Average: Notes और MCQ Practice

आसान भाषा में:

Average का सीधा मतलब होता है 'Equal Distribution' यानी सब में बराबर-बराबर बांट देना। अगर किसी group में सबके पास अलग-अलग values हैं, तो Average वो single number है जो सबको represent करता है। SSC CGL में formulas रटने से ज्यादा 'Net Deviation Zero' का concept दिमाग में रखना जरूरी है।

Important Formulas & Rules:

• Basic Formula: Average = Sum of observations ÷ Number of observations
• Sum: Sum = Average × Number of observations
• Consecutive Numbers (AP series): Average = (First Term + Last Term) ÷ 2
• First 'n' Natural Numbers: Average = (n + 1) ÷ 2
• First 'n' Even Numbers: Average = n + 1
• First 'n' Odd Numbers: Average = n
• Average Speed (जब distance equal हो): (2 × S1 × S2) ÷ (S1 + S2)

2 Quick Tricks:

1. Deviation Method: बड़ी calculations से बचने के लिए किसी भी round figure को Assumed Average मान लो। फिर हर observation का deviation (कमी/बेशी) देखो।

• New Average = Assumed Average + (Net Deviation ÷ Total count)

1. Inclusion/Replacement Shortcut:

• New Member = Replaced Member ± (Total Members × Increase/Decrease in Average)
• अगर नया आने वाला Average बढ़ा रहा है, तो वो definitely average से ज्यादा value लेकर आया है।

2 Common Mistakes to Avoid:

1. Average Speed में Simple Mean लेना: दूरी बराबर होने पर कभी भी speeds का simple average (S1 + S2) ÷ 2 मत निकालना, यह गलत है। Total Distance ÷ Total Time ही basic rule है।
2. New Count (n) को update न करना: जब group में कोई नया member शामिल होता है या छोड़ता है, तो Total Observations n + 1 या n − 1 हो जाते हैं। Students अक्सर पुराने 'n' से ही divide कर देते हैं।

1. Concept

Average is the central value of a set of numbers. It represents the 'mean' or the typical value that the data points share if they were distributed equally.

Think of it this way: If you have a total amount of money and you want to divide it equally among a group of people, the amount each person gets is the average.

• Sum: The total of all values.
• Count: The number of values.
• Average: The result of dividing the Sum by the Count.

In exams, you rarely need to calculate the exact average. Instead, you need to find the Sum or the Count using the given Average.

2. Important formulas / rules / facts

• Basic Formula: Average = (Sum of Observations) ÷ (Number of Observations)
• Sum Formula: Sum of Observations = Average × Number of Observations
• Count Formula: Number of Observations = (Sum of Observations) ÷ Average
• Weighted Average: If Group 1 has $n_1$ items with average $A_1$ and Group 2 has $n_2$ items with average $A_2$, the combined average is:

Combined Average = $(n_1 \times A_1 + n_2 \times A_2) \div (n_1 + n_2)$

• Arithmetic Progression (AP) Rule: The average of an AP is equal to its middle term. If the number of terms is even, it is the average of the two middle terms.
• Example: Average of 1, 2, 3, 4, 5 is 3 (middle term).
• Example: Average of 1, 2, 3, 4 is $(2+3) \div 2 = 2.5$.
• Sum of First $n$ Natural Numbers: Average = $(n + 1) \div 2$
• Sum of First $n$ Even Numbers: Average = $n + 1$
• Sum of First $n$ Odd Numbers: Average = $n$
• Effect of Adding/Subtracting a Constant: If you add $k$ to every number in a set, the average also increases by $k$. If you multiply every number by $k$, the average also multiplies by $k$.
• Effect of Wrong Value: If a value is recorded incorrectly, the new average changes. The change in average = (Error in value) ÷ (Number of items).

3. Solved Examples

Example 1: Basic Calculation

Question: The average of 5 numbers is 20. If one number, 15, is removed, what is the average of the remaining 4 numbers?

Solution:

1. Find the total sum of the original 5 numbers.

Sum = Average × Count = $20 \times 5 = 100$

1. Subtract the removed number from the total sum.

New Sum = $100 - 15 = 85$

1. Calculate the new average for the remaining 4 numbers.

New Average = New Sum ÷ New Count = $85 \div 4 = 21.25$

Answer: 21.25

Example 2: Weighted Average (Mixture Concept)

Question: The average age of 10 boys is 15 years and the average age of 15 girls is 12 years. What is the average age of all 25 students?

