Q.Calculate standard deviation of the following data.
Marks | 10-20 | 20-30 | 30-50 | 50-70 | 70-80
Frequency | 5 | 8 | 16 | 8 | 3

Arithmetic Mean in Economics — A First Look

You already use the arithmetic mean every day without thinking about it. If your friend says "my average speed was 40 km/h" or "the average price of a burger in this city is ₹80," they are talking about the arithmetic mean. It is the most natural way to find a "typical" value when you have a list of numbers.

The Intuition

Imagine you have five friends, and their monthly pocket money (in ₹) is: 500, 600, 700, 800, 900. If you wanted one number that "represents" all of them, you would add them all up and divide by the number of friends. That gives you ₹700. That ₹700 is the arithmetic mean — it is the value each person would get if all the pocket money were pooled and shared equally.

In economics, this idea is everywhere. When we talk about "average income" of a country, "average price" of a basket of goods, or "average product" of a worker, we are using the arithmetic mean to summarise a large set of data into a single, understandable number.

The Precise Meaning

The arithmetic mean is defined as the sum of all observations divided by the number of observations. For a set of $n$ values $x_1, x_2, x_3, \dots, x_n$, the arithmetic mean $\bar{x}$ (read as "x-bar") is:

Or, more compactly:

Where:

• $\sum$ (sigma) means "sum of"
• $x_i$ represents each individual observation
• $n$ is the total number of observations

Why It Matters in Economics

Economics deals with aggregates — total production, total income, total expenditure. But a single total tells you nothing about the typical experience. If a country's total income is ₹100 lakh crore, is that because everyone is rich or because a few people are extremely rich? The arithmetic mean of income gives you a first approximation, though as you will learn later, it has limitations (the median is often better for income data because the mean is pulled by extreme values).

Here are the key places you will meet the arithmetic mean in your Class 11/12 syllabus:

1. Average Product (AP) in Production

When a firm hires workers, the total output changes. The average product of labour is the output per worker:

Where $TP$ is total product (total output) and $L$ is the number of workers. This is simply the arithmetic mean of output per worker.

2. Average Revenue (AR) or Price

A firm sells different quantities at possibly different prices. The average revenue is total revenue divided by quantity sold:

Since $TR = P \times Q$ for a single price, $AR = P$. But if prices vary, $AR$ is the arithmetic mean price.

3. Average Cost (AC)

Total cost divided by output:

This tells you the cost per unit — again, an arithmetic mean.

4. Simple Index Numbers (Price Index)

When constructing a simple (unweighted) price index, you take the arithmetic mean of price relatives:

Where $P_1$ is the current year price, $P_0$ is the base year price, and $n$ is the number of commodities. This is a direct application of the arithmetic mean formula. …