Important Formula & Tips: Averages
What are Averages?
Averages represent the central or typical value of a set of data. They are used to summarise a collection of numbers by a single representative value.
- An average is obtained by dividing the sum of all values in a data set by the number of values.
- An average gives an idea of the middle value of data such as ages, marks, money, runs, heights, etc.
Example
What is the average of the first five consecutive odd numbers?
Solution:
The first five consecutive odd numbers are 1, 3, 5, 7, 9.
The number of observations is 5.
The sum of these numbers is 25.
The average is
\[ \text{Average} = \frac{25}{5} = 5 \]
Basic Formulae to Remember
- Simple average (arithmetic mean) = sum of observations ÷ number of observations.
- Arithmetic mean: \(\displaystyle \text{Arithmetic Mean} = \frac{a_1 + a_2 + a_3 + \cdots + a_n}{n}\)
- Weighted average = (sum of each value × its weight) ÷ (sum of weights).
- Geometric mean (for n positive numbers \(x_1,\dots,x_n\)) = \(\bigl(x_1 x_2 \cdots x_n\bigr)^{1/n}\).
- Harmonic mean (for n positive numbers \(x_1,\dots,x_n\)) = \(\dfrac{n}{\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}}\).
- For two numbers a and b:
Median and Mode
- Median: Arrange observations in ascending (or descending) order and pick the middle value. If the number of observations is even, the median is the average of the two middle values.
- Mode: The value that occurs most frequently in the data.
- Relation (empirical) between mode, median and mean:
\[ \text{Mode} = 3\times\text{Median} - 2\times\text{Mean} \]
Note:
Sum of first \(n\) natural numbers = \(\dfrac{n(n+1)}{2}\).
Average of first \(n\) natural numbers = \(\dfrac{n+1}{2}\).
Important Points (Practical Formulae)
- When one person replaces another:
If the average increases by \(d\) when a person leaves and another joins, then
New person's value = Value of person who left + \(d\times\) (total number of persons).
If the average decreases by \(d\), then
New person's value = Value of person who left - \(d\times\) (total number of persons).
- When a new person joins the group:
If the average increases by \(d\) on adding a new member, then
New member's value = Previous average + \(d\times\) (new total number of members).
If the average decreases by \(d\), then
New member's value = Previous average - \(d\times\) (new total number of members).
- In an arithmetic progression (AP):
If the number of terms is odd, the average equals the middle term.
If the number of terms is even, the average equals the average of the two middle terms.
Averages: Important Formulae
| Average of first \(n\) natural numbers | \(\dfrac{n+1}{2}\) |
| Average of squares of first \(n\) natural numbers | \(\dfrac{(n+1)(2n+1)}{6}\) |
| Average of cubes of first \(n\) natural numbers | \(\dfrac{n(n+1)^2}{4}\) |
| Average of first \(n\) even numbers | \(n+1\) |
| Average of squares of first \(n\) even numbers | \(\dfrac{2(n+1)(2n+1)}{3}\) |
| Average of cubes of first \(n\) even numbers | \(2n(n+1)^2\) |
| Average of first \(n\) odd numbers | \(n\) |
| Average of squares of first \(n\) odd numbers | \(\dfrac{(2n+1)(2n-1)}{3}\) |
| Average of cubes of first \(n\) odd numbers | \(n(2n^2-1)\) |
Shortcut Techniques
Many average problems can be solved quickly using simple shortcuts. These are especially useful in competitive exams to save time.
1. To find the new average after adding one new observation
Example 1.
The average of a batsman in 16 innings is 36. In the 17th innings he scores 70 runs. What will be his new average?
- 44
- 38
- 40
- 48
Solution:
Conventional method:
Total after 16 innings = \(16\times36\).
New total = \(16\times36 + 70\).
New average =
\[ \frac{16\times36 + 70}{17} = 38 \]
Shortcut method (difference method):
Take the difference between the new score and the old average: \(70-36=34\).
Spread this excess over the new total number of innings, 17:
\[ \frac{34}{17} = 2 \]
Thus the average increases by 2 and the new average is \(36+2=38\).
Example 2.
The average marks of 19 children are 50. A new student with marks 75 joins the class. What is the new average?
Solution (shortcut):
Increase in marks = \(75-50=25\).
This increase is spread over 20 students:
\[ \frac{25}{20} = 1.25 \]
New average = \(50 + 1.25 = 51.25\)
2. When the average decreases (example)
Example 3.
The average age of Mr. Mark's 3 children is 8 years. A new baby is born (age 0). Find the new average age of all his children.
Solution:
Difference = \(0-8 = -8\).
This change is spread over 4 children:
\[ \frac{-8}{4} = -2 \]
New average = \(8-2 = 6\) years.
3. To find a new value when the new average is given
Example.
The average age of 29 students is 18. After including the teacher the average becomes 18.2. Find the age of the teacher.
- 28
- 32
- 22
- 24
Solution:
Conventional method:
Total age of 29 students = \(29\times18\).
Total age after including teacher = \(30\times18.2\).
Teacher's age = \(30\times18.2 - 29\times18\) = 24.
Shortcut method:
Change in average = \(18.2-18 = 0.2\).
This change applies to 30 people, so the teacher's age exceeds the class average by:
\[ 30\times 0.2 = 6 \]
Hence teacher's age = \(18 + 6 = 24\).
