Reading graphs and spotting misleading scales

Start with what the graph represents and what you are asking

A graph places values in a visual form so you can read cases, compare quantities or examine a relationship. First ask what is represented: a measured quantity, a count, a stipulated mathematical model or selected records? A title helps identify the question, but it does not replace the axis labels or the description of the data.

This lesson uses an illustrative tank table with exact stipulated records: day zero has 100 L, day one 102 L, day three 104 L and day six 106 L. These are given example values, not a claim about an actual measured tank. The table supplies four cases; it does not supply a continuous filling rule or explain why the amount changes.

Read axes, units, range and supporting labels

Identify the horizontal and vertical quantities, their units, increasing directions and tick intervals. Here time t is measured in days horizontally and volume V in litres vertically. Check the visible minimum and maximum on each axis. The origin (0, 0) may be outside a zoomed view; the bottom of a plot is not automatically numerical zero.

If a graph contains several series, read the legend before assigning meaning to a colour, marker or line. Also check where the values came from, the period covered and any stated conditions. A displayed upper limit of 120 L is an axis choice, not evidence that the tank’s capacity is 120 L. Distinguish the graphical frame from information about the object.

A labelled zoom changes the view, not the values

The same tank records can be drawn with volume from zero to 120 L or with a zoom from 99 to 107 L. The second view makes differences easier to inspect, while the first gives more context relative to zero. A line or point plot can use a clearly labelled zoom; a nonzero lower limit is not by itself evidence of an incorrect graph.

The explorer keeps the same plot height in those two views. The 6 L endpoint difference occupies 12 drawing units in the full view and 180 in the zoom, fifteen times as much space. The recorded values remain 100 and 106 L and their difference remains 6 L. Hiding the range or describing the enlarged visual gap as a larger numerical change would misrepresent the same information.

Separate a difference from a percentage of a named base

Absolute change is final value minus initial value, with the quantity’s unit. From 100 L to 106 L, the increase is 106 L − 100 L = 6 L. Relative percentage change divides that difference by a stated nonzero reference value and multiplies by 100%. Using the original 100 L gives (6 L)/(100 L) × 100% = 6%.

The final amount is 106% of the original amount; that differs from an increase of 106%. Reversing the comparison also changes the reference: from 106 L back to 100 L, the decrease is (6/106) × 100% = (300/53)%, about 5.66%, rather than six percent. Do not use the plot’s lower limit, such as 99 L, as the reference unless that is explicitly the quantity the question asks you to compare with.

Choose a meaningful baseline when bar length carries meaning

When bar lengths are used to compare the magnitudes of quantities, the baseline normally needs to represent zero. Bars for 100 L and 106 L drawn from zero have lengths in the ratio 100:106. If instead both start at 99 L, their visible lengths represent one and seven litres above that offset, not the full volumes. Reading those lengths as whole-volume magnitudes would wrongly make the second look seven times the first.

An explicitly named excess above a reference can be a different quantity to display, but explain the reference and label that quantity. For signed values, zero can lie inside the plot: positive bars extend one way from it and negative bars the other. This reasoning concerns what bar length encodes; it is not a rule that every point or line graph must begin its vertical axis at zero. The explorer uses points and optional segments, not bars.

Give unequal time intervals their correct widths

On a linear time axis, equal screen distances must represent equal elapsed times. The sample days zero, one, three and six have successive durations one, two and three days. Their horizontal gaps must therefore have the ratio 1:2:3. Placing them at equally spaced positions while still labelling the axis as linear elapsed time misrepresents those durations.

Each successive volume increase is 2 L, but the interval-average change rates differ: (2 L)/(1 day) = 2 L/day, (2 L)/(2 days) = 1 L/day, and (2 L)/(3 days) = 2/3 L/day. The overall net change rate from day zero to day six is (6 L)/(6 days) = 1 L/day. These are net interval averages; they do not establish a constant filling rate or every fluctuation between records.

