Handling data: units, scales, rates and uncertainty

Units, prefixes and the ladder biology lives on

Biology spans about nine factors of ten between a membrane and a person, and it handles that span with prefixes rather than with long strings of zeros. Every prefix is a power of ten and every conversion is a multiplication, so the only thing that can go wrong is the direction.

1 mm = 1000 µm    1 µm = 1000 nm    1 dm³ = 1000 cm³three conversions carry most of the marks in this topic

The habit that stops errors is writing the unit on every line of working rather than only on the answer: if the units either side of an equals sign do not match, you have found the mistake before the marker did. It also catches conversions done backwards. A micrometre is smaller than a millimetre, so a length in micrometres is a bigger number, and a conversion that made the number smaller went the wrong way.

Describe the drawing

A vertical axis marked at seven equally spaced ticks running from one nanometre at the bottom to one millimetre at the top, each tick a factor of ten above the one below it. Six biological structures are marked along it at their own sizes: a membrane, a ribosome, a virus, a bacterium, a red blood cell and a plant palisade cell.

Standard form writes anything very small or very large as a value between 1 and 10 multiplied by a power of ten: 0.000 007 m becomes 7 × 10⁻⁶ m, and 128 000 becomes 1.28 × 10⁵. Two things go wrong with it. The first digit must lie between 1 and 10, so 12 × 10⁴ is the right size and not standard form. And a negative index means a small number, not a negative one. 10⁻⁶ is a millionth.

Where significant figures come from

A calculator gives you ten digits. The measurement gave you two or three, and writing down the ten is a claim about your apparatus that is not true. The number of significant figures records how finely you measured.

Start with the instrument. A metre rule reads to the nearest millimetre, so a leaf measured as 43 mm is known to two significant figures and one measured as 143 mm to three. A balance reading 4.20 g has three, and the trailing zero is significant because the balance printed it. Writing 4.2 g throws away information you were given.

Significant figures

The digits in a measurement that carry information about its size, counted from the first non-zero digit.

Leading zeros

Never significant. 0.0043 has two significant figures, because the zeros only locate the decimal point.

Trailing zeros after a decimal point

Always significant. 4.20 has three, and says the hundredths digit was measured.

The rule for multiplying and dividing is that the answer carries the same number of significant figures as the least precise measurement that went into it, so a rate found by dividing a volume known to two figures by a time known to four is known to two. Adding and subtracting follow a different convention: the result keeps the decimal places, or absolute resolution, of the least resolved measurement rather than its significant figures. Round at the end rather than in the middle, because rounding twice drags the answer away from where it belongs. Two things sit outside the rule. Exact numbers impose no limit at all, such as the 100 in a percentage or the number of discs you counted, and an instruction in the question about how many figures to give always wins.

Decimal places are a different count. Measure a leaf at 43 mm with a millimetre rule and you have two significant figures; write the same measurement in metres and it becomes 0.043 m, which has three decimal places and still two significant figures, because moving the point cannot improve the ruler. Significant figures record how well you measured. Decimal places record which unit you happened to use.

Ratios, percentages, and telling the two percentages apart

A ratio compares two quantities of the same kind by division, and because the units cancel it has none of its own. That single fact is why magnification is written ×12 000 and never ×12 000 µm, and why an Rf value has no unit either.

magnification = image size ÷ actual sizeboth sizes in the same unit, always, before you divide

The two percentages are different calculations answering different questions.

percentage change = (final − initial) ÷ initial × 100the denominator is where you started, and the sign is part of the answer

percentage change relative to a reference value = (difference between the two) ÷ (the reference value) × 100state which value was used as the reference

Percentage change follows one thing over time. Keep the sign: a chip that loses mass has changed by −10%, and writing 10% throws away the direction the investigation was about.

Comparing two separate values depends on which is used as the denominator. If a treated group has a mean of 24 and a control 20, the treated group is 20% higher than the control while the control is 16.7% lower than the treated group. Both figures are correct and they answer different questions, so state which value was the reference and use the control where there is one. For a symmetric percentage difference between two values, divide the absolute difference by their mean and multiply by 100. If the examination paper supplies a formula, use that convention.

Notice what the percentage adds. A chip going from 4.20 g to 3.78 g lost 0.42 g and a second going from 6.30 g to 5.67 g lost 0.63 g, which looks worse, and yet both changed by −10.0%. The two chips behaved identically, and only the percentage says so.

Rates are gradients

A rate is a change in something divided by the time it took, so its unit always ends in 'per unit time' and its value is always a gradient on a graph of that something against time. Both halves of that sentence are examined.

rate = change in the quantity ÷ time takenunits: cm³ s⁻¹, mg min⁻¹, bubbles per minute, arbitrary units per second

On a straight line the gradient is the rate, and any two points give it. Take them far apart, since a triangle spanning most of the line is read more accurately than a small one, and take the values off the axes rather than counting squares. On a curve the gradient changes from moment to moment and you need a tangent: a straight line touching the curve at the point of interest and running along it rather than cutting across it. The gradient of the tangent is the rate at that instant.

