Continuous Variable

Introduction

Continuous variables are the ones which in between any two numeric values have an infinite number of values. Variables in statistics can be quantitative variables, which define the quantity of measurement. Quantitative measurement can further be divided into continuous variables and discrete variables. Such variables are used widely in statistical research and tools. In this tutorial, we will learn about variables and its types, continuity, and continuous variables along with some solved examples.

Variables

In statistics, a variable may be defined as a term that represents an unknown value which is not fixed and is in numerical form. Variables thus can be used for easy calculation of large data. Variety of variables are used in various domains. Variables can be of two types: continuous variable and discrete variable. Variables can also be categorised into several other types such as quantitative variables, qualitative variables, dependent and independent variables and many others.

Continuity

In nature continuity is seen all around us. For example, the flow of rivers, time etc. are some of the real-life examples of continuity in nature. In statistics also we have continuity of functions having numerical values. Graphically, a continuous function corresponds to a continuous graph. Algebraically, a function f(a) can be called continuous at a point a=x only if it satisfies the following conditions:

• f(x) exists (which means the value of f(x) is finite.

• $\mathrm{\lim_{a \rightarrow x}f(a)}$ exists (i.e., both sides of the limit are equal, and both of them are finite)

• $\mathrm{\lim_{a \rightarrow x}f(a)\:=\:f(x)}$

The function f(a) is said to be continuous if the three conditions mentioned above are satisfied for every point in an interval.

Continuous Variables

Continuous variables may be defined as variables which have numerical value which is determined with the help of measurements. Continuous variables can have incremental value in fractions or decimals. Continuous variables can take up any numerical value that can be divided into increments. Such variables can have infinite values in a range. If a variable is discrete around a value and the variable is non-infinite on either side, then the variables are called discrete variables. Continuous variables are generally used to measure scales such as length, mass, or temperature.

Continuous variables can be used to calculate various statistical operations such as mean, mode, median, etc. Continuous variables which have infinite values in between two numerical values. Various calculus tools are used in problems involving continuous variables. Different techniques of calculus are used in continuous optimization problems in which the variables are also continuous. There are two types of continuous variables namely, instant variables and ratio variables.

The table given below shows the difference between two types of variables, i.e., discrete and continuous variables.

Discrete Variable	Continuous Variable
Value of a discrete variable can be determined by counting.	Value of a continuous variable cannot be determined by measuring.
Examples − Number of days in a week Number of employees in an office	Examples − Number of trees in the world Temperature of a city
Come under a specific range which is finite.	Come under a specific range which is infinite.
Can be used to assume a specific or distinct value	Can be used to assume any value within a range.

Solved Examples

1)Below are two cases of variables. Determine the type of variable in each case and explain why?

Case 1 − The length of a polar bear

Case 2 − Age of a polar bear

Answer −

Case 1 − As we know that the length is a continuous variable. Because the length of the polar bear cannot be measured exactly. Each measurement cannot precisely define the length of a bear. This is why the length is a continuous variable.

Case 2 − Age can be considered a continuous variable in some cases and discrete in some cases. Because the age is continuously incrementing with time we cannot fully define it, therefore we consider age a continuous variable. Age is defined usually in terms of years, in such cases age is considered a discrete variable.

2)For a function f(a) =

$\mathrm{5\:-\:2a\:for\:a<1}$

$\mathrm{3\:for\:a\:=\:1}$

$\mathrm{a\:+\:2\:for\:a>1}$

Prove that the function is continuous for all values of a?

Answer − As the function is linear the function is a straight line on a graph, which means the function is continuous for all a≠ 1. Now for a = 1, we have to check the conditions.

Left-hand limit −

$\mathrm{=\:\lim_{a \rightarrow 1^{-}}\:f(a)}$

$\mathrm{=\:\lim_{a \rightarrow 1^{-}}\:f(5\:-\:2a)}$

$\mathrm{=\:5\:-\:2\times\:1}$

$\mathrm{=\:3}$

Right hand limit −

$\mathrm{=\:\lim_{a \rightarrow 1^{+}}\:f(a)}$

$\mathrm{=\:\lim_{a \rightarrow 1^{+}}\:f(a\:+\:2)}$

$\mathrm{=\:1\:+\:2}$

$\mathrm{=\:3}$

At value a=1

𝑓(1) = 3

As all the conditions are satisfied, we can say that the given function is continuous for all a.

3)Look into the scenarios given below and determine which one of them is continuous

• Number of pets

• Age of a person

• Temperature of a city

• Outcomes from flipping a coin

From the scenarios given above temperature of a city is continuous variable because temperature can be divided further into smaller units. Age of a person is considered both discrete and continuous. All the other scenarios the variable cannot be divided into further smaller units.

Conclusion

Variable may be defined as a term that represents an unknown value which is not fixed and is in numerical form. Variables can be of two types; continuous variable and discrete variable. In nature continuity is seen all around us. For example, the flow of rivers, time etc. are some of the real life examples of continuity in nature. Continuous variables may be defined as variables which have numerical value which is determined with the help of measurements.

Continuous variables can be used to calculate various statistical operations such as mean, mode, median, etc. There are two types of continuous variables namely, instant variables and ratio variables. If a variable is discrete around a value and the variable is non-infinite on either side then the variables are called discrete variables. For example, the number of days in a month, number of computers in the classroom.

FAQs

1. Can gender be a continuous variable?

No, gender cannot be considered a continuous variable because it is fixed. It is a discrete variable.

2. What are instant variables?

Instant variables are variables which have a static value. For example, temperature in Delhi in winters.

3. What are two types of important variables other than continuous and discrete variables?

The two other types of variables are dependent and independent variables.

4. What are quantitative variables?

Variables which represent a certain quantity of values are called quantitative variables. Two types of quantitative variables are: Continuous variables and discrete variables.

5. What are the real-life examples of continuity around us?

Some of the real-life examples of continuity are −

• The flow of air in our atmosphere is continuous because it never stops.

• The age of a person or animal or any living being is continuously increasing.

• The growth of hair in our body is a continuous process.