Graphs and data analysis

A table of numbers hides the pattern in an experiment; a graph shows it at a glance. A best-fit line through the points evens out small reading errors, and its gradient and intercept tell you how the quantities are related. The pendulum readings from measuring and recording practical data show how to process the readings, plot them, find a gradient and state a conclusion the data support.

Process the readings

Each valid elapsed time is for 20 complete oscillations, so divide the mean elapsed time by 20 to find the period T, then square the unrounded period.

In the constructed example, the observer confirmed that the 26.9 s timing at 0.500 m counted only 19 oscillations. It remains in the raw-data record with that note; it is excluded from the mean of valid 20-oscillation timings. A third, correctly counted run gave 28.2 s. A point being inconvenient for a graph is never enough reason to remove it.

The processed table uses the valid timings. A dash means no third reading was supplied, not a zero time.

T² is worked out from the unrounded period. For 0.300 m, T = 21.9 ÷ 20 = 1.095 s and T² = 1.199 ≈ 1.20 s². Squaring the rounded 1.10 would give 1.21 instead.

Plot a graph and read it

Exam questions usually tell you which graph to plot. For these example readings, plotting T² against l gives an approximately straight pattern.

• Put the quantity you changed (l) on the horizontal axis unless the question says otherwise.
• Label each axis with the quantity and unit.
• Choose a simple scale (1, 2 or 5 units per large square, or those times a power of ten) so that the points fill more than half the grid in each direction.
• Plot each point as a small, sharp cross.
• If the pattern is straight, draw one best-fit line with points roughly balanced on both sides. Do not join the dots or automatically force the fit through the origin. Use a smooth curve when the data indicate a curved relationship.

Period squared against pendulum length

Period squared T² in s² plotted against pendulum length l in m. Six example points from 0.200 m to 0.700 m lie close to a straight line through the origin. The gradient triangle runs horizontally from l = 0.10 m to 0.70 m at T² = 0.402 s², then vertically to T² = 2.814 s². The table lists each plotted coordinate.

Scroll across the figure to read all labels.

Open full-size graph

What to noticeThese are constructed practice readings, not reported measurements. The fit is consistent with direct proportionality over the tested range; do not force an experimental fit through the origin. The gradient triangle is a guide, not extra data.

View figure data

Figure data: Period squared against pendulum length

Series	Length l (m)	T² (s²)
Example readings	0.20	0.801
Example readings	0.30	1.200
Example readings	0.40	1.610
Example readings	0.50	2.000
Example readings	0.60	2.400
Example readings	0.70	2.820
Best-fit line	0.00	0.000
Best-fit line	0.75	3.015
Gradient triangle	0.10	0.402
Gradient triangle	0.70	0.402
Gradient triangle	0.70	2.814

Figure data: Period squared against pendulum length

Series	Length l (m)	T² (s²)
Example readings	0.20	0.801
Example readings	0.30	1.200
Example readings	0.40	1.610
Example readings	0.50	2.000
Example readings	0.60	2.400
Example readings	0.70	2.820
Best-fit line	0.00	0.000
Best-fit line	0.75	3.015
Gradient triangle	0.10	0.402
Gradient triangle	0.70	0.402
Gradient triangle	0.70	2.814

The example points are close to a straight line through the origin, so they are consistent with T² being directly proportional to l over the lengths tested. The drawn fit has a gradient of about 4.02 s²/m. The right-angled triangle shows its rise and run; its corners are points on the line, not additional readings.

A straight line with a non-zero intercept does not show direct proportionality. Check whether an offset is supported by the data instead of moving the line to the origin. A short range or scattered readings also limits how confidently you can describe the relationship.

To find the gradient, draw a large right-angled triangle on the line, using at least half of its length. Read two points on the line, not data points.

You can also read values from the line. Answer these from the graph.

Anomalous points. If one point lies clearly off an otherwise good line, check its raw readings and repeat it if you can. Retain the raw reading and plotted point. Explain any decision to exclude it using the method and further evidence, not simply because it lies away from your preferred line. Without enough evidence, describe the uncertainty and investigate further.

Common mistakes

• Rounding the period before squaring it.
• Joining the dots instead of drawing one best-fit line or smooth curve.
• Finding a gradient from a small triangle, or from data points instead of points on the line.
• Leaving out the gradient’s unit: the vertical-axis unit divided by the horizontal-axis unit.
• Forcing the line through the origin when the data do not support it.
• Removing a point only because it is off the line, without evidence from the method.