Random Variable

A random variable, as defined by J. Susan Milton and Jesse Arnold, is an assigned value, usually a real number, to all individual possible outcomes of an experiment. This mapping from outcome to value is a one-to-one relationship, meaning that for every experimental outcome, only one numeric value is associated with it.

Take for example the tossing of two coins. The outcomes of such an experiment will have the following possibilities: heads, heads; heads, tails; tails, heads; and tails, tails. For this experiment, the random variable will map those categorical outcomes to numerical values, such as by letting the random variable X be the number of heads observed. Preserving the above order, the possible outcomes for X are 2, 1, 1, and 0. This particular set lists each element with an equal probability; in this case, the four outcomes each have a 25% chance of occurring.

The listing of all possible outcomes of an experiment is called the sample space. In the case of the random variable X, which counts the number of heads, the sample space can be simplified to the set (2, 1, 0) where the values 2 and 0 each have a 25% chance of occurring and the value 1 has a 50% chance, since the outcomes (heads, tails) and (tails, heads) both have a value of 1 in terms of X.

A random variable is the unpredictable occurrence of a subset of outcomes from the aforementioned sample space. There are two types of random variables, namely, qualitative and quantitative, with quantitative random variables being the more widely used version in both applied and theoretical settings.

Qualitative random variables’ defining characteristic is that they can be categorized into specific groups. The focus of the variable is on a “quality” that cannot be mathematically measured. Some examples of qualitative variables are characteristics such as hair color, eye color, or agreeability toward statistics. These variables contain outcomes that are not arithmetically comparable to one another, and they are also referred to as nominal variables.

Quantitative random variables are assigned an actual numeric value for each element that occurs in the sample space, as in the coin-tossing example above. A prerequisite for quantitative variables is that the numeric values will have a mathematical relationship with other values. This relationship may be as rudimentary as one value’s being greater than another. Examples of quantitative variables include a person’s height, age, or the position in which they finished a race. Quantitative variables can be further classified into interval, ratio, and ordinal scales.

In an interval scale, the key principle is that the distance between any value and its neighboring value will be the same distance found between any other two adjacent values on the scale. This scale does not contain an absolute zero point, so it does not make sense to construct a ratio of interval values. For example, the temperature scale of Fahrenheit is an interval scale, and [Page 1356]it is not proper to say that a 100-degree day is twice as hot as a day with a high of 50 degrees Fahrenheit.

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