Types of variables

Scientists rarely just watch. Most research works by changing the size of one physical quantity on purpose and then observing how a second physical quantity responds. Anything in an experiment that changes, or could be changed, is called a variable, because its value varies from one trial to the next. Each variable has a name and is usually given a short symbol, so that the relation between variables can be written as a mathematical formula.

Not every variable plays the same part. Some are changed by the experimenter, some change as a result, and some have to be held fixed so that they do not spoil the result. Variables are therefore usually sorted into three types: the independent variable, the dependent variable and the controlled variable.

One simple experiment shows all three types clearly. A rubber band is fixed to a catapult, and a small folded piece of paper is used as a bullet. The band is pulled back by a measured amount and released, and the distance the paper bullet travels is measured.

The band is stretched by 4 cm first, then by 6 cm, 8 cm and 10 cm. Each time, the distance travelled by the bullet is written in a table like this one:

If the stretch is written as e and the distance as x, the results show that the distance grows as the stretch grows. This is written \( x \propto e \) and read as 'x is directly proportional to e'. The experiment also contains two quantities that could have been changed but were not: the thickness of the rubber band and the size of the paper bullet. These are variables too, and they matter just as much as the other two.

In every experiment the researcher changes one variable by a chosen amount and measures the change this brings in another variable. The change made by the experimenter is called the causative factor, and the change that follows in the other variable is called the effect.

In the catapult experiment, the stretched band throws the bullet. The extension of the band is the causative factor and the distance travelled by the bullet is the effect. Nobody tells the experimenter how far to stretch the band. They are free to choose 4 cm, 7 cm or 10 cm. A variable that the experimenter can set freely in this way is called the independent variable.

The causative factors in an experiment are its independent variables. When studying how a plant grows, the amount of fertilizer, the amount of sunlight and the amount of water can each be the independent variable. The experimenter picks just one of them, varies it, and watches what happens to the plant.

Before letting go of the band, the experimenter cannot say exactly how far the bullet will fly. That distance is not something they choose. It depends on how far the band was stretched, and it can only be found by measuring it after the launch. A variable like this is called the dependent variable.

In the catapult experiment, the distance covered by the paper bullet is the dependent variable. In the same way, when observing the effect of sunlight on the growth and development of a plant, the amount of sunlight is the independent variable and the height or growth of the plant is the dependent variable. There is a quick test for the dependent variable: ask yourself 'which quantity am I waiting to find out?' The answer is the dependent variable.

Many experiments involve more than two variables, and each of them can influence the result. Suppose the first trial uses a thin rubber band, and the second uses a thick band with a bigger stretch as well. If the bullet flies further, nobody can say whether the extra stretch or the thicker band caused it. It becomes hard to draw a conclusion, and any conclusion drawn may not be valid or reliable.

So every variable other than the chosen independent and dependent variables is kept under control from the first trial to the last. This makes the finding of the experiment valid, reliable and accurate. The variables held in check throughout the experiment in this way are called controlled variables.

In the catapult experiment, the same rubber band of the same thickness and paper bullets of the same size are used every time. If a different band were used in each trial, the result would not be valid. So the thickness of the rubber band and the size of the paper bullet are the controlled variables of this experiment.

When observing the effect of light on plant growth, all the plants should be the same size at the start. The air, water and manure given to them should also be kept equal throughout the experiment. So the initial size of the plants and the air, water and manure supplied to them are the controlled variables here.

When studying how the surface area of limestone affects the rate of its chemical reaction with an acid, the acid used each time should have the same concentration, and the quantity and weight of limestone should also be kept the same. In this experiment the surface area is the independent variable, the rate of reaction is the dependent variable, and the concentration of the acid and the quantity and weight of the limestone are the controlled variables.

Keeping a variable steady does not happen by luck. It needs special arrangements while the experiment is being designed, such as taking all the limestone from one bag, or watering every pot with the same measuring cup.

Setting out the variables of the three experiments in one table makes their roles easy to tell apart. Every row has exactly one independent variable and one dependent variable, but there can be more than one controlled variable.

The same three types of variable turn up in every piece of research, in the laboratory and in daily life. The table below shows the variables of four more investigations.

Look at the tap. The further the knob is turned, the more water flows out each minute. But if the water pressure in the pipe changed during the experiment, the flow would change even with the knob held still, so the pressure must stay the same. In the sugar investigation the water is always 100 grams, because more water would dissolve more sugar whatever the temperature. In the exercise investigation, the longer someone waits after exercising before counting, the more the heart settles down, so that waiting time is also kept the same every time.

Notice that every measured quantity in the table carries its unit: degree, litre, ampere, degree celsius, grams, minutes. A number without its unit means nothing, so a variable is always written with the unit it is measured in.

Five points need attention whenever variables are handled. The first three apply while the experiment is being planned:

• There should be only one independent variable in research or an experiment.
• There should be only one dependent variable in research or an experiment.
• Except for the specified independent and dependent variables, all other variables should be controlled.

The reason behind all three is the same. When only one cause is changed and only one effect is measured, any change in the result can be traced with confidence to that one cause. The other two rules are about how the variables are written, in an equation and on a graph.

When the relation between variables is written as an equation, the dependent variable is usually written on the left side and the independent variable on the right side. For this reason the independent variable is sometimes called the right variable and the dependent variable the left variable.

When we study how the distance covered by a moving object varies with time, the equation of their relationship is:

\[ s = vt \]

Here the distance travelled, s, is the dependent variable, so it sits on the left. The time, t, is the independent variable, so it sits on the right. The speed, v, is kept the same while the distance is measured at different times, so speed is the controlled variable. The catapult result \( x \propto e \) follows the same pattern: the dependent distance x on the left and the independent stretch e on the right.

When the relationship between variables is plotted on a graph, the dependent variable is always plotted on the y-axis, the vertical line, and the independent variable on the x-axis, the horizontal line. That is why the dependent variable is sometimes called the vertical variable or the y-variable, and the independent variable the horizontal variable or the x-variable.

A graph always shows how a change in the independent variable brings a change in the dependent variable. Read it from left to right: as you move along the x-axis to a bigger stretch, the line climbs to a longer distance.