What are variables in statistics? Explain categorical and numerical Variables with Example. Explain each random variable of probability distribution?
SOLUTION....
1. Variables in Statistics
In statistics, a variable is any characteristic, number, or quantity that can be measured or counted. It represents something that can vary from one individual to another or within the same individual at different times.
For example:
The height of students in a classroom.
The age of employees in a company.
The favorite color of people in a survey.
Each of these can take on different values, so they are called variables.
2. Types of Variables
(a) Categorical Variables
Categorical variables represent data that can be divided into groups or categories. They do not have numerical meaning; instead, they describe qualities or attributes.
Examples:
Gender: Male, Female, Other
Blood Group: A, B, AB, O
Marital Status: Single, Married, Divorced
Further divisions of categorical variables:
Nominal Variables: Categories without any natural order. Example: Eye color (blue, brown, green).
Ordinal Variables: Categories with a natural order. Example: Education level (Primary, Secondary, Graduate, Postgraduate).
(b) Numerical Variables
Numerical variables represent data that can be measured or counted in numbers. They show a measurable quantity.
Examples:
Height in cm: 160, 175, 182
Marks in exams: 45, 78, 92
Temperature: 25°C, 30°C
Further divisions of numerical variables:
Discrete Variables: Countable values (usually whole numbers). Example: Number of children in a family (0,1,2,3).
Continuous Variables: Measured values that can take any value in a range. Example: Weight of a person (65.2 kg, 72.8 kg).
3. Random Variables in Probability Distribution
In probability and statistics, a random variable is a variable whose possible values are the outcomes of a random phenomenon. It assigns numbers to outcomes in a probability experiment.
Types of Random Variables:
(a) Discrete Random Variable
Can take only a finite or countable number of values.
Example:
Rolling a die → outcomes: {1, 2, 3, 4, 5, 6}.
Number of heads when flipping three coins → outcomes: {0, 1, 2, 3}.
Probability Distribution Example (Discrete):
When tossing a fair coin once,
Random variable X = number of heads.
Possible values: {0,1}.
Probability distribution: P(X=0)=0.5, P(X=1)=0.5.
(b) Continuous Random Variable
Can take any value in a given interval (infinite possibilities).
Example:
The exact time taken by a student to complete an exam (120.5 minutes, 121.3 minutes, etc.).
The weight of newborn babies in a hospital.
Probability Distribution Example (Continuous):
Suppose X = weight of apples in a basket.
Possible values: between 100g and 250g.
Probability is represented using a Probability Density Function (PDF) instead of simple counts.
Example: P(100 ≤ X ≤ 150) gives the probability that apple weighs between 100g and 150g.
(c) Other Random Variable Concepts
Bernoulli Random Variable: Only two outcomes (success/failure, yes/no, 1/0).
Binomial Random Variable: Counts the number of successes in a fixed number of trials.
Poisson Random Variable: Represents the number of occurrences of an event in a fixed interval (e.g., number of calls received in an hour).
Normal Random Variable: Continuous variable that follows the normal (bell-curve) distribution.