PSLE Science Reality Lab Vol No.073 | “Both Lines Start at 100” — Were Their Real Starting Values the Same?

PSLE-SCI-REALITY-0073

Wait, What? Two lines can both begin at 100 even when one real starting measurement is ten times larger than the other.

Look at a chart. Two lines begin together at exactly 100. Over time, the blue line rises to 150 while the orange line rises to 110.

It is very tempting to think the two things started equal, then the blue one became larger.

But perhaps the chart has been rebased. The original blue measurement might have started at 20 units while the original orange measurement started at 200 units. The chart maker simply defined each series’ own starting value as 100 so that their relative changes could be compared.

In that case, “100” is not the original measurement. It is a reference point.

Reality Lab Vol No.073 teaches a powerful evidence habit: before comparing the heights of two lines, find out what the vertical numbers represent.

Quick Answer

If a chart says “base = 100,” “index = 100,” “normalised to 100,” or shows several series all starting at 100, do this:

Find the base period or starting condition.
Ask what original quantity each series measured.
Remember that 100 means “this series’ own reference level,” not necessarily 100 real units.
Use the index to compare relative change.
Do not use equal index levels to claim equal original absolute levels unless the original data justify that comparison.

Reality Lab rule: Equal indexed starting points can be a drawing convention, not evidence that the real starting quantities were equal.

What This Guide Owns

This guide owns one real-world evidence-transfer job: how to evaluate a chart that rescales or rebases different scientific series to a common starting index such as 100, without confusing relative change with original absolute level.

It does not take over the general owners for reading graph axes, baselines, units or percentages. It applies those skills to a special communication object: the rebased or indexed chart.

The Mystery Chart

Imagine two ponds being observed for algae concentration. These numbers are invented for teaching.

Day	Pond A original concentration	Pond B original concentration
Day 0	20 units	200 units
Day 10	24 units	210 units
Day 20	30 units	220 units

The original table immediately tells you Pond B has the higher absolute concentration at every time point.

Now rebase each pond’s Day 0 value to 100.

Day	Pond A index	Pond B index
Day 0	100	100
Day 10	120	105
Day 20	150	110

Now Pond A’s line rises much faster. That is a real feature of the relative change: Pond A increased by 50% from its own starting level, while Pond B increased by 10%.

But at Day 20, Pond A’s original value is still 30 units while Pond B’s is 220 units.

The indexed chart answers, “Which changed more relative to where it started?” It does not automatically answer, “Which has the larger absolute value?”

Observed, Represented and Inferred

Layer	What is happening?
Observed data	Original scientific measurements such as 20, 24 and 30 units.
Representation	Each series is divided by its own base value and expressed relative to 100.
Valid inference	Pond A changed more, proportionally, from its own starting level.
Invalid inference	Pond A and Pond B started at equal absolute levels because both index lines start at 100.

The chart is not lying. The reader has to keep the representation rule attached to the numbers.

Why Rebase Data at All?

Rebasing can be extremely useful when two series use different starting levels and the question is about how much each has changed relative to its own starting point.

The U.S. Bureau of Labor Statistics, for example, explains index numbers by setting a reference or base period equal to 100 and then showing movement relative to that base. The same representational idea can be used outside economics: a researcher can rebase plant height, sensor response, particle concentration or another scientific quantity to compare proportional change.

The key word is relative.

How a Base-100 Index Is Built

A simple base-100 index can be formed like this:

index = current value ÷ base value × 100

If a plant is 40 cm tall at the base measurement, 40 cm becomes index 100. If it later becomes 50 cm:

50 ÷ 40 × 100 = 125.

Index 125 means the plant’s measured height is 25% above its base measurement. It does not mean the plant is 125 cm tall.

Index value ≠ original measurement unit.

The Base Period Is Part of the Meaning

If the chart does not tell you which day, year, trial or condition was assigned 100, you are missing an important part of the representation.

Suppose a temperature-response index uses Day 1 as 100. Another chart uses Day 5 as 100. The same original data can produce different-looking index paths because the reference point changed.

That does not make either chart wrong. It means you must know the reference before interpreting the shape.

