Questions on handling data: units, scales, rates and uncertainty

Question 1

A potato chip is weighed at 5.00 g, left in a sucrose solution for an hour and weighed again at 4.60 g, on a balance reading to plus or minus 0.01 g. Calculate the percentage change in mass, and calculate the percentage uncertainty in that percentage change.

Accepted Answer

M1 change in mass = 4.60 − 5.00 = −0.40 g, and because the two masses were subtracted the absolute uncertainties add to give plus or minus 0.02 g A1 percentage change = −0.40 ÷ 5.00 × 100 = −8.0 per cent, the sign showing that the chip lost mass M1 percentage uncertainty in the change = 0.02 ÷ 0.40 × 100 = 5.0 per cent, and in the initial mass = 0.01 ÷ 5.00 × 100 = 0.20 per cent A1 the two were divided, so the percentage uncertainties add: about 5.2 per cent

Question 2

A bacterial culture is counted every hour for ten hours and the counts are plotted with the logarithm to base 10 of the count on the vertical axis. Explain why a logarithmic axis is used here, and explain what a straight line on such a plot tells you.

Accepted Answer

B1 a population that multiplies repeatedly spans several factors of ten, so on a linear axis the early counts lie flattened against the baseline where nothing can be read from them B1 on a logarithmic axis equal distances represent equal multiplications, so the steps 1, 10, 100 and 1000 are evenly spaced and every hour is as readable as every other B1 each hour multiplies the population by the same factor and therefore adds the same amount to the logarithm, so constant growth plots as a straight line B1 a straight line is therefore evidence that the growth factor never changed over that period, which is what exponential growth means, and a rise of 3 on the axis is a multiplication of the population by 1000

Question 3

The table shows the time an enzyme-catalysed reaction took to reach a colour change at five temperatures. Calculate the mean time to the colour change at 50 °C, saying what you do with the third reading and why, and then calculate the rate of reaction at 40 °C in s⁻¹.

Accepted Answer

M1 identifies 122 s as an anomaly, because it is more than twice either of the other two readings at that temperature while every other set agrees within 2 s A1 excludes it and gives the mean of 51 and 49, which is 50 s M1 rate = 1 ÷ mean time; the mean at 40 °C is (40 + 41 + 39) ÷ 3 = 40 s A1 1 ÷ 40 = 0.025 s⁻¹

Question 4

A treated group of seedlings has a mean height of 24 cm and a control group has a mean height of 20 cm. Calculate the percentage by which the treated group is taller than the control group, and calculate the percentage by which the control group is shorter than the treated group, using each one as its own reference value.

Accepted Answer

M1 difference between the means = 24 − 20 = 4 cm A1 treated relative to control: 4 ÷ 20 × 100 = 20% taller M1 control relative to treated: 4 ÷ 24 × 100, using a different reference value A1 = 17% shorter

Question 5

A student measures an electron micrograph with a ruler. Its scale bar, labelled 500 nm, comes to 20 mm, and a ribosome in the same image comes to 1.0 mm across. Calculate the magnification the image was printed at, and calculate the actual diameter of the ribosome in nanometres.

Accepted Answer

M1 converts the scale bar's page length to the same unit as its labelled length: 20 mm = 20 000 000 nm M1 magnification = image size ÷ actual size = 20 000 000 ÷ 500 = ×40 000, with no unit M1 converts the ribosome's image size to the same units: 1.0 mm = 1 000 000 nm, then actual = image ÷ magnification A1 1 000 000 ÷ 40 000 = 25 nm, matching the expected size of a ribosome

Question 6

A student collects 24 cm³ of gas in 20.0 s and writes the rate as 1.2 cm³ s⁻¹. Explain why the answer is given to two significant figures, and explain why counting the decimal places in the measurements is the wrong guide.

Accepted Answer

B1 an answer carries the same number of significant figures as the least precise measurement that went into it, and the volume of 24 cm³ carries only two B1 significant figures record how finely the instrument measured, so writing more digits claims a precision the syringe never had B1 decimal places depend only on the unit chosen: 24 cm³ written as 0.024 dm³ has three decimal places and still two significant figures, because changing the unit cannot improve the measurement

Question 7

A tangent is drawn at the start of a curve of volume of oxygen collected against time. The tangent passes through the points (0 s, 0 cm³) and (30 s, 18.0 cm³). Calculate the initial rate of oxygen production and give its unit.

Accepted Answer

M1 the rate is the gradient of the tangent, which is the rise divided by the run, with both values read off the axes rather than counted in squares of paper M1 rise = 18.0 − 0 = 18.0 cm³ and run = 30 − 0 = 30 s A1 initial rate = 18.0 ÷ 30 = 0.60 cm³ s⁻¹

Question 8

An osmosis investigation reports a percentage change in the mass of a potato chip carrying a percentage uncertainty of over 30 per cent, even though the balance reads to plus or minus 0.01 g. Suggest why the uncertainty is so large, and suggest one change to the method that would reduce it.

