Q.If the Arithmetic Mean and Median from a data are respectively 25 and 20, what will be the value of mode ?

Mode of Discrete Series – A First Look

Imagine you run a small tea stall near a school. Every day, you note how many cups each customer buys. Most customers buy one cup, some buy two, and very rarely someone buys three. If you had to answer the question "What is the typical number of cups a customer buys?", you wouldn't say the average (which might be 1.3) — you'd say one cup, because that's what most people do. That instinct — "the most common value" — is exactly what the mode captures.

In statistics, the mode is the value that appears most frequently in a data set. For a discrete series (where data is given as individual values or as a frequency distribution with distinct, separate values), the mode is simply the value with the highest frequency.

Why the Mode Matters in Economics

The mode is not just a mathematical curiosity. It tells you what is typical in a real sense. Consider these examples:

• A shoe shop wants to stock the most popular size. The average size might be 8.2, but if size 9 is bought by 40% of customers, the shop should stock size 9. The mode gives that answer.
• A government studying household income might find that the most common income bracket is ₹2–3 lakh per year. That tells them where the largest group of people sits — more useful for targeting welfare schemes than the average, which could be pulled up by a few very rich households.
• In a factory, the mode of the number of defective items per batch tells the manager the most frequent defect count — useful for quality control.

The mode is the only measure of central tendency that can be used for qualitative data (like "most common brand of soap" or "most frequent complaint type"). You cannot calculate a mean or median for categories, but you can always find the mode.

How to Find the Mode in a Discrete Series

A discrete series is a set of values where each value is distinct and countable — like the number of children per family (0, 1, 2, 3, ...) or the shoe sizes (5, 6, 7, 8, ...). The data is usually given as two columns: the value ($x$) and its frequency ($f$).

Step 1: Look at the frequency column.

Step 2: Identify the highest frequency.

Step 3: The value corresponding to that highest frequency is the mode.

Example:

The highest frequency is 12, which corresponds to $x = 1$. So the mode is 1 book per month.

What If Two Values Have the Same Highest Frequency?

That's called a bimodal distribution. Both values are modes. For example:

Here, sizes 8 and 9 both appear 35 times. The data is bimodal — both 8 and 9 are modes. If more than two values tie, it's multimodal. …