Average: AFCAT Numerical Ability

Formulas (4)

• Average, total and count · Arithmetic mean
  $$\text{Average} = \dfrac{\text{Sum of the values}}{\text{Number of values}}$$
• A missing value, or the score needed for a target · The missing value
  $$x = A \times n - (\text{sum of the other } n - 1 \text{ values})$$
• When every value changes · A change applied to every value
  $$x_i \to a\,x_i + b \;\Rightarrow\; \bar x \to a\,\bar x + b$$
• Evenly spaced numbers: first plus last, halved · Average of an evenly spaced list
  $$\bar x = \dfrac{\text{first term} + \text{last term}}{2}$$

Reference tables (1)

Watch out for (9)

• An average of averages
  → Average, total and count
• Zero is a value
  → Average, total and count
• Count the test still to come
  → A missing value, or the score needed for a target
• Order matters
  → When every value changes
• Some values, not all
  → When every value changes
• Check the spacing first
  → Evenly spaced numbers: first plus last, halved
• Find the right first term
  → Evenly spaced numbers: first plus last, halved
• Even numbers average n + 1, not n
  → The first n natural, odd and even numbers
• Squares are not evenly spaced
  → The first n natural, odd and even numbers

Formulas (4)

• An odd count: the average is the middle number · Ends of an odd-length run
  $$\text{largest} = A + \dfrac{n - 1}{2}\,d, \qquad \text{smallest} = A - \dfrac{n - 1}{2}\,d$$
• An even count: halfway between the two middle numbers · Ends of an even-length run
  $$\text{largest} = A + \dfrac{n - 1}{2}\,d \quad (n \text{ even, so } \tfrac{n - 1}{2} \text{ ends in } .5)$$
• Pairs from the two ends · End pairs and spread
  $$\text{first} + \text{last} = 2\bar x, \qquad \text{last} - \text{first} = (n - 1)\,d$$
• Letters, products, and a run that changes · A run centred on its average
  $$x - 2,\; x - 1,\; x,\; x + 1,\; x + 2 \quad (\text{five terms averaging } x)$$

Watch out for (4)

• Use the right step
  → An odd count: the average is the middle number
• Check the parity
  → An even count: halfway between the two middle numbers
• The spread does not depend on the average
  → Pairs from the two ends
• Centre the run, do not start it
  → Letters, products, and a run that changes

Formulas (5)

• One more value: a new day, a teacher, a manager · One value added
  $$\text{new average} = \dfrac{nA + x}{n + 1}$$
• One value removed · One value removed
  $$\text{new average} = \dfrac{nA - x}{n - 1}$$
• Finding the value that joined or left · The newcomer
  $$x = (n + 1)A' - nA = A' + n(A' - A)$$
• Replacing a member · One member replaced
  $$x_{\text{new}} = x_{\text{old}} + n\,d$$
• A value misread or miscopied · Correcting one value
  $$\text{correct average} = A + \dfrac{\text{right value} - \text{wrong value}}{n}$$

Watch out for (5)

• The count goes up too
  → One more value: a new day, a teacher, a manager
• A low value leaving raises the average
  → One value removed
• The newcomer pays for everyone
  → Finding the value that joined or left
• No one was added
  → Replacing a member
• Which way round
  → A value misread or miscopied

Formulas (4)

• Combining groups · Combined average of two groups
  $$\bar x = \dfrac{n_1A_1 + n_2A_2}{n_1 + n_2}$$
• The rest of the group · Average of the rest
  $$A_{\text{rest}} = \dfrac{nA - n_1A_1}{n - n_1}$$
• Overlapping groups: the shared value counted twice · The shared value
  $$\text{shared value(s)} = n_1A_1 + n_2A_2 - nA$$
• Pair averages: add all three · Three values from three pairs
  $$a + b + c = \dfrac{(a + b) + (b + c) + (c + a)}{2}$$

Watch out for (4)

• Not the plain mean of the averages
  → Combining groups
• Check that the groups cover everyone once
  → The rest of the group
• The overlap may be more than one number
  → Overlapping groups: the shared value counted twice
• Halve the sum
  → Pair averages: add all three

