Introduction to Average

In everyday life, we often want to summarize a collection of numbers by a single number that best represents them. This "middle value" or typical value is called Average. Imagine you have marks scored by five students in a test - instead of remembering all five scores, you can calculate their average to get a sense of their overall performance.

The average helps in understanding patterns and making comparisons. For example, you might want to know your average monthly expense, average speed during a trip, or average rainfall in a month. In all such scenarios, average tells us about the general trend of data rather than individual details.

Formally, average is a measure of central tendency, which means it indicates the center or typical value of a data set.

Arithmetic Mean

The most common type of average is the Arithmetic Mean. This is simply the sum of all values divided by the total number of values.

Mathematically, if you have numbers \( x_1, x_2, x_3, \ldots, x_n \), their arithmetic mean (average) is:

Example: Find the average of 4, 7, 10, 15, and 20.

• Step 1: Add all numbers: \(4 + 7 + 10 + 15 + 20 = 56\)
• Step 2: Count how many numbers: 5
• Step 3: Divide the sum by count: \( \frac{56}{5} = 11.2 \)

So, the average is 11.2.

Weighted Average

Sometimes, certain values are more important or occur more frequently than others. In such cases, the simple average does not tell the whole story. Instead, we use a Weighted Average, where each value is multiplied by its weight or frequency before averaging.

For example, suppose a student scores 80 marks in a test worth 30% of the final grade, and 90 marks in a test worth 70%. The final average won't be simply the mean of 80 and 90, but weighted by the respective percentages.

The formula is:

Consider the following table:

Calculate weighted average:

\[\text{Weighted Average} = \frac{2400 + 6300}{30 + 70} = \frac{8700}{100} = 87\]

The final average score is 87, reflecting the importance of each test.

Effect of Adding or Removing Data Points

In exams and real scenarios, data sets can change. You may add new values or remove some. How does this affect the average?

Let's understand these effects using the total sum concept.

If the original average is \( \bar{x} \) with \( n \) values, then total sum is:

\[\text{Total Sum} = \bar{x} \times n\]

Now, if a new value \( a \) is added, the new average \( \bar{x}_{new} \) is:

If a value \( r \) is removed, new average becomes:

graph TD  A[Start: Known Average and Count] --> B[Calculate total sum: Average x Count]  B --> C{Data Added or Removed?}  C -->|Added| D[New Total = Old Total + New Value]  C -->|Removed| E[New Total = Old Total - Removed Value]  D --> F[New Count = Old Count + 1]  E --> G[New Count = Old Count - 1]  F --> H[New Average = New Total / New Count]  G --> H  H --> I[End: Updated Average Known]

Summary

Formula Bank

Worked Examples