Average

Introduction to Average

In everyday life, we often want to find a single value that represents a group of numbers. This representative value is called the average. The average helps us understand the "central" or "typical" value of a set of data. For example, if you want to know the average marks scored by students in a class or the average monthly expense of a family, you use averages.

There are different types of averages, but the two most common ones you will learn here are the simple mean and the weighted average. The simple mean treats all data points equally, while the weighted average gives different importance to different data points.

Understanding averages is important not only for exams like the Assam Direct Recruitment Examination (ADRE) but also for making informed decisions in daily life.

Mean (Simple Average)

The mean, often called the simple average, is the sum of all values divided by the number of values. It gives a single number that best represents the entire data set when each value is equally important.

Formula for Mean:

Simple Mean

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

Sum of all data values divided by the number of values

\(x_i\) = Each data value

n = Total number of data values

Let's see how to calculate mean with a simple example.

**Daily Temperatures (°C) Over 5 Days**
Day	Temperature (°C)
1	30
2	32
3	31
4	29
5	28
Total	150

To find the average temperature:

Add all temperatures: 30 + 32 + 31 + 29 + 28 = 150
Count the number of days: 5
Divide the total by the count: \( \frac{150}{5} = 30 \)

So, the average temperature over 5 days is 30°C.

Weighted Average

Sometimes, not all data points are equally important. For example, if you want to find the average marks of a student where different subjects have different maximum marks or importance, you use the weighted average.

In a weighted average, each value is multiplied by a weight that shows its importance or frequency. Then, the sum of these weighted values is divided by the sum of the weights.

Formula for Weighted Average:

Weighted Average

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

Sum of weighted values divided by sum of weights

\(x_i\) = Each data value

\(w_i\) = Weight of each data value

Let's understand this with a table:

**Weighted Average Calculation Example**
Value (x_i)	Weight (w_i)	Weighted Value (w_i x x_i)
80	2	160
90	3	270
70	1	70
Totals	6	500

Weighted average = \( \frac{500}{6} \approx 83.33 \)

This means the average value, considering the weights, is approximately 83.33.

Worked Example 1: Calculating Simple Average of Monthly Expenses

Example 1: Average Monthly Expenses Easy

A family spends the following amounts on groceries over 4 months: Rs.12,000, Rs.15,000, Rs.13,500, and Rs.14,500. Find the average monthly expense on groceries.

Step 1: Add all monthly expenses:

Rs.12,000 + Rs.15,000 + Rs.13,500 + Rs.14,500 = Rs.55,000

Step 2: Count the number of months: 4

Step 3: Divide the total by the number of months:

Average expense = \( \frac{55,000}{4} = Rs.13,750 \)

Answer: The average monthly expense on groceries is Rs.13,750.

Worked Example 2: Finding Weighted Average of Test Scores

Example 2: Weighted Average of Test Scores Medium

A student scored 75 out of 100 in Maths, 80 out of 50 in Science, and 90 out of 150 in English. Find the weighted average score considering the maximum marks as weights.

Step 1: Identify values and weights:

Maths: Score = 75, Weight = 100
Science: Score = 80, Weight = 50
English: Score = 90, Weight = 150

Step 2: Calculate weighted scores:

Maths: 75 x 100 = 7,500
Science: 80 x 50 = 4,000
English: 90 x 150 = 13,500

Step 3: Sum weighted scores and weights:

Total weighted score = 7,500 + 4,000 + 13,500 = 25,000
Total weights = 100 + 50 + 150 = 300

Step 4: Calculate weighted average:

Weighted average = \( \frac{25,000}{300} \approx 83.33 \)

Answer: The weighted average score is approximately 83.33.

Worked Example 3: Average Speed Calculation Using Weighted Average

Example 3: Average Speed for a Journey Hard

A car travels 60 km at 40 km/h and then 90 km at 60 km/h. Find the average speed for the entire journey.