Formulas (4)

• A group joins · A group joins
  $$\bar x' = \dfrac{nA + mB}{n + m}$$
• A group leaves · A group leaves
  $$\bar x' = \dfrac{nA - mB}{n - m}$$
• Some leave, others join · Leave and join together
  $$\bar x' = \dfrac{nA - mB + kC}{n - m + k}$$
• Finding the group that joined or left · The joining group's total
  $$mB = (n + m)A' - nA$$

Watch out for (4)

• The size of the joining group matters
  → A group joins
• Leaving can raise the average
  → A group leaves
• Two changes to the count
  → Some leave, others join
• Divide by the group, not by everyone
  → Finding the group that joined or left

Formulas (4)

• Weights given as numbers or ratios · Weighted average
  $$\bar x = \dfrac{w_1x_1 + w_2x_2 + \dots}{w_1 + w_2 + \dots}$$
• Averages of rates: speed, pass rate, return · Average speed
  $$\text{average speed} = \dfrac{\text{total distance}}{\text{total time}}$$
• Percentage shares as weights, and the missing share · Overall average from percentage shares
  $$\bar x = \dfrac{p_1x_1 + p_2x_2 + \dots}{100}$$
• Values a percentage above or below the average · A value relative to the average
  $$\text{value} = A\left(1 \pm \dfrac{p}{100}\right)$$

Watch out for (4)

• Divide by the weights, not the count
  → Weights given as numbers or ratios
• Speeds do not average directly
  → Averages of rates: speed, pass rate, return
• Turn the percentage into people
  → Percentage shares as weights, and the missing share
• Which average?
  → Values a percentage above or below the average

Formulas (3)

• Removing the highest and the lowest · The two extremes together
  $$H + L = nA - (n - 2)A'$$
• One extreme alone: the largest or the smallest · One value taken out
  $$x = nA - (n - 1)A'$$
• Highest and lowest from their sum and difference · Two numbers from sum and difference
  $$H = \dfrac{(H + L) + (H - L)}{2}, \qquad L = \dfrac{(H + L) - (H - L)}{2}$$

Watch out for (3)

• Count what is left
  → Removing the highest and the lowest
• Check that it really is the extreme
  → One extreme alone: the largest or the smallest
• Halve at the end
  → Highest and lowest from their sum and difference

Formulas (3)

• Batting average and the next innings · Batting average
  $$\text{batting average} = \dfrac{\text{runs}}{\text{innings}} \quad \left(\text{or } \dfrac{\text{runs}}{\text{dismissals}}\right)$$
• Runs needed for a target, and how many innings · Score needed in the next innings
  $$x = A'(n + 1) - An$$
• Run rate, required rate and economy · Run rates
  $$\text{run rate} = \dfrac{\text{runs}}{\text{overs}}, \qquad \text{required rate} = \dfrac{\text{target} - \text{runs so far}}{\text{overs left}}$$

Watch out for (3)

• Not-outs change the divisor
  → Batting average and the next innings
• Count the innings still to come
  → Runs needed for a target, and how many innings
• Do not average two rates
  → Run rate, required rate and economy

Formulas (4)

• Moving a family through time · A family's total after t years
  $$\text{total after } t \text{ years} = \text{total now} + n\,t$$
• A child or a baby joins the family · A child joins
  $$\text{new average} = \dfrac{\text{family total} + \text{child's age}}{n + 1}$$
• Finding the baby's age · The baby's age
  $$\text{baby} = (n + 1)A_{\text{now}} - \left(nA_{\text{then}} + n\,t\right)$$
• Births at different times · Today's total
  $$\text{total now} = \text{past total} + n\,t + (\text{each child's age now})$$

Watch out for (5)

• The gap stays, the ratio does not
  → Moving a family through time
• Age the family first
  → A child or a baby joins the family
• Age the older members first
  → Finding the baby's age
• A just-born baby is 0
  → Finding the baby's age
• Children age from their own birth
  → Births at different times