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Average and Weighted Average

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Introduction to Average and Weighted Average

When you hear the word average, it usually means finding a single value that best represents a group of numbers. Imagine you took five tests and want to know how you did overall. Instead of looking at all scores separately, you calculate the average score to find a typical or central value.

This "typical value" helps simplify complex data sets and allows easier comparison. For example, we often want to know the average temperature of a week or the average price of apples in a market.

There are two main types of averages we will explore:

  • Simple average, where all values weigh equally.
  • Weighted average, where some values have more importance or frequency than others.

Let's quickly think about daily life: if you scored 80% in Maths, 70% in Science, and 90% in English, finding the simple average gives your overall percentage. But what if Maths is twice as important as the other subjects? Then, we use the weighted average to emphasize Maths score more.

Basic Average

The average (also called arithmetic mean) of a set of numbers is found by adding all the numbers together and then dividing by how many numbers there are.

This is expressed by the formula:

\[ \text{Average} = \frac{\sum_{i=1}^n x_i}{n} \]

where \(x_i\) are the individual values and \(n\) is the number of values.

For example, if you have scores of 10, 15, 20, 25, and 30, you add all these (which equals 100) and divide by 5 (the number of scores), resulting in an average of 20.

Average Calculation Table
Data Points Values
110
215
320
425
530
Sum100
Count (n)5
Average = Sum / Count100 / 5 = 20

Key points about averages:

  • The average is sensitive to very high or very low values (called outliers), which may skew it.
  • It gives a quick summary but sometimes hides data details-for example, the average of 1 and 99 is 50, which doesn't represent any real value in between.
  • Used widely in sports, academics, finance, and daily life to measure performance, prices, speed, and more.

Weighted Average

In many practical cases, some values in data are more important or occur more frequently than others. This calls for the weighted average, where each value is multiplied by its weight, reflecting its importance or frequency.

The formula for weighted average is:

\[ \text{Weighted Average} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} \]

where \(x_i\) are the values and \(w_i\) their corresponding weights.

For example, suppose a student has marks in subjects with different credit hours. The credit hours act as weights because subjects with more credit hours matter more for the overall grade.

65 85 50 70 Heights represent weights

Here, the height of each bar illustrates the weight (importance). Notice that the weighted average "leans" towards the values with greater heights/weights, adjusting the final average accordingly.

Worked Examples

Example 1: Calculating Average of Test Scores Easy
Find the average marks of a student who scored 70, 75, 80, 85, and 90 in five subjects.

Step 1: Add all the marks: \(70 + 75 + 80 + 85 + 90 = 400\).

Step 2: Count the number of subjects: \(5\).

Step 3: Calculate average: \(\frac{400}{5} = 80\).

Answer: The average mark is 80.

Example 2: Finding Weighted Average of Prices Medium
Calculate the weighted average price per kg if 3 kg of fruit is bought at Rs.40/kg and 5 kg at Rs.50/kg.

Step 1: Multiply each price by the quantity (weight):

  • \(3 \text{ kg} \times Rs.40 = Rs.120\)
  • \(5 \text{ kg} \times Rs.50 = Rs.250\)

Step 2: Sum of prices weighted by quantities: \(Rs.120 + Rs.250 = Rs.370\).

Step 3: Total quantity: \(3 + 5 = 8 \text{ kg}\).

Step 4: Weighted average price: \(\frac{Rs.370}{8} = Rs.46.25 \text{ per kg}\).

Answer: The weighted average price is Rs.46.25 per kg.

Example 3: Weighted Average for Grade Point Calculation Hard
A student has the following grades and credits:
Subject A: Grade 9, Credit 3
Subject B: Grade 8, Credit 4
Subject C: Grade 7, Credit 2
Calculate the GPA considering credits as weights.

Step 1: Multiply each grade by its credit:

  • Subject A: \(9 \times 3 = 27\)
  • Subject B: \(8 \times 4 = 32\)
  • Subject C: \(7 \times 2 = 14\)

Step 2: Sum weighted grades: \(27 + 32 + 14 = 73\).

Step 3: Sum credits: \(3 + 4 + 2 = 9\).

Step 4: GPA = \(\frac{73}{9} \approx 8.11\).

Answer: The weighted GPA is approximately 8.11.

Example 4: Average Speed Using Weighted Average Medium
A car travels 60 km at 40 km/h and 90 km at 60 km/h. Find the average speed.

Step 1: Calculate total distance: \(60 + 90 = 150 \text{ km}\).

Step 2: Calculate time for each segment:

  • Time for 60 km at 40 km/h = \(\frac{60}{40} = 1.5 \text{ hours}\)
  • Time for 90 km at 60 km/h = \(\frac{90}{60} = 1.5 \text{ hours}\)

Step 3: Total time = \(1.5 + 1.5 = 3 \text{ hours}\).

Step 4: Average speed = \(\frac{\text{Total Distance}}{\text{Total Time}} = \frac{150}{3} = 50 \text{ km/h}\).

Answer: The average speed is 50 km/h.

Example 5: Solving a Competitive Exam Question on Weighted Average Hard
Two classes have average marks of 75 and 85 respectively. The first class has 40 students and the second 60 students. Calculate the combined average marks.

Step 1: Multiply each class's average marks by number of students (weights):

  • Class 1: \(75 \times 40 = 3000\)
  • Class 2: \(85 \times 60 = 5100\)

Step 2: Add total marks: \(3000 + 5100 = 8100\).

Step 3: Total students: \(40 + 60 = 100\).

Step 4: Weighted average marks = \(\frac{8100}{100} = 81\).

Answer: The combined average marks are 81.

Formula Bank

Simple Average
\[ \text{Average} = \frac{\sum_{i=1}^n x_i}{n} \]
where: \(x_i\) = individual values, \(n\) = number of values
Weighted Average
\[ \text{Weighted Average} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} \]
where: \(x_i\) = individual values, \(w_i\) = corresponding weights

Tips & Tricks

Tip: Multiply each value by its weight (or frequency) first to simplify calculations when repeats occur.

When to use: Repeated values or data with different levels of importance

Tip: Always check that sum of weights is correct and used properly in denominator.

When to use: Weighted averages involving percentages or counts

Tip: For average speed problems, use distances as weights, never time or speed directly.

When to use: Problems where speed changes over different distances

Tip: Simplify fractions early in your calculations to avoid messy math.

When to use: Any weighted average involving fractions or decimals

Tip: Recognize when weighted average is needed by identifying varying importances or quantities associated with values.

When to use: Situations where data points do not have equal significance

Common Mistakes to Avoid

❌ Calculating weighted average without dividing by total sum of weights.
✓ Always divide the weighted sum by the total of all weights.
Why: Missing this step leads to wrong averages because denominator should reflect total importance.
❌ Using simple average when values represent different quantities or importance.
✓ Use weighted average when data points have different weights.
Why: Ignoring weights assumes equal importance incorrectly.
❌ Confusing weights with the values themselves.
✓ Clearly separate data values from the weights that multiply them.
Why: Mixing them leads to misunderstanding the problem and calculation errors.
❌ Using time as weights for average speed instead of distance.
✓ Use distances as weights in average speed problems.
Why: Average speed formula requires total distance divided by total time, so distance weights are correct.
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