Method of Deviation (Assumed Mean) to Find the Average
The assumed-mean (deviation) method reduces computation when dealing with large data sets. We take a convenient value as an assumed mean and work with deviations from that value.
Illustrative example:
In a class of 30 students the average age is 12 years. When the teacher is included the new average becomes 13 years. Find the age of the teacher.
Standard approach:
Total age of 30 students = \(30\times12 = 360\).
Total age including teacher = \(31\times13 = 403\).
Teacher's age = \(403-360 = 43\) years.
Deviation method (intuition):
Think of each student having 12 units (say chocolates). After inclusion of the teacher and redistribution each of the 31 people has 13 units. The teacher must have brought the extra units given to all 30 students plus his own 13 units:
Extra given to students = \(30\times1 = 30\).
Teacher's final share = 13.
Teacher's total = \(30+13 = 43\) (same result).
Important Facts About Averages
- If each number in a data set is increased (or decreased) by a constant \(n\), the mean also increases (or decreases) by \(n\).
- If each number is multiplied (or divided) by a constant \(n\), the mean is also multiplied (or divided) by \(n\).
- If the same value is added to half the observations and the same value is subtracted from the other half, the overall mean remains unchanged.
Weighted Mean / Weighted Average
The weighted arithmetic mean for values \(x_1,x_2,\dots,x_n\) with weights \(w_1,w_2,\dots,w_n\) is
That is,
\[ \text{Weighted Mean} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} \]
Example. In a class there are 25 boys with average height 150 cm and 15 girls with average height 140 cm. Find the average height of the whole class.
Standard approach:
Total height of boys =
Total height of girls =
Average height =
Deviation approach (intuitive shortcut):
- Assume each boy has 150 units and each girl 140 units.
- Reduce each boy's amount by 10 so both groups have 140; the excess removed from boys is
- Distribute this excess equally among all 40 students. Each student gets
Hence the class average = \(140 + 6.25 = 146.25\) cm.
Try yourself: Suppose that a marketing firm conducts a survey of 1,000 households to determine the average number of TVs each household owns. The data show a large number of households with two or three TVs and a smaller number with one or four. Every household in the sample has at least one TV and no household has more than four. Find the mean number of TVs per household.
Tips and Tricks to Solve Questions Based on Averages
- Tip 1:
Average = (Sum of observations) ÷ (Number of observations). - Tip 2:
The average of any consecutive arithmetic sequence equals its middle term (median). Example: for 8, 10, 12 the middle term 10 is also the average.
Solved Examples
Q1: The average goal scored by 15 selected players in EPL is 16. The maximum number of goals scored by a player is 20 and the minimum is 12. The goals scored by players are between 12 and 20. What can be the maximum number of players who scored at least 18 goals?
a) 10
b) 5
c) 9
d) 6
e) None of these
Solution: Option (c)
To maximise the count of players scoring at least 18 while keeping the average 16, assume extreme values for the fewest players: one player scores the maximum 20 and to balance that a player must score the minimum 12. That uses up 2 players, leaving 13 players.
To keep the average 16, for every two players scoring 18, we can include one player scoring 12 to balance. One arrangement that attains the maximum number scoring ≥18 is: eight players scoring 18, four players scoring 12, one player scoring 20 and one player scoring 16. That gives 8 players with 18, 1 player with 20 - total 9 players scoring at least 18. Thus the maximum is 9.
Q2: The average weight of a group of 8 girls is 50 kg. If 2 girls R and S replace P and Q, the new average weight becomes 48 kg. The weight of P = Weight of Q and the weight of R = Weight of another girl T is included in the group and the new average weight becomes 48 kg. Weight of T = Weight of R. Find the weight of P?
a) 48 kgs
b) 52 kgs
c) 46 kgs
d) 56 kgs
Solution: Option (d)
Let initial total weight of 8 girls = \(8\times50 = 400\).
After replacing P and Q by R and S the new total becomes \(8\times48 = 384\).
So \(R + S - P - Q = 384 - 400 = -16\).
Given \(P=Q\) and \(R=S\), let \(P=Q=p\) and \(R=S=r\). Then \(2r - 2p = -16\) so \(r-p = -8\) or \(p - r = 8\).
Later when another girl T with weight equal to R joins, the new average becomes 48 for 9 girls. So total for 9 girls = \(9\times48 = 432\).
The total after replacing (384) plus T's weight \(= 432\), so \(T = 48\). Thus \(r = 48\).
Therefore \(p = r + 8 = 48 + 8 = 56\) kg.
Q3: The average number of goals scored by 15 selected players in EPL is 16. The maximum number of goals scored by a player is 20 and the minimum is 12. The goals scored by players is between 12 and 20. What can be the maximum number of players who scored at least 18 goals?
a) 10
b) 5
c) 9
d) 6
Solution: Option (c)
Reasoning is identical to Q1: put one player at the maximum 20 and one at the minimum 12 to allow as many players as possible to score ≥18 while keeping the average 16. Arranging values as explained for Q1 yields at most 9 players with 18 or more goals.
FAQs on Important Formula & Tips: Averages
| 1. What's the basic formula for calculating average and when should I use it? |  |
| 2. How do I find the new average when a number is added or removed from a group? | |
| 3. Why do weighted averages give different results than simple averages? | |
| 4. What's the trick to solve average problems involving replacement or substitution quickly? | |
| 5. How can I use the average formula to check if my answer is reasonable in UPSC CSAT questions? | |