Solution:

1. Calculate the total age of boys.

Sum of Boys = $10 \times 15 = 150$ years

1. Calculate the total age of girls.

Sum of Girls = $15 \times 12 = 180$ years

1. Calculate the total age of all students.

Total Sum = $150 + 180 = 330$ years

1. Calculate the total number of students.

Total Count = $10 + 15 = 25$

1. Calculate the combined average.

Average = $330 \div 25 = 13.2$ years

Answer: 13.2 years

Example 3: Finding the Missing Number

Question: The average of 7 numbers is 40. The average of the first 3 numbers is 35 and the average of the last 3 numbers is 45. What is the 4th number?

Solution:

1. Find the total sum of all 7 numbers.

Total Sum = $7 \times 40 = 280$

1. Find the sum of the first 3 numbers.

Sum of First 3 = $3 \times 35 = 105$

1. Find the sum of the last 3 numbers.

Sum of Last 3 = $3 \times 45 = 135$

1. Subtract the known sums from the total sum to find the middle (4th) number.

4th Number = Total Sum - (Sum of First 3 + Sum of Last 3)

4th Number = $280 - (105 + 135) = 280 - 245 = 35$

Answer: 35

Example 4: Correction of Error

Question: The average of 20 students was calculated as 60. Later, it was found that one student's score was recorded as 50 instead of 70. What is the correct average?

Solution:

1. Calculate the incorrect total sum.

Incorrect Sum = $20 \times 60 = 1200$

1. Calculate the error in the sum.

Error = Correct Value - Wrong Value = $70 - 50 = 20$

1. Add the error to the incorrect sum to get the correct sum.

Correct Sum = $1200 + 20 = 1220$

1. Calculate the correct average.

Correct Average = $1220 \div 20 = 61$

Answer: 61

Example 5: Average of Consecutive Numbers

Question: What is the average of the first 100 even natural numbers?

Solution:

1. Use the formula for the average of the first $n$ even numbers: Average = $n + 1$.
2. Here, $n = 100$.
3. Average = $100 + 1 = 101$.

Alternative Method (Verification):

• First even number = 2
• 100th even number = $2 \times 100 = 200$
• Average of an AP = (First Term + Last Term) ÷ 2
• Average = $(2 + 200) \div 2 = 202 \div 2 = 101$

Answer: 101

4. Shortcuts and Tricks

• Alligation Method: For weighted averages (like mixing two groups), use the Alligation rule to find the ratio of quantities without calculating total sums.
• Ratio of Quantities = (Difference between Mean and Higher Average) : (Difference between Lower Average and Mean)
• Deviation Method: If numbers are close to a base value, assume the base as the average. Calculate the deviations (differences) from the base. The average deviation will be the adjustment to the base.
• Example: Find average of 101, 103, 99, 102, 95.
• Base = 100.
• Deviations: +1, +3, -1, +2, -5.
• Sum of deviations = $1+3-1+2-5 = 0$.
• Average = Base + (Sum of Deviations ÷ Count) = $100 + 0 = 100$.
• Middle Term Rule: For any Arithmetic Progression (consecutive numbers, consecutive evens, consecutive odds), the average is simply the middle term. If there is no single middle term (even count), average the two middle numbers.
• Fractional Averages: If the average is a fraction, the denominator often gives a clue about the number of items. For example, if the average is $x/3$, the number of items is likely a multiple of 3.

5. Common Mistakes to Avoid

• Confusing Sum and Average: Always remember that Average = Sum ÷ Count. Do not add the averages directly when combining groups; you must weight them by the number of items in each group.
• Ignoring the Count: When a new item is added or removed, the count changes. Ensure you update the denominator in your division.
• Wrong Formula for Even/Odd Numbers:
• Average of first $n$ odd numbers is $n$ (not $n+1$).
• Average of first $n$ even numbers is $n+1$ (not $n$).
• Sign Errors in Deviation Method: Be careful with positive and negative signs when calculating deviations from a base value.
• Assuming Symmetry: The average is only the middle term if the numbers are in an Arithmetic Progression (equally spaced). If the numbers are random, you must calculate the sum and divide by the count.
• Unit Consistency: Ensure all values are in the same unit (e.g., all in years, all in rupees) before calculating the average.