Equal at 100 Does Not Mean Equal in Reality

Consider two plant shoots:

Plant	Starting height	Indexed starting value
X	5 cm	100
Y	50 cm	100

The common “100” has deliberately erased the ten-fold difference in starting height from the vertical scale. That erasure is useful if you want to compare proportional growth. It is harmful if you forget it happened.

This is why representation checks matter. Every graph highlights some relationships and hides others.

Relative Growth Can Reverse the Visual Winner

Plant X grows from 5 cm to 10 cm. Plant Y grows from 50 cm to 60 cm.

Absolute growth:

Plant X gains 5 cm.
Plant Y gains 10 cm.

Relative growth from the starting point:

Plant X doubles: index 200.
Plant Y rises by 20%: index 120.

Which plant “grew more” depends on the scientific question. Plant Y gained more centimetres. Plant X changed more as a percentage of its own starting height.

The indexed chart is useful because it forces you to ask which meaning of “more” is being shown.

Worked Case 1: Battery Capacity Retention

Two fictional batteries begin with different measured capacities. Battery A starts at 1,000 units. Battery B starts at 2,000 units. A chart sets each starting capacity to 100.

After repeated use, A is at index 90 and B is at index 80.

The correct reading is that A retained a larger fraction of its own starting capacity: 90% versus 80%.

You cannot conclude A has the larger remaining absolute capacity. In original units, A has 900 while B has 1,600.

Worked Case 2: Sensor Signal Change

Sensor A begins at 2 signal units. Sensor B begins at 20. Both are normalised to 100. Later A reaches index 150 and B reaches index 120.

A increased more proportionally. But the raw signals are now 3 units and 24 units respectively. If the scientific question depends on absolute signal level, the original values still matter.

Worked Case 3: Two Experiments With Different Units

One series measures plant height in centimetres. Another measures leaf count. A graphic rebases both to 100 and plots them together.

This can help compare relative change, but it does not make centimetres and leaf counts the same physical quantity. The common index scale is a mathematical representation layered on top of two different measurements.

Do not say, “Both reached 130, so plant height equals leaf count.” Index equality is not physical equality.

Worked Case 4: A Convenient Starting Date

A report compares two environmental measurements by rebasing both at a date when one series happens to be unusually low. That series then shows a dramatic rise from 100.

The index calculation may be correct. The communication question is whether the chosen base period is representative and whether the conclusion changes if a reasonable nearby period is used.

This is not an invitation to reject every chosen baseline. Every index needs one. It is a reminder that the baseline is part of the claim’s meaning and should not be invisible.

Representation Check: What Did the Chart Preserve, and What Did It Hide?

Usually preserved	Often hidden or transformed
Direction of change relative to base	Original absolute starting level
Proportional change	Original measurement units
Timing of rises and falls	Absolute difference between series
Relative growth or decline	Whether two equal index values mean equal physical quantities

Every representation is a trade. Good scientific reading asks whether the information hidden by the transformation matters to the claim being made.

What Evidence Would Strengthen an Indexed-Chart Claim?

The chart clearly says it is indexed or normalised.
The base period or base condition is stated.
The meaning of index 100 is explained.
The original measured quantities and units are available nearby when absolute levels matter.
The chosen base period has a scientific reason or is at least not hidden.
The caption states whether the comparison concerns relative change, absolute amount or both.
Different series remain clearly labelled so the common scale does not imply they are the same physical quantity.
The conclusion does not silently switch from “grew faster relative to baseline” to “became larger in absolute amount.”

What Would Weaken It?

The axis simply says “100, 120, 140” with no explanation of what 100 means.
Several lines start at 100 and the caption encourages readers to assume equal starting measurements.
Original units disappear even though the conclusion depends on absolute amount.
A convenient extreme base date makes one series look unusually dramatic without disclosure.
The chart compares different quantities on one index and then treats equal index values as equal real-world quantities.
A line ending higher is described as “having more” when the chart only supports “increased more relative to its own base.”

How Far Can the Conclusion Travel?

From the pond example, you can say:

Pond A increased more relative to its own Day 0 concentration than Pond B did over the same period.

