Login
OR
Don't worry. We'll fix this for you!
Have you ever wanted to find a single number that represents a group of numbers? For example, if you want to know the typical marks a student scores in exams or the usual speed of a vehicle over a trip, you use the concept of average. The average gives us a central or typical value that summarizes the entire set of numbers.
Imagine you and your friends went shopping and spent different amounts of money. To understand how much each person spent on average, you add all the amounts and divide by the number of friends. This simple idea helps us compare, summarize, and understand data easily.
There are different types of averages such as mean, median, and mode. In this chapter, we will focus mainly on the arithmetic mean, often called just the average, which is the most commonly used average in daily life and exams.
The arithmetic mean is the sum of all the numbers divided by how many numbers there are. It tells us the central value of a data set.
Suppose you have the following marks scored by a student in 5 subjects: 72, 85, 90, 65, and 78. To find the average marks, you add all these marks and then divide by 5 (because there are 5 subjects).
| Subject | Marks |
|---|---|
| 1 | 72 |
| 2 | 85 |
| 3 | 90 |
| 4 | 65 |
| 5 | 78 |
| Sum | 390 |
Now, calculate the average:
Average marks = \(\frac{72 + 85 + 90 + 65 + 78}{5} = \frac{390}{5} = 78\)
This means the student scored an average of 78 marks across all subjects.
The arithmetic mean of \(n\) numbers \(x_1, x_2, \ldots, x_n\) is given by:
\[ \text{Average} = \frac{\sum_{i=1}^n x_i}{n} \]
where:
Sometimes, all numbers do not have the same importance or weight. For example, if you buy 3 kg of mangoes at Rs.60 per kg and 2 kg of bananas at Rs.40 per kg, the average price per kg is not just the simple average of Rs.60 and Rs.40. Instead, it depends on the quantity (weight) of each fruit bought.
In such cases, we use the weighted average, where each value is multiplied by its weight (importance), and then the sum of these weighted values is divided by the total weight.
| Fruit | Price per kg (Rs.) | Quantity (kg) | Weighted Value (Price x Quantity) |
|---|---|---|---|
| Mangoes | 60 | 3 | 180 |
| Bananas | 40 | 2 | 80 |
| Total | 5 | 260 |
Calculate the weighted average price per kg:
Weighted Average Price = \(\frac{180 + 80}{3 + 2} = \frac{260}{5} = 52\) Rs. per kg
This means the average price per kg for the fruits bought is Rs.52.
For values \(x_1, x_2, \ldots, x_n\) with weights \(w_1, w_2, \ldots, w_n\), the weighted average is:
\[ \text{Weighted Average} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} \]
where:
Step 1: Add all the marks: \(68 + 74 + 82 + 90 + 76 = 390\)
Step 2: Count the number of subjects: 5
Step 3: Calculate the average marks:
\(\text{Average} = \frac{390}{5} = 78\)
Answer: The average marks obtained by the student is 78.
Step 1: Calculate weighted values:
Step 2: Total weight = \(4 + 6 = 10\) kg
Step 3: Total weighted value = \(320 + 300 = 620\)
Step 4: Calculate weighted average price:
\(\text{Weighted Average} = \frac{620}{10} = 62\) Rs. per kg
Answer: The average price per kg of the fruits is Rs.62.
Step 1: Find the total marks of the 10 students:
\(75 \times 10 = 750\)
Step 2: Add the new student's marks:
\(750 + 85 = 835\)
Step 3: Total number of students now = \(10 + 1 = 11\)
Step 4: Calculate new average:
\(\text{New Average} = \frac{835}{11} \approx 75.91\)
Answer: The new average marks after adding the new student is approximately 75.91.
Step 1: Since distances are equal, use the formula for average speed:
\[ \text{Average Speed} = \frac{2 \times S_1 \times S_2}{S_1 + S_2} \]
where \(S_1 = 40\) km/h and \(S_2 = 60\) km/h.
Step 2: Calculate:
\[ \frac{2 \times 40 \times 60}{40 + 60} = \frac{4800}{100} = 48 \text{ km/h} \]
Answer: The average speed for the entire journey is 48 km/h.
Step 1: Add all daily expenditures:
\(500 + 600 + 550 + 650 + 700 + 620 + 580 = 4200\)
Step 2: Number of days = 7
Step 3: Calculate average daily expenditure:
\(\frac{4200}{7} = 600\)
Answer: The average daily expenditure is Rs.600.
When to use: When a vehicle travels equal distances at different speeds.
When to use: When different quantities or importance are assigned to observations.
When to use: When a new value is added to an existing data set.
When to use: In problems involving measurements like speed, distance, or weight.
When to use: To avoid cumulative rounding errors in multi-step problems.
Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.
Go to practice →
Your Score
Your Rank
| Rank | Name | Score |
|---|
