An index number in statistics is a tool used to track changes in a variable or a group of related variables, typically over time. Index Numbers condense the complex data into a single number (expressed as a percentage) for easier comparison between different periods or situations.

**Example:** A factory manager may wish to compare this month’s per-unit production cost with that of the past 6 months.

An index number measures how much a variable changes over time.

### Simple Relatives Index Numbers

A simple relative is a ratio of the value of a variable in a given period to its value in the base (or reference) period.

If $x_0$ and $x_n$ are the values of a variable during the base period and a given period, respectively, then the simple relative, denoted by $x_{0n}$ is

$$x_{0n}=\frac{x_n}{x_0}$$

A relative is usually expressed as a percentage by multiplying by 100.

#### Simple Price Relative

If $p_0$ and $p_n$ are the prices of a commodity \texturdu{مفید شے، مال اسباب} during the base period and a given period, respectively, then the simple price relative, denoted by $p_{0n}$ is

$$p_{0n}=\frac{p_n}{p_0}$$

The price is generally defined as “money per unit quantity” and is usually taken as the average price for a period because the prices are not constant throughout a period.

#### Simple Quantity (Volume) Relative

If $q_0$ and $q_n$ are quantities of a commodity (produced, consumed, purchased, sold, exported, or imported, etc.) during the base period and a given period, respectively, then the simple quantity relative, denoted by $q_{0n}$ is

$$q_{0n}=\frac{q_n}{q_0}$$

#### Value

If $p$ is the price of a commodity and $q$ is its quantity during a period, then the value $v$ is given by $v=p\,q$. For example, if a quantity of 560kg of a commodity is purchased at the rate of Rs. 5 per Kg then

$$v=pq=5\times 560 = 2800$$

#### Simple Value Relative

If $v_0$ and $v_n$ are the values of a commodity during the base period and a given period, respectively, then the simple value relative, denoted by $v_{0n}$ is

$$v_{0n}=\frac{v_n}{v_0}=\frac{p_nq_n}{p_0q_0}=\frac{p_n}{q_n}\times \frac{q_n}{q_0}=p_{0n}\times q_{0n}$$

### Uses of Index Number in Statistics

- Functions: Measure changes in variables like prices, production levels, or stock values.
- Benefits:
- Simplifies complex data comparisons
- Tracks trends over time
- Provides a benchmark for analysis (often using a base period as a reference point at 100)

- Examples:
- Consumer Price Index (CPI) tracks inflation by measuring changes in the prices of a basket of goods and services.
- Stock market indices like the S&P 500 track the overall performance of a specific stock market section.

Note that there are various types of index numbers used for different purposes. Computing the index numbers involves specific formulas and functions that take into account the chosen base period and the way different variables are weighted within the index.