If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable. A numerically valued variable is said to be continuous if, in any unit of measurement, whenever it can take on the values a and b. If the random variable X can assume an infinite and uncountable set of values, it is said to be a continuous random variable. Suppose, we are interested in measuring the heights of a large number of plants and if our unit of measurement is very fine, there would be no point along the scale of measurement (between the extreme values of the heights) at which we may not find the height of the plant, no more matters, how finely we divide the scale.

### Few points to be considered

- A continuous variable is defined as the function of the interval.
- The continuous variable may be qualitative or maybe a quantitative.
- The continuous variable is the measurable characteristics or attributes for the different subjects in the variables.
- The continuous variable has the value of the uncountable set of values.
- The continuous variable has an infinite set of data.
- The probability for any specific value in the continuous random variable is zero.

The ranges for the continuous variable may change for each function.

### Continuous Random Variable:

Quite unlike a discrete probability distribution, the continuous probability distribution cannot be presented or formed in the tabular form. It has either a formula form or a graphical form. A frequency polygon gets smoother and smoother as the sample size gets bigger, and the class intervals become more and more numerous and narrower. Finally, the density polygon becomes a smooth curve better known as the density curve. The function that defines the curve is called a probability density function.

If f is a distribution function of the continuous random variable X and if a < b then P(a < X ≤ b) = F(b) – F(a)

P(a ≤ X ≤ b) = P(X = a) + F(b) – F(a)

P(a < X < b) = F(b) – F(a) – P(X = a)

P(a ≤ X < b) = F(b) – F(a) + P(X = a) – P(X = b)

F(x) ≤ F(y) if x