For a discrete random variable the expected value is calculated by summing the product of the value of the random variable and its associated probability, taken over all of the values of the random variable.

random variable has infinitely many values associated with measurements. of a discrete random variable represents the mean value of the outcomes. The normal distribution is a continuous probability distribution being used as an approximation to the binomial distribution which is a discrete probability distribution.

The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi.

The probability function P(X=x) is also denoted by f(x) and is called the pmf of the discrete random variable X. Some authors refer f(x) as the frequency function or a probability function.

In general, quantities such as pressure, height, mass, weight, density, volume, temperature, and distance are examples of continuous random variables.

A continuous variable is a variable whose value is obtained by measuring, ie one which can take on an uncountable set of values. For example, a variable over a non-empty range of the real numbers is continuous, if it can take on any value in that range.

Age is measured in units that, if precise enough, could be any number. Therefore the set they come from is infinite. For example, someone could be 22.32698457 years old or 22.32698459 years old. We could be infinitly accurate and use an infinite number of decimal places, therefore making age continuous.

Mondal[1] suggests that age can be viewed as a discrete variable because it is commonly expressed as an integer in units of years with no decimal to indicate days and presumably, hours, minutes, and seconds.

Eye color is solely considered a discrete variation rather than continuous variation. It is a polygenic, hereditary trait. Since eyecolor is unchangable and, as a variable, contains a finite amount of variance, it is not able to become continuous (in the way that height, weight, and salary are continuous variables).

And clearly hair color, skin color and eye color all fall under the definition of a continuous trait, because even though they dont seem to be affected by the environment, they are definitely polygenic traits and show a gradation, so they’re definitely continuous traits.

Nominal scales Gender, handedness, favorite color, and religion are examples of variables measured on a nominal scale.

Quantitative variables

Quantitative Variables – Variables whose values result from counting or measuring something. Qualitative Variables – Variables that are not measurement variables. Their values do not result from measuring or counting. Designator values usually do not repeat in a table, but variable values often do repeat.