A variable is a characteristic that can take different values across individuals — for example height, gender, score, or eye colour. Something that does not vary (the same for everyone) is called a constant.

The four measurement scales, from simplest to most sophisticated:

• Nominal — categories with no inherent order: gender, country of birth, colour. Only equality/difference can be assessed; frequencies may be counted.
• Ordinal — categories ranked in order, but the gaps between ranks are not equal: satisfaction rating (low–medium–high), competition placement.
• Interval — ordered with equal gaps, but the zero point is arbitrary: temperature in Celsius, calendar year. Ratios are meaningless (\(20^\circ\) is not 'twice as hot' as \(10^\circ\)).
• Ratio — like interval but with a true absolute zero: height, weight, time, income. Ratios are meaningful (\(8\) kg is twice \(4\) kg).

Discrete vs. continuous: a discrete variable takes separate, countable values (number of children, number of cars), while a continuous variable can take any value within a range and is measured (height, time).

Variable roles in research: the independent variable is the factor the researcher manipulates or examines (the 'explanatory' variable), and the dependent variable is the outcome that is measured. A causal relationship means that a change in one variable causes a change in the other — unlike a mere correlation, which may arise from a third confounding variable.

Example 1: Identifying the Measurement Scale

Problem: A researcher recorded for each participant: (a) phone number, (b) judo belt rank (white, yellow, black), (c) body temperature in Celsius, (d) weight in kg. Classify each according to its measurement scale.

Solution:

1. (a) A phone number is an identifying label with no order or quantity — nominal scale.
2. (b) Belt ranks have a clear order but the gaps between ranks are not numerically equal — ordinal scale.
3. (c) Temperature in Celsius: equal gaps exist but zero is arbitrary (\(0^\circ\) does not mean 'no heat') — interval scale.
4. (d) Weight: there is a true absolute zero (\(0\) kg = no mass) and ratios are meaningful — ratio scale.

Answer: Nominal, ordinal, interval, ratio — respectively.

Example 2: Discrete or Continuous

Problem: Classify each variable as discrete or continuous: (a) number of text messages a person sends per day, (b) running time for 100 metres in seconds, (c) number of passengers on a bus.

Solution:

1. (a) Number of messages is a whole number that is counted; '3.5 messages' is impossible — discrete.
2. (b) Running time can be measured to any level of precision (\(11.43\) s, \(11.431\) s) — continuous.
3. (c) Number of passengers is counted and always a whole number — discrete.

Answer: Discrete, continuous, discrete — respectively.

Example 3: Dependent and Independent Variables

Problem: A study investigates whether the number of hours of sleep per night affects the next-day exam score. What is the independent variable and what is the dependent variable?

Solution:

1. We ask: which is the factor suspected of having an effect, and which is the measured outcome?
2. Hours of sleep is the influencing factor — it is the independent variable.
3. The exam score is the outcome measured as a result of sleep — it is the dependent variable.
4. Memory tip: the dependent variable 'depends' on the independent variable, just as the score depends on hours of sleep.

Answer: Independent variable: hours of sleep. Dependent variable: exam score.

Example 4: Causal Relationship or Confounding Variable

Problem: It was found that children with larger shoe sizes read better. Does shoe size cause better reading? What is the likely explanation?

Solution:

1. There appears to be a positive correlation between shoe size and reading ability, but correlation does not imply causation.
2. We look for a confounding variable that explains both variables simultaneously.
3. Age is the confounding variable: older children have larger feet and also read better.
4. Conclusion: this is a spurious correlation (correlation only), not a causal relationship between shoe size and reading.

Answer: Not a causal relationship; age is the confounding variable that explains the correlation.

Example 5: Permitted Operations by Scale

Problem: A student calculated the 'average' eye colour by assigning numbers (blue=1, green=2, brown=3) and obtained 2.1. Is the calculation valid?

Solution:

1. Eye colour is a variable on a nominal scale — the numbers are merely identifying labels.
2. On a nominal scale the only valid operation is counting frequencies (and identifying the mode); arithmetic is not permitted.
3. Computing a mean requires at least an interval scale, where the gaps between values are equal and meaningful.
4. Therefore the value \(2.1\) is meaningless — there is no such thing as an 'average colour'.

Answer: The calculation is invalid; computing a mean on a nominal scale is not permitted.

✗ Common mistake: Inferring a causal relationship from the mere existence of a correlation ('X is related to Y, therefore X causes Y').

✓ The correct way: Correlation is not causation. To claim causality, confounding variables must be ruled out — ideally through a controlled experiment in which only the independent variable is manipulated.

✗ Common mistake: Confusing interval and ratio scales and computing ratios for Celsius temperatures ('\(30^\circ\) is three times hotter than \(10^\circ\)').

✓ The correct way: Ratios are only valid on a ratio scale that has a true absolute zero. On an interval scale (Celsius, calendar year) the zero is arbitrary, so only differences are meaningful — not ratios.

✗ Common mistake: Classifying 'number of children' as continuous because it involves a number, or 'height' as discrete because it was recorded as a whole number.

✓ The correct way: The question is whether values are counted (discrete) or measured along a continuum (continuous), not whether a number appears. Number of children is counted → discrete; height is measured along a continuum → continuous, even when rounded.

Measurement scales:

• Discrete = counted; continuous = measured.
• Independent = the cause; dependent = the effect.
• Correlation \(\ne\) causation — beware of confounding variables.