Designing Experiments and Controlling Variables

A well-designed physics experiment investigates how one quantity affects another by changing the independent variable, measuring the dependent variable, and keeping all other (control) variables constant, with repeated measurements to improve reliability. In brief: the steps are to state a question or hypothesis, identify the variables, choose the range and number of values of the independent variable (at least five, evenly spread), select instruments with suitable resolution, plan the procedure and safety measures, repeat the measurements, and record the data in a table with headings and units. A fair test changes only one variable at a time.

Experimental design determines whether the results can answer the question. A poorly controlled experiment gives data that cannot be interpreted. This guide explains the types of variable, how to plan the investigation and the table of results, how repeated measurements improve reliability, compares the types of variable in a table, works through a complete example with a pendulum and corrects common mistakes.

What are the steps of an investigation?

1. Question or hypothesis: a testable statement, such as “the period of a simple pendulum depends on its length, but not on the mass of the bob”. A hypothesis should predict a result that could be shown to be wrong (falsifiability).
2. Variables: decide what to change, what to measure and what to keep constant.
3. Apparatus and method: choose the instruments (the resolution should be suitable for the size of the quantity), draw a labeled diagram, describe the procedure in enough detail for another person to repeat it, and decide the range of values (for example, five to ten values of the independent variable, over as large a range as practical).
4. Risk assessment: identify hazards (hot objects, electricity, heavy masses falling, lasers), and control measures.
5. Data collection: record all the raw measurements, with units and uncertainties, in a prepared table, immediately.
6. Analysis: calculate, plot graphs, find relationships, and estimate uncertainties.
7. Conclusion and evaluation: state what the results show, compare them with the hypothesis and accepted values, and discuss the limitations and improvements.

What are the types of variables?

• Independent variable: the quantity that the experimenter deliberately changes (usually plotted on the x-axis).
• Dependent variable: the quantity that is measured, expected to respond to the change (usually plotted on the y-axis).
• Control (controlled) variables: all the other quantities that could affect the result and must be kept constant, so that any change in the dependent variable can be attributed to the independent variable. If they are not controlled, they become confounding variables.

A control experiment (a comparison with a standard condition) may also be included. Variables can be categoric (type of material), discrete (number of coils) or continuous (length, temperature). Controlled variables can be kept constant by using the same apparatus, the same position, the same temperature (using a water bath), the same starting conditions and the same person for timing.

Why repeat measurements?

Random errors cause readings to vary around the true value. Repeating each measurement (at least three times) and taking the mean reduces the effect of random errors, shows how consistent the results are (reliability: similar results are obtained when the experiment is repeated), and allows anomalous results to be identified and, if there is a reason, rejected. The spread of repeated readings gives an estimate of the uncertainty (half the range). Another way of reducing the uncertainty in a short time interval is to measure many events together (the time for 10 oscillations divided by 10), as the same absolute timing error is then divided by 10. Repeating the whole experiment with a different method or apparatus checks for systematic errors (validity).

Variables compared

Worked example: a pendulum investigation

Question: how does the period of a simple pendulum depend on its length, and what is the value of g?

1. Prediction from theory: T = 2π√(L/g), so T² = (4π²/g)L: a graph of T² against L should be a straight line through the origin with gradient 4π²/g.
2. Method: a small dense bob (a metal sphere) is hung on a thin string from a clamp. The length L is measured from the point of suspension to the center of the bob. The bob is displaced by a small angle (about 5° to 10°) and released; the time for 10 complete oscillations is measured, starting the stopwatch as the bob passes the equilibrium position (a fiducial marker is placed there to reduce errors in timing). The timing is repeated twice more for each length and the mean is taken.
3. Table of results (mean values):

   Results for 10 oscillations

   L (m)	Time for 10 oscillations (s)	T (s)	T2 (s2)
   0.200	9.0	0.900	0.810
   0.400	12.7	1.270	1.613
   0.600	15.6	1.560	2.434
   0.800	17.9	1.790	3.204
   1.000	20.1	2.010	4.040

4. Why these choices? Five lengths are enough to see a trend and draw a line; timing 10 oscillations reduces the percentage uncertainty from the reaction time; keeping the amplitude small means that T does not depend on it; using the same bob controls the mass (which in theory does not matter, but should be checked in a separate test by changing the mass at constant L).
5. Safety: make sure the clamp is stable and secured to the bench, and keep feet away from under the bob.

The analysis of these data is continued in the articles on uncertainties, graphs and linearization (these data give g ≈ 9.8 m/s²).

How did the scientific method develop?

Galileo introduced the systematic use of controlled experiments and mathematics in physics in the early 1600s (for example, studying pendulums and falling bodies on inclined planes). Francis Bacon argued for induction from observation (1620), and Karl Popper emphasized falsifiability as the criterion for scientific theories (1934). Ronald Fisher developed the statistical design of experiments in the 1920s and 1930s. Modern data loggers and sensors automate measurements and allow many repeated readings.

Which mistakes are common?

Mistake 1: Changing more than one variable at a time

Then it cannot be known which change caused the effect.

Mistake 2: Confusing the dependent and independent variables

The independent variable is what you choose to change. The dependent variable is what you measure.

Mistake 3: Using too few values or too small a range

With two or three points, the pattern of the relationship cannot be seen. Use at least five values over a wide range.

Mistake 4: Not recording the units, resolution or repeated values

Data without units and uncertainties are incomplete, and the repeated readings are needed for assessing reliability.