Step 1: Identify distances and speeds:

Distance 1 (d₁) = 60 km, Speed 1 (v₁) = 40 km/h
Distance 2 (d₂) = 90 km, Speed 2 (v₂) = 60 km/h

Step 2: Calculate time taken for each part:

Time 1 = \( \frac{60}{40} = 1.5 \) hours
Time 2 = \( \frac{90}{60} = 1.5 \) hours

Step 3: Total distance = 60 + 90 = 150 km

Step 4: Total time = 1.5 + 1.5 = 3 hours

Step 5: Average speed = \( \frac{\text{Total distance}}{\text{Total time}} = \frac{150}{3} = 50 \) km/h

Answer: The average speed for the entire journey is 50 km/h.

Worked Example 4: Average Cost of Items Bought

Example 4: Average Cost of Items Easy

A shopper buys 3 items costing Rs.120, Rs.150, and Rs.180. Find the average cost of the items.

Step 1: Add the costs:

Rs.120 + Rs.150 + Rs.180 = Rs.450

Step 2: Number of items = 3

Step 3: Average cost = \( \frac{450}{3} = Rs.150 \)

Answer: The average cost of the items is Rs.150.

Worked Example 5: Weighted Average Price of Mixed Quantities

Example 5: Weighted Average Price per kg Hard

A shopkeeper buys 10 kg of rice at Rs.40/kg and 15 kg at Rs.50/kg. Find the average price per kg of the rice.

Step 1: Identify quantities and prices:

Quantity 1 (w₁) = 10 kg, Price 1 (x₁) = Rs.40/kg
Quantity 2 (w₂) = 15 kg, Price 2 (x₂) = Rs.50/kg

Step 2: Calculate weighted prices:

10 x 40 = Rs.400
15 x 50 = Rs.750

Step 3: Sum weighted prices and quantities:

Total price = Rs.400 + Rs.750 = Rs.1,150
Total quantity = 10 + 15 = 25 kg

Step 4: Average price per kg = \( \frac{1,150}{25} = Rs.46 \)

Answer: The average price per kg of rice is Rs.46.

Formula Bank

Simple Mean

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

where: \( x_i \) = individual data values, \( n \) = number of data points

Weighted Average

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

where: \( x_i \) = individual data values, \( w_i \) = weights associated with each data value

Tips & Tricks

Tip: Always add all weights first before dividing in weighted average problems.

When to use: To avoid mistakes in weighted average calculations.

Tip: Use rough estimation to check if your average answer is reasonable.

When to use: After solving problems quickly during exams to verify answers.

Tip: Convert all units to metric before calculating averages.

When to use: When data involves mixed units like km and meters or INR and paise.

Tip: For average speed problems, use weighted average weighted by distances, not by time.

When to use: When speeds vary over different distances in a journey.

Tip: In multiple-choice questions, eliminate options far from your rough estimate to save time.

When to use: During entrance exams with time pressure.

Common Mistakes to Avoid

❌ Adding weights directly instead of multiplying weights with values in weighted average

✓ Multiply each value by its weight before summing

Why: Students confuse weights as additive factors rather than multipliers, leading to wrong totals.

❌ Dividing by the number of data points instead of the sum of weights in weighted average

✓ Divide by the sum of weights, not the count of data points

Why: Misunderstanding the role of weights causes incorrect averages.

❌ Ignoring units or mixing units in calculations

✓ Convert all measurements to consistent metric units before calculation

Why: Mixing units leads to incorrect averages and confusion.

❌ Using simple mean formula for average speed problems without weighting by distance

✓ Use weighted average weighted by distances traveled

Why: Average speed depends on total distance and total time, not just speeds.

❌ Forgetting to check if data values and weights correspond correctly

✓ Always verify pairs of values and weights before calculation

Why: Misalignment causes wrong results and confusion.

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Introduction to Average

Mean (Simple Average)

Simple Mean

Weighted Average

Weighted Average

Worked Example 1: Calculating Simple Average of Monthly Expenses

Worked Example 2: Finding Weighted Average of Test Scores

Worked Example 3: Average Speed Calculation Using Weighted Average

Worked Example 4: Average Cost of Items Bought

Worked Example 5: Weighted Average Price of Mixed Quantities

Formula Bank

Tips & Tricks

Common Mistakes to Avoid

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