Averages: SSC CGL Quantitative Aptitude

Formulas (3)

• Average, sum and count · Average, sum and count
  $$\text{Average} = \frac{\text{Sum}}{\text{Count}} \qquad \text{Sum} = \text{Count} \times \text{Average}$$
• Average from an assumed value · Average from an assumed value
  $$\bar{x} = A + \frac{\sum (x - A)}{n} \qquad \sum (x - \bar{x}) = 0$$
• Pair averages: add all the statements · Total from three pair sums
  $$a + b + c = \frac{(a + b) + (b + c) + (c + a)}{2}$$

Watch out for (6)

• Averaging two averages of unequal groups
  → Average, sum and count
• Forgetting part of the count
  → Average, sum and count
• Dropping the sign of a deviation
  → Average from an assumed value
• Adding the deviation total without dividing
  → Average from an assumed value
• Forgetting to halve the total of the pair sums
  → Pair averages: add all the statements
• Reading an average as a sum
  → Pair averages: add all the statements

Formulas (2)

• A new member joins the group · A member joins
  $$x = A + (n + 1)(A' - A) \qquad A' = A + \frac{x - A}{n + 1}$$
• A member leaves the group · A member leaves
  $$x = A + (n - 1)(A - A') \qquad \text{Leavers' total} = nA - (n - k)A'$$

Watch out for (4)

• Multiplying the rise by the wrong count
  → A new member joins the group
• Taking the new average as the newcomer's value
  → A new member joins the group
• Multiplying the fall by the old count
  → A member leaves the group
• Thinking a fall means the leaver was low
  → A member leaves the group

Formulas (2)

• Replacing members, or changing every value · Replacement and shifts
  $$\text{New} = \text{Old} + n \times \text{rise} \qquad \overline{x + c} = \bar{x} + c \qquad \overline{cx} = c\,\bar{x}$$
• Correcting a wrong entry · Correcting an average
  $$\text{Correct average} = \text{Wrong average} + \frac{\text{Correct value} - \text{Wrong value}}{n}$$

Watch out for (4)

• Using n + 1 for a replacement
  → Replacing members, or changing every value
• Multiplying the rise by the number replaced
  → Replacing members, or changing every value
• Subtracting the wrong way round
  → Correcting a wrong entry
• Moving the average by the whole error
  → Correcting a wrong entry

Formulas (3)

• The members left after the known parts · Leftover total
  $$\text{Leftover total} = nA - (n_1A_1 + n_2A_2 + \cdots)$$
• Overlapping parts share a member · Shared member of two overlapping parts
  $$\text{Shared} = n_1A_1 + n_2A_2 - nA$$
• A member tied to the average of all · Average of all when one member is tied to it
  $$a = \bar{x}_{\text{others}} + \frac{k}{n - 1}$$

Watch out for (3)

• Applying a percentage to the wrong quantity
  → The members left after the known parts
• Missing the shared member
  → Overlapping parts share a member
• Adding k to the others' average
  → A member tied to the average of all

Formulas (4)

• Combined average of two groups · Combined average
  $$\bar{x} = \frac{n_1A_1 + n_2A_2}{n_1 + n_2} \qquad \bar{x} = A_1 + \frac{n_2}{n_1 + n_2}(A_2 - A_1)$$
• Weights given as percentages or a ratio · Weighted average
  $$\bar{x} = \frac{w_1x_1 + w_2x_2 + w_3x_3}{w_1 + w_2 + w_3}$$
• Finding a missing group's average · Missing group's average
  $$A_2 = \frac{nA - n_1A_1}{n - n_1} \qquad n_1(A_1 - A) = n_2(A - A_2)$$
• Alligation: group sizes from the averages · Rule of alligation
  $$\frac{n_1}{n_2} = \frac{A_2 - A}{A - A_1}$$

Watch out for (6)

• Averaging the two averages
  → Combined average of two groups
• Choosing an option outside the two averages
  → Combined average of two groups
• Dropping the sign of a loss
  → Weights given as percentages or a ratio
• Ignoring the weights
  → Weights given as percentages or a ratio
• Dividing the missing total by the whole count
  → Finding a missing group's average
• Giving each group its own distance
  → Alligation: group sizes from the averages

Formulas (3)

• Average of numbers in equal steps · Average of an arithmetic progression
  $$\text{Average} = \frac{\text{first} + \text{last}}{2} \qquad \text{Sum} = \text{count} \times \text{middle term}$$
• Average of the multiples of a number in a range · Multiples of k in a range
  $$\text{Average} = \frac{\text{first multiple} + \text{last multiple}}{2} \qquad \text{Count} = \frac{\text{last} - \text{first}}{k} + 1$$
• Averages of the first n natural numbers and their squares · Sums and averages of the first n natural numbers
  $$\frac{1 + 2 + \cdots + n}{n} = \frac{n + 1}{2} \qquad \frac{1^2 + 2^2 + \cdots + n^2}{n} = \frac{(n + 1)(2n + 1)}{6} \qquad \frac{1^3 + 2^3 + \cdots + n^3}{n} = \frac{n(n + 1)^2}{4}$$

Watch out for (5)

• Expecting a whole-number average from an even count of consecutive integers
  → Average of numbers in equal steps
• Taking a middle term that does not exist
  → Average of numbers in equal steps
• Averaging the ends of the range instead of the multiples
  → Average of the multiples of a number in a range
• Using n ÷ 2 for the average of 1 to n
  → Averages of the first n natural numbers and their squares
• Giving the sum of squares as the average
  → Averages of the first n natural numbers and their squares