Lesson 5 of 8
Practical skills: tables, graphs and errors
I can choose and explain measuring instruments and methods with suitable range and precision, including the linked quantity-specific determinations.Syllabus
Syllabus K323, 1(e). Choose and explain measuring instruments and methods with suitable range and precision, including the linked quantity-specific determinations.
Make a guess. You are not marked.
You time 10 swings of a pendulum three times and get 12.6 s, 12.4 s and 12.8 s. What value should you record for the time of 10 swings?
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12.6 s, the mean of the three readings.
Repeated readings scatter because of reaction time. Their mean, (12.6 + 12.4 + 12.8) / 3 = 12.6 s, is the best value to record.
Paper 3, the practical paper, is worth 20% of your grade. Most of its marks reward the same few habits in every experiment. Learn them once and use them every time.
1. Record each reading to the precision of the instrument
Write every reading to the smallest division or display step of the instrument you used. A rule marked in millimetres reads to 0.1 cm, so a length that ends exactly on the 12 cm mark is 12.0 cm, not 12 cm. For a digital instrument, write exactly what the display shows.
| Instrument | Record to | Example |
|---|---|---|
| Metre rule (mm markings) | 0.1 cm | 45.0 cm |
| Digital calipers | 0.01 mm, as displayed | 12.34 mm |
| Digital stopwatch | 0.01 s, as displayed | 15.62 s |
| Digital ammeter or voltmeter | Every digit displayed | 0.38 A |
Readings from one instrument should have the same number of decimal places all the way down a column.
2. Draw the table so that it can be marked
- Heading = quantity / unit. Write l / cm or R / Ω at the top of each column. Put units only in the heading, never next to each number.
- Take at least 5 sets of readings. Spread them evenly over the widest range the apparatus allows.
- Repeat and average when readings vary, for example timing a swing or a falling object. Keep the raw readings in the table as well as the mean.
3. Give calculated values to sensible significant figures
A calculated value should have the same number of significant figures as the least precise reading you used, or one more. Keep extra digits while you calculate. Round only the final answer.
Worked example
How many significant figures?
A voltmeter reads 2.46 V (3 significant figures) and an ammeter reads 0.38 A (2 significant figures).
R = V / I = 2.46 / 0.38 = 6.4736... Ω
Write 6.5 Ω (2 s.f.) or 6.47 Ω (3 s.f.).
Writing 6.4736842 Ω claims far more precision than a 2-figure ammeter reading can give. Most mark schemes accept 2 or 3 significant figures in a final answer.
Try it step by step
Work it out in steps: Calculator value of R = V / I in ohm, then Fewest significant figures in the data, then R to that number of sf in ohm.
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Problem 1.
Calculator value of R = V / I in ohm: R = 3.72 / 0.45 = 8.2666... ohm
Fewest significant figures in the data: 3.72 V has 3 sf and 0.45 A has 2 sf, so the data justify 2 sf
R to that number of sf in ohm: R = 8.3 ohm (2 sf)
Problem 2.
Calculator value of R = V / I in ohm: R = 5.1 / 0.235 = 21.702... ohm
Fewest significant figures in the data: 5.1 V has 2 sf and 0.235 A has 3 sf, so 2 sf
R to that number of sf in ohm: R = 22 ohm (2 sf)
Problem 3.
Calculator value of R = V / I in ohm: R = 2.84 / 0.62 = 4.5806... ohm
Fewest significant figures in the data: 2.84 V has 3 sf and 0.62 A has 2 sf, so 2 sf
R to that number of sf in ohm: R = 4.6 ohm (2 sf)
4. Plot a graph the way it is marked
- Plot the right way round. "Plot R against l" means R goes up the vertical axis and l goes along the horizontal axis.
- Label both axes with the quantity and unit, written the same way as the table heading, such as R / Ω.
- Choose an easy scale. Let each large square stand for 1, 2 or 5 units (or 10, 20, 50 and so on). Never use 3, 6 or 7: these make points hard to plot and read.
- Fill at least half the grid in both directions. The axes do not need to start at zero unless the question asks for it.
- Plot each point carefully, to within half a small square, as a small cross or a dot with a circle round it.
- Draw one thin line of best fit. Use a ruler for a straight line, or draw one smooth curve. Leave about as many points above the line as below it. Never join the dots point to point.
5. Find a gradient from a large triangle
Draw a right-angled triangle on your best-fit line that uses more than half the length of the line. Read the two corners from the line itself, not from your table, unless a data point lies exactly on the line. Write both coordinates in your working.
Worked example
Gradient of a resistance-length graph
A student measured the resistance R of different lengths l of one wire.
| l / cm | R / Ω |
|---|---|
| 20.0 | 3.1 |
| 30.0 | 4.4 |
| 40.0 | 6.1 |
| 50.0 | 7.4 |
| 60.0 | 9.0 |
| 70.0 | 10.6 |
Her best-fit straight line passes through the origin. She picks two points on the line, (20.0 cm, 3.0 Ω) and (70.0 cm, 10.5 Ω), which span the whole drawn line.
Gradient = (10.5 - 3.0) / (70.0 - 20.0)
= 7.5 / 50.0 = 0.150 Ω/cm
The gradient has a unit: the y-unit divided by the x-unit. Here it is the resistance of each centimetre of wire. A straight line through the origin shows that R is directly proportional to l.
6. Name a real source of error and a matching improvement
"Human error" or "repeat the experiment" on its own scores nothing. Name the specific problem, say how it affects the reading, and give an improvement that fixes that problem.
| Source of error | Matching improvement |
|---|---|
| Reaction time is large compared with the time of one swing. | Time 20 oscillations and divide by 20. |
| Parallax when reading a scale from an angle. | Place the eye directly in line with the mark, perpendicular to the scale. |
| The wire heats up, so its resistance rises during the experiment. | Use a small current and open the switch between readings. |
| The hot liquid loses energy to the surroundings. | Insulate the beaker and cover it with a lid. |
You will also be asked to plan an experiment of your own. Planning an experiment shows the full structure, with a worked plan.
Try it step by step
Work it out in steps: Choose the y-axis scale, then Gradient from a large triangle in cm/N, then Intercept on the length axis in cm.
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Problem 1.
Choose the y-axis scale: Use 2 cm per large square, 0 to 12 cm: the points then fill the whole grid and every reading is easy to plot.
Gradient from a large triangle in cm/N: Draw the best-fit line and a triangle covering most of it: gradient = (12.0 - 4.0) / (5.0 - 0) = 1.6 cm/N
Intercept on the length axis in cm: The line meets the length axis at F = 0: intercept = 4.0 cm, the unstretched length
Problem 2.
Choose the y-axis scale: Use 2 cm per large square, 0 to 16 cm, which fills the grid.
Gradient from a large triangle in cm/N: Gradient = (16.0 - 6.0) / (4.0 - 0) = 2.5 cm/N
Intercept on the length axis in cm: Intercept = 6.0 cm
Problem 3.
Choose the y-axis scale: 2 cm per large square, 0 to 12 cm: the points fill most of the grid and the scale is easy to read.
Gradient from a large triangle in cm/N: Gradient = (11.0 - 3.0) / (10.0 - 0) = 0.80 cm/N
Intercept on the length axis in cm: Intercept = 3.0 cm
Check your understanding
A student measures the length of a spring with a metre rule marked in millimetres. The spring ends exactly at the 12 cm mark. How should she record the length in her table?
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12.0 cm
The rule is marked in millimetres, which is 0.1 cm. So record every reading to 0.1 cm, even when the last digit is zero.