Accepted Answer

B1 the percentage change rests on the difference between two similar masses, and subtracting keeps the whole of both absolute uncertainties while leaving only a small value behind B1 so the percentage uncertainty in that small difference is large even though each individual weighing was good to a fraction of one per cent B1 leaving the chips immersed for longer, or using a solution further from the potato's own water potential, produces a larger change in mass and shrinks the percentage uncertainty without changing the balance

Question 9

An enzyme-catalysed reaction is timed to a colour change at five temperatures, three repeats each; one repeat at 50 °C (122 s, against 51 s and 49 s) was left out of the mean as anomalous. The times fall from 20 °C to 40 °C and then rise again by 60 °C. Explain what this shows about the rate of the reaction, explain how the rate column is obtained from the times and why its unit is s⁻¹, and explain why the anomalous reading should be repeated rather than simply deleted.

Accepted Answer

B1 a shorter time means a faster reaction, so the rate rises to its highest measured value at 40 °C and falls after it, so the optimum lies between 30 °C and 50 °C, probably near 40 °C B1 each rate is 1 ÷ the mean time, an amount of change per second, so its unit is s⁻¹: 1 ÷ 40 s = 0.0250 s⁻¹ at 40 °C against 1 ÷ 95 s = 0.0105 s⁻¹ at 20 °C, which is why the rates run the opposite way up from the times B1 deleting a reading assumes it was a mistake; repeating it tests that assumption, and if the repeat agrees with the original the result is real and the explanation has to account for it

Question 10

State the number of nanometres in one micrometre, the number of micrometres in one millimetre, and the unit that a ratio, such as a magnification, is given in.

Accepted Answer

B1 1 µm = 1000 nm B1 1 mm = 1000 µm B1 a ratio has no unit at all, because the units of the two quantities cancel when one is divided by the other

Question 11

A student divides a total colour-change time by exactly 6 replicate tubes to calculate a mean, and wonders whether the 6 limits how many significant figures the mean can be given. Explain why the count of 6 tubes does not limit the significant figures of the mean, and explain what does.

Accepted Answer

B1 6 is an exact, counted number rather than a measurement, so it carries no uncertainty and imposes no limit on the number of significant figures of a result B1 the number of significant figures is instead set by the least precise instrument used to take the actual measurements that were averaged, such as the resolution of the stopwatch B1 the mean should therefore be rounded to match the significant figures of the timings themselves, not the number of readings that were combined

Question 12

Compare percentage change with the symmetric percentage difference between two separate values, referring to what each is calculated from and the question each one answers.

Accepted Answer

B1 percentage change tracks a single quantity over time and divides by its initial value, keeping the sign of the change, whereas percentage difference compares two separate values measured at the same time and divides by their mean B1 percentage change answers 'how much did this one thing change', whereas a percentage difference or comparison answers 'how do these two separate values compare' B1 which value is used as the reference must be stated for a comparison between two values, because comparing a treated value to a control gives a different percentage from comparing the control to the treated

Question 13

A student calculates the resting heart rate of a healthy adult human from their data and obtains an answer of 400 beats per minute. Suggest why this result should immediately be recognised as an error, and suggest one likely source of the mistake in the calculation.

Accepted Answer

B1 a healthy adult resting heart rate is around 60 to 100 beats per minute, so 400 is several times too large to be plausible for this organism B1 an order-of-magnitude check like this catches an error without redoing the whole calculation B1 a likely source is a unit not converted, such as working in seconds where minutes were needed, or a value multiplied where it should have been divided

Question 14

A student times how long it takes a catalase-soaked filter-paper disc to float in hydrogen peroxide solution at five different pH values. Explain how the student should calculate a rate from each time, and explain why the times themselves should not be plotted directly as if they were the rate.

Accepted Answer

B1 rate is calculated as 1 ÷ t, in a unit such as per second, because the same fixed amount of reaction, the disc floating, has happened in every trial, so a shorter time means a faster reaction B1 the times themselves are not the rate: plotting time directly gives a graph the wrong way up compared with a graph of rate, since a longer time would appear to mean more when it actually means slower B1 the correct graph, of 1/t against pH, rises where the reaction is faster and falls where it is slower, matching how a rate is normally interpreted

Question 15

State what is meant by the resolution of a measuring instrument, and state the uncertainty carried by a single reading taken from a millimetre rule.

Accepted Answer

B1 resolution is the smallest change an instrument can register, such as 1 mm on a metre rule or 0.01 g on a two-decimal-place balance B1 a single reading off a scale is taken as half the smallest division, so plus or minus 0.5 mm

Temperature / °C	Repeat 1 / s	Repeat 2 / s	Repeat 3 / s
20	96	94	95
30	62	60	61
40	40	41	39
50	51	49	122
60	88	90	89

Temperature / °C	Mean time / s	Rate / s⁻¹
20	95	0.0105
30	61	0.0164
40	40	0.0250
50	50	0.0200
60	89	0.0112