to the full-range view. Calling that close-up the plot of all six observations would be incorrect. The valid 24-hour value has not ceased to be part of the data because it is outside the chosen limits.
A Practical Scale-Selection Check
Before interpreting a scatterplot, check whether its display supports the intended reading. A concise scale check is especially useful when comparing plots or preparing a graph for others.
Make sure the horizontal and vertical axes name the quantities being plotted and give their units where applicable.
Verify that the limits include all observations intended for the display. Identify any valid values that a close-up would leave out.
Include enough of the relevant measurement scale to answer the question, without using a range so broad that the observations become unreadable.
Choose regular tick spacing that suits the range, and label ticks and units clearly.
Ask whether the limits could make small differences look large, hide useful variation, or conceal observations. State the limits clearly enough for a reader to judge.
Common Mistakes
- Assuming every scatterplot must start at zero. A zero lower limit may be useful for a question about comparison with zero, but it can compress variation that matters within the observed range. Choose limits for the purpose and make them explicit.
- Zooming in without noticing the effect. A narrow range makes a given difference occupy a larger fraction of the axis. That can be appropriate, but the reader needs clear limits and context to judge the size of the change.
- Cutting off an inconvenient point. A valid observation should not disappear simply because it stretches the scale. Show it on a full-range plot or clearly identify the scope of any additional close-up.
- Using tick marks that are hard to interpret. Crowded, irregular, or unlabeled ticks make it difficult to estimate values. Use regular spacing and state the measurement units.
- Reading screen angle as a data property. Apparent steepness changes when the axis ranges or plot dimensions change. Read both scales before describing how large a change is.
Check Your Understanding
Try these questions using the scale principles from this tutorial.
- A set of measurements runs from 48 to 53 units. What information should you check before choosing limits from 45 to 55 rather than 0 to 60?
- A difference of 6 units is shown on an axis with range 30 units. What fraction of the axis does it occupy? How would that fraction change if the range were 15 units?
- Why should a valid observation of 24 hours not simply be omitted from the display of study times that otherwise range from 1 to 5 hours?
- What should a reader know when a scatterplot uses a vertical axis whose lower limit is not zero?
- Why can two scatterplots of the same observations appear to have different trend angles if their axis ranges or physical plot dimensions differ?