Calculate a rate instead of judging its drawn angle alone

A segment’s visual angle depends on how much screen space represents each horizontal and vertical unit. Stretching the plot vertically, narrowing it horizontally or zooming its volume range can make the same numerical change look steeper. The interval’s rate still comes from change in volume divided by change in time, using the axis values and units.

Compare the first tank interval in the full and zoom views. It remains a 2 L change over one day, hence 2 L/day. In the zoom it occupies more vertical space because each litre is drawn larger. In the incorrect-spacing example, equal horizontal gaps instead conceal the increasing elapsed durations. Correct labels and proportional time spacing are needed to interpret the slopes.

Label estimates between or beyond the records

On correctly scaled axes, joining samples with straight segments is a linear interpolation convention between the records. It assumes straight-line change within each interval for the purpose of estimating unrecorded values. Between day one at 102 L and day three at 104 L, that convention estimates 103 L on day two. The estimate is not an additional observation supplied by the table.

Extrapolation extends an assumption beyond the recorded range. Continuing the last interval’s average rate of 2/3 L/day from day six would predict 108 L on day nine, but the table does not establish that continuation. A tank could change behavior or have limits not given here. State the chosen assumption, distinguish the prediction from a record, and check whether the context permits the extension.

Keep an absent value separate from zero and uncertainty

No record is supplied for day two. That absence does not mean the tank contained zero litres, and it does not automatically mean the interpolated 103 L was measured. In a graph of observations, mark missing information appropriately or leave a gap; explain any rule used to connect across it. A genuine zero is a recorded or otherwise justified numerical value, not a replacement for a blank.

Real measurements can also have uncertainty. Error bars might describe a range, variation among observations, a confidence interval or a stated measurement uncertainty; read their defined meaning rather than treating these as interchangeable. Missing error bars do not prove that measurements are exact. Here the four values are exact stipulated examples, and no measurement uncertainty is supplied or inferred from their whole-number notation.

A plotted association does not supply its cause

The tank table records larger stipulated volumes at later selected days. This comparison establishes those four values and their differences; it does not identify an inflow, an outflow or a reason for the changes. Time on an axis organizes the records, but a rising picture is not by itself an explanation of a mechanism.

More generally, two plotted quantities can vary together without the graph proving that one causes the other. Other influences, shared conditions or the way cases were selected can matter. Use the graph to describe the represented association, then keep any causal explanation separate and ask what additional information would support it. Calculating a correct slope does not settle that question.

Change the presentation while keeping the records

Start with the full view, day six selected and interpolation off. The table and point plot retain the exact records (0 days, 100 L), (1 day, 102 L), (3 days, 104 L) and (6 days, 106 L). The selected endpoint shows an increase of 6 L and six percent relative to the starting 100 L. No sample-to-sample segments are drawn until you choose interpolation.

Choose the full volume range 0 to 120 L, the labelled zoom from 99 to 107 L, or the explicitly incorrect-spacing example. The first two retain proportional time positions for days zero, one, three and six. The incorrect example deliberately gives their unequal durations equal horizontal spacing and is marked as incorrect wherever the presentation is described. The correct table remains visible for comparison.

Three presentations, four selected records and interpolation on or off give 3 × 4 × 2 = 24 states. Each control retains the other choices and all supplied values. In the two correct time views, optional straight segments are labelled interpolation between samples, not an observed rule or an instantaneous rate. In the incorrect-spacing example, connecting segments cannot correctly represent interpolation over actual elapsed time because the time axis is distorted. Compare both scales and interval durations, then verify your numerical conclusion in the table.

Finish with a numerical claim and its conditions

State the quantity, values, units, interval and reference that support your conclusion. For this example, the net increase is 6 L over six days, equal to six percent of the original 100 L. The labelled zoom changes its visual size, while equal spacing of unequal elapsed times changes the graph’s time representation. Keep interpolation, missing information and possible explanations distinct from the given records.

This completes the foundation path from quantities and numbers to formulas, coordinates and graph reading. The next planned lesson, Linear equations and equivalent transformations, develops equation solving further while preserving allowed values and valid operations. The same habit continues: read the relationship and its conditions, calculate carefully, and check what the result actually establishes.

