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# Quantity Index Numbers

To measure the growth and progress of an economy, economists and scientists use many statistical tools. One such very important tool are index numbers. They help reveal the trends and tendencies of the economy and also help in the formulation of economic policies and laws.

There are broadly three types of index numbers – price index numbers, value index numbers, and quantity index numbers. Let us learn about the latter.

## What is an Index Number?

Very simply put, index numbers help us observe the change in some quantity that we cannot otherwise easily observe or measure. For example, we cannot directly measure the growth of business activity in an economy. However, we can study the changes in factors that influence this business activity.

So an index number is a tool to measure the change in a variable quantity that has happened over a defined period of time. These index numbers are not directly measurable, they are represented as percentages which express the relative changes in quantity.

## Quantity Index Numbers

Now we will specifically understand what are quantity index numbers. Quantity index numbers measure the change in the quantity or volume of goods sold, consumed or produced during a given time period. Hence it is a measure of relative changes over a period of time in the quantities of a particular set of goods.

Just like price index numbers and value index numbers, there are also two types of quantity index numbers, namely

• Unweighted Quantity Indices
• Weighted Quantity Indices

Let us take a look at the various methods, formulas, and examples of both these types of quantity index numbers.

### Unweighted Index: Simple Aggregate Method

Here we do a simple and direct comparison of the aggregate quantities of the current year, with those of the previous year. We express this index number as a percentage. No weights are assigned, it is the simplest calculation. The formula is as follows,

$$Q_{01}Â =Â \frac{\Sigma{Q_1}}{\Sigma{Q_0}} \times 100$$

where, Q1 is the quantity of the current year, and Q0 is the quantity of the previous year,

### Unweighted Index: Simple Average of Quantity Method

In this method, we take the aggregate quantities of the current year as a percentage of the quantity of the base year. Then to obtain the index number, we average this percentage figure. So the formula under this method is as follows,

$$Q_{01}Â =Â \frac{\Sigma{Q_1}}{\Sigma{Q_0}}Â \times 100Â \div N$$

where N is the total number of items

### Weighted Index: Simple Aggregative Method

There are a few various methods for calculating this index number. We will take a look at some of the most important ones.

#### 1]Â Laspeyres Method

In this method, the base price is taken as the weight. We only use the price of the base year (P0), not the current year. The formula is as follows,

$$Q_{01}Â =Â \frac{\Sigma{Q_1 P_0}}{\Sigma{Q_0 P_0}} \times 100$$

#### 2]Â Paascheâ€™s Method

Here, the current year price (P1) of the commodity is taken as the weight.

$$Q_{01}Â =Â \frac{\Sigma{Q_1 P_0}}{\Sigma{Q_0 P_0}} \times 100$$

#### 3]Â Dorbish & Bowleyâ€™s Method

$$Q_{01}Â = \frac{\Sigma{Q_1 P_0}}{\Sigma{Q_0 P_0}}Â +Â \frac{\Sigma{Q_1 P_1}}{\Sigma{Q_0 P_1}} \div 2$$

### Weighted Index : Weighted Average of Relative Method

In this method, we use the arithmetic mean for averaging the values. The formula is a little more complex as seen below,

$$Q_{01}=Â Â \frac{\Sigma Q V}{\Sigma V}$$

where $$Q =Â \frac{\Sigma{q1}}{\Sigma{q0}}$$

and $$V = q_0Â p_0$$

## Solved Question onÂ Quantity Index Numbers

Q: All circumstances remaining same, a company sold 23000 tonnes of grain in the current year compared to 21500 tonnes in the last year. Find the simple unweighted quantity index number.

Ans: The formula is as follows,

$$Q_{01} = \frac{\Sigma{Q_1}}{\Sigma{Q_0}} Â \times 100$$

$$Q_{01} = \frac{\Sigma{23000}}{\Sigma{21500}} \timesÂ 100 =Â 106.97$$

This means the quantities saw an increase of 6.97%

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