22.3 - Processing Data - Means, Percentages, Rates & Significant Figures

What a calculation question looks like

• Almost every Cambridge practical paper gives you some readings and asks you to do a calculation with them - a mean, a percentage, a percentage change, a difference, or a rate.
• The examiner isn't marking whether the biology is right, because the numbers are handed to you - they mark the calculation instead.
• The marks are awarded in stages, with each stage worth its own mark.

A typical question

A student measured the rate of photosynthesis of some pondweed at two carbon dioxide concentrations. The rate fell from 20 bubbles per minute to 17 bubbles per minute.

Calculate the percentage change in the rate. [2]

The two marks

1. Pick the right numbers out of the data and set the calculation up - here, the change of 3 over the original 20 (1 mark).
2. Work the answer out correctly and round it to the precision the question asks for - here, a 15% decrease (1 mark).

Two consequences of staged marking

1. Units can be their own mark - If the answer has a unit, that unit is usually worth a separate mark, so a correct number with the unit missing can still drop a mark.
2. Error carried forward (ecf) - If you get an early value wrong but then use it correctly, the later marks still count - as long as you've shown your working.

This is why you should write the calculation out rather than just the final number: it lets the examiner give you the marks you've earned even when one value is off.

Because the same handful of calculations comes up every series, this is a reliable set of marks once you know the five calculations and how each is rounded.

Common data-processing calculations

Five calculations cover almost every one you'll be asked for. Learn the formula and method for each, then slot the exam numbers straight in.

The five calculations

Getting the numbers the right way round

• Percentage change - Always divide by the original value (the one you started from), not the value you ended on. State whether it is an increase or a decrease.
• Rate - Divide the change by the time, not the other way round. Its unit is "something per minute" or "something per second".

Rounding, significant figures, and precision

• Rounding to the precision the question states is a separate marking point from the calculation itself.
• A question saying "give your answer to two significant figures" or "to one decimal place" tells you exactly how to round.
• The mark scheme gives one mark for the correct calculation and a second mark for the correct rounding - so a perfect sum left as a long string of digits still drops a mark.

Significant figures vs decimal places

These are counted differently, and don't always give the same answer - so read which one the question asks for:

• Significant figures - Counted from the first non-zero digit. So 47.316 to three significant figures is 47.3.
• Decimal places - Counted after the point. So 47.316 to one decimal place is also 47.3, but 6.028 to one decimal place is 6.0.

Three things examiners flag every series

1. Keep paired answers consistent - If a question asks for two values, round them both to the same number of decimal places or significant figures. Giving one answer as 1.3 and its pair as 1.25 is marked as inconsistent rounding, even when both numbers are right.
2. Do not round part-way through - Carry the full figures through the calculation and round only the final answer. Rounding an intermediate value and carrying it on can push the final answer outside the range the mark scheme accepts.
3. Give the unit - Where the answer has a unit, it is often a mark of its own. For a calculation such as 150 × 0.92 = 138, the number scores one mark and writing cm³ after it scores the other - so an answer with no unit is only half marked.

Remember that all of this rides on showing your working, because of error carried forward: if an earlier value was wrong but you rounded and used it correctly, the rounding and unit marks still stand.

Common exam questions

Question 1

A calculation on the paper is marked in stages. What are the stages?

Select the right data and set the calculation up; work it out correctly; and round the answer to the precision the question states, giving the unit where there is one.

---

Question 2

What does "error carried forward" mean, and why should you always show your working?

It means that if an early value is wrong but you then use it correctly, the later marks still count. You only get those marks if your working is written down, so the examiner can see the wrong value was used correctly.

---

Question 3

A student's reaction times at one temperature were 42, 45, and 48 seconds. Calculate the mean.

$\frac{42+45+48}{3}=45$ seconds.

---

Question 4

The rate of a reaction fell from 40 units to 30 units. Calculate the percentage change.

$\frac{40-30}{40}\times 100=25\%$ decrease. The change is divided by the original value, 40, and it is a decrease.

---

Question 5

A question asks for an answer to two significant figures and a student writes 47.316. What might this lose?

The rounding mark, because the answer was not given to the stated precision; 47.316 should be written as 47.

---

Question 6

Why should two answers calculated in the same question be rounded to the same number of decimal places?

Because examiners mark inconsistent rounding of a paired answer as wrong, so giving one to two decimal places and the other to none can lose a mark even when both numbers are correct.