Worked examples

Try it yourself

1. A graph labels horizontal time in days and vertical volume in litres, with a visible volume range zero to 120 L. Does 120 L establish the tank’s capacity? What else should you read before interpreting a coloured line?

   Show the answer and reasoning

   No: 120 L is the displayed axis limit, not stated tank capacity. Read the legend, axis tick values and units, the data description, period and any conditions or source information.

   The graphical frame and object properties are different information. A colour or line needs its labelled meaning before it can support a numerical claim.

2. From 100 L to 106 L, find the increase and distinguish six percent gained from 106% of the original. If the comparison is reversed, is the decrease also exactly six percent?

   Show the answer and reasoning

   The increase is 6 L, equal to 6% of the original, while the final volume is 106% of it. Reversing the comparison uses 106 L as the base: the decrease is (6/106) × 100% = (300/53)%, about 5.66%.

   Percentage change depends on the reference denominator. The same absolute difference can have different relative percentages when the base changes.

3. Is a clearly labelled point-plot zoom from 99 to 107 L automatically incorrect? For bars intended to compare full volumes 100 L and 106 L, why would visible lengths measured from 99 L need a different interpretation?

   Show the answer and reasoning

   The labelled point-plot zoom can legitimately reveal small differences. Bars starting at 99 L have visible lengths representing excesses of one and seven litres, not the full volumes; zero-based lengths normally support the intended magnitude comparison.

   Choose the scale according to what the marks encode and label it honestly. A bar-length comparison and a point or line plot do not impose one universal zero-baseline rule.

4. Find the three successive elapsed durations for days zero, one, three and six and their volume change rates. What is the overall average from day zero to six? Does that prove the volume rose by one litre every day?

   Show the answer and reasoning

   The durations are one, two and three days, with rates 2, 1 and 2/3 L/day. The overall average is (6 L)/(6 days) = 1 L/day. It does not prove a uniform daily increase.

   Divide each net change by its own elapsed duration. An average over an interval does not supply all intermediate values or an instantaneous rate.

5. The 6 L endpoint increase takes 12 drawing units in the full view and 180 in the zoom. Find the display enlargement factor. Have the volumes, original-base percentage or first interval’s 2 L/day rate changed?

   Show the answer and reasoning

   The enlargement factor is 180/12 = 15. The recorded volumes, six-percent increase and first interval’s 2 L/day average rate remain unchanged.

   Vertical range and aspect ratio affect the visual impression. Numerical differences and rates come from values and units, not just the drawn angle or gap.

6. No record is supplied for day two. Is its volume zero or a measured 103 L? If a different graph includes error bars, can you infer their meaning solely from their appearance? Does omitting them prove measurements exact?

   Show the answer and reasoning

   Neither: day two is unrecorded; 103 L is an estimate only under straight interpolation. Error-bar meaning must be stated, and their absence does not establish exact measurements.

   Keep missing information, a justified zero, modeled estimates and measurement uncertainty separate. Whole-number labels or familiar-looking bars do not supply omitted definitions.

7. The four stipulated tank values are larger at each later selected day. Does the table establish continuous filling, its cause or day-nine volume? What information is justified, and what requires an additional assumption?

   Show the answer and reasoning

   It establishes the four values, their differences and the corresponding interval averages. It does not establish every intermediate value, a cause or a future volume. Continuous segments and future predictions require stated assumptions.

   Describe the represented association at its actual scope. Correct arithmetic and a rising picture do not turn an interpolation convention or possible explanation into an observed fact.

What to carry forward

• Read the quantities, units, ticks, ranges, legend and context before making a claim from a graph.
• Calculate differences and rates from values; a percentage also needs its named nonzero reference base.
• A labelled line or point zoom can be useful. Bar lengths need a meaningful baseline, and linear time axes must preserve unequal elapsed spacing.
• Interpolation, extrapolation, missing values, justified zeros and defined measurement uncertainty are different kinds of information.
• Visual steepness and association alone do not establish a changed numerical rate, a full between-record rule or a causal explanation.