Describe the drawing

The tangent construction built four times on the same curve of oxygen collected against time. First a dashed straight line joins the first and last points, giving an average rate over the whole run. Then a tangent is drawn touching the curve at time zero, with a dashed right-angled triangle beneath it showing a rise of 18 cubic centimetres over a run of 20 seconds, a gradient of 0.90 cubic centimetres per second. The same construction is repeated at 20 seconds and at 50 seconds, where the tangent is visibly shallower and the gradients are 0.49 and 0.20 cubic centimetres per second.

The initial rate matters more than any other in enzyme work, and the reason is worth writing out in an answer. At t = 0 the substrate concentration is the one you chose, the product concentration is zero and nothing the enzyme made has inhibited it yet. Every later gradient reflects the conditions you set and how much substrate has since been used up, so a rate measured at three minutes measures two things at once.

There is a second route, used whenever the thing you can measure is a time. Time an amylase reaction until the iodine stops going blue-black, or a catalase-soaked paper disc until it floats, and the rate is 1 ÷ t, in units of per second. It works because the same amount of reaction has happened in every trial, so a shorter time is a faster reaction. Plot 1 ÷ t rather than the time itself: a graph of time against temperature is inverted relative to a graph of rate against temperature, so a graph of time described as a graph of rate states the opposite of what it shows.

Why the axis is sometimes logarithmic

Two quantities in biology span so many factors of ten that a linear axis cannot show them, and both get a logarithmic one. On a logarithmic axis, equal distances represent equal multiplications rather than equal additions, so the steps 1, 10, 100, 1000 are evenly spaced.

Describe the drawing

Two graphs side by side of the same bacterial counts taken every hour for eight hours. On the left the vertical axis is linear and the plotted points hug the bottom of the graph for five hours before rising almost vertically. On the right the vertical axis is logarithmic and the same points lie on a straight line of constant gradient.

Population growth is the first case. A population doubling every hour from 500 cells reaches 128 000 in eight hours, and on a linear axis the first five hours lie flattened against the baseline where nothing can be read from them. Take the logarithm and the curve becomes a straight line, because each hour multiplies the population by the same factor and so adds the same amount to the logarithm. That the line is straight is itself the evidence that growth was exponential.

pH is the second, and it catches people out because the logarithm has a minus sign in front of it.

pH = −log₁₀[H⁺]so a lower pH means a higher hydrogen ion concentration

Each whole pH unit is a factor of ten in hydrogen ion concentration. Moving a buffer from pH 7 to pH 5 does not double anything; it multiplies the hydrogen ion concentration by a hundred, which is why an enzyme can be untouched at pH 7 and denatured at pH 5, and why describing that change as 'a small change in pH' in an answer is wrong. The stomach at pH 2 holds a hydrogen ion concentration more than a hundred thousand times that of blood plasma at pH 7.4.

Uncertainty, and how it travels through a calculation

Every measurement has an uncertainty: a range within which the true value lies. It is not a mistake.

For a single reading off a scale, the uncertainty is usually taken as half the smallest division, so a millimetre rule gives ±0.5 mm. But a length is two readings, one at each end, so a measured length carries ±1 mm, and the same doubling applies to a thermometer read before and after or a burette read at start and finish. A digital instrument is quoted to its last displayed digit: a balance showing 4.20 g carries ±0.01 g.

percentage uncertainty = uncertainty ÷ measurement × 100which is why the same instrument gives a small percentage uncertainty on a big reading and a large one on a small reading

Describe the drawing

A bar chart of three percentage uncertainties against a vertical scale marked from one to five per cent. The bars for the initial and final masses are so short as to be barely visible above the baseline, while the bar for the difference between them rises to nearly five per cent, roughly twenty times as tall.

Combining uncertainties follows two rules, and which one you use depends on the arithmetic you did:

The subtraction rule is the one with teeth: subtracting two similar numbers keeps the whole of both uncertainties and throws away most of the value. That is why a percentage change built on a small mass change is far shakier than either weighing was, and why an osmosis investigation is designed with a long enough immersion to produce a change worth measuring.

One last habit. When a calculation is finished, ask whether the answer is the right size for the living thing in the question. A cell 2 mm across, a healthy adult human with a resting heart rate of 400 beats per minute (a shrew's heart genuinely beats that fast; a person's does not), a respiration rate that would consume a mouse's body mass in oxygen every hour. An order-of-magnitude check catches slips like these.