You cannot use the indexed chart alone to say:

Pond A and Pond B had equal starting concentrations;
Pond A had the higher final concentration;
index 150 means 150 concentration units;
the same percentage change has the same ecological importance in both ponds;
the chosen base day was the only reasonable baseline.

The chart may be excellent for the question it was designed to answer. Problems begin when it is used to answer a different question.

PSLE-Style Transfer Case

Two seedlings are measured on Day 1. Seedling P is 4 cm tall. Seedling Q is 12 cm tall. A student converts both Day 1 heights to index 100. On Day 8, P is 6 cm and Q is 15 cm.

P’s index is 150. Q’s index is 125.

The student writes: “P is taller than Q on Day 8 because 150 is greater than 125.”

That conclusion is wrong. The indexes describe relative change from each seedling’s own starting height. The original Day 8 heights are 6 cm and 15 cm, so Q is still taller.

A correct statement is: Seedling P increased by a larger percentage from its own Day 1 height, while Seedling Q remained taller in absolute height.

Tempting Reasoning That Fails

“Both start at 100, so they started equal.” Each series may have been divided by a different original base value.
“Index 150 means 150 real units.” The index is dimensionless unless the chart explicitly says otherwise.
“The higher indexed line has the higher absolute value.” It may only have changed more relative to its own baseline.
“Rebasing changes the real data.” It transforms the representation, not the underlying observations.
“A normalised chart is misleading.” It can be exactly the right tool for comparing proportional change—if labelled and interpreted correctly.
“Because both lines use the same index scale, they measure the same thing.” Different physical quantities can be converted to a common relative scale without becoming the same quantity.

Practice 1: Recover the Meaning of 125

A series is indexed so its starting measurement equals 100. It later reaches 125. What can you say?

Answer: The measured value is 25% above its base value, assuming a simple base-100 index. You cannot determine the original physical value without knowing the base measurement.

Practice 2: Same Index, Different Quantity

Series A starts at 8 units and Series B starts at 80 units. Both later reach index 150. Are their final absolute values equal?

Answer: No. A becomes 12 units while B becomes 120 units. Both increased by 50% relative to their own starting values.

Practice 3: Which Question Does the Chart Answer?

Two plants have different starting masses. A chart rebases both to 100. Is it better for comparing final mass or proportional growth?

Answer: It is directly useful for proportional growth. To compare final absolute mass, recover or display the original mass values.

Practice 4: The Missing Caption

A graph shows three scientific lines all beginning at 100 but gives no explanation. What is the first question to ask?

Answer: Ask what 100 represents—what base period or condition was used, and whether each line was normalised to its own starting value.

Delayed Independent Return: The RAW → BASE → INDEX Audit

The next time several lines mysteriously begin at the same value, reconstruct three layers:

RAW — What original quantity and unit was measured?
BASE — Which original value or period was chosen as the reference?
INDEX — What relative change does the transformed number show?

Then ask a fourth question: Does the conclusion belong to the raw level or the relative-change level?

Teaching Guide for Parents and Tutors

This lesson works well with two strips of paper. Write a starting value of 10 on one strip and 100 on another. Tell the learner that both starting points will now be called “100.”

Increase the first real value from 10 to 15 and the second from 100 to 110. Ask the learner to calculate the relative indexes: 150 and 110.

Now ask two separate questions:

Which changed more proportionally?
Which has the larger absolute final value?

The two correct answers are different. That is the conceptual payoff.

Next, change the context without changing the mathematics: plant height, leaf count, sensor signal, battery capacity or turbidity. If the learner can keep “relative change” and “absolute amount” separate across contexts, the reasoning has transferred.

Do not teach “normalised charts are deceptive.” Teach the more accurate habit: every representation has a job; read it according to the job it was built to do.

Authoritative Sources

The Quiet Return

A graph does not merely hold data. It makes a choice about which relationship to make easy to see.

Setting every series to 100 can make relative change beautifully clear. It can also make very different starting magnitudes disappear from sight.

So when two scientific lines begin together at 100, do not assume the world began equal. Find the raw measurements. Find the base. Then read the index for what it actually knows.

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