Statistics: AFCAT Numerical Ability

Formulas (2)

• Data and class intervals · Class width from the range
  $$w = \dfrac{H - L}{k}, \qquad \text{last boundary} = L + kw$$
• Pie charts and square diagrams · Angle of a sector
  $$\theta = \dfrac{v}{T} \times 360^\circ, \qquad \text{side} = \sqrt{\dfrac{v}{s}}$$

Watch out for (5)

• Every limit moves by the same half-gap
  → Data and class intervals
• Census figures are secondary
  → Data and class intervals
• Out of 360, not out of 100
  → Pie charts and square diagrams
• The side is a square root
  → Pie charts and square diagrams
• A ratio needs no total
  → Pie charts and square diagrams

Formulas (4)

• Mean of a list and of a frequency table · Mean of a frequency table
  $$\bar x = \dfrac{\sum f x}{\sum f}$$
• How the mean moves when the data change · Correcting a wrong value
  $$\bar x_{\text{correct}} = \dfrac{n \bar x - a + b}{n}$$
• Combined mean of groups · Combined mean
  $$\bar x = \dfrac{n_1 \bar x_1 + n_2 \bar x_2}{n_1 + n_2}$$
• Spread: range, variance and standard deviation · Standard deviation
  $$\sigma = \sqrt{\dfrac{\sum (x - \bar x)^2}{n}}$$

Watch out for (7)

• Divide by the total frequency
  → Mean of a list and of a frequency table
• A class counts at its mid-point
  → Mean of a list and of a frequency table
• A wrong value does not change the count
  → How the mean moves when the data change
• Adding and multiplying act differently
  → How the mean moves when the data change
• Do not average the averages
  → Combined mean of groups
• Adding a constant does not change the spread
  → Spread: range, variance and standard deviation
• Take the square root at the end
  → Spread: range, variance and standard deviation

Formulas (5)

• Median and mode of a list · Position of the median
  $$\text{median} = \left(\dfrac{n + 1}{2}\right)^{\text{th}} \text{ value} \quad (n \text{ odd})$$
• Median of grouped data · Median of grouped data
  $$M = l + \dfrac{\tfrac N2 - F}{f} \times h$$
• Mode of grouped data · Mode of grouped data
  $$Z = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$
• Geometric mean, and numbers from their averages · Geometric mean
  $$G = \sqrt[n]{x_1 x_2 \cdots x_n}$$
• Mean, median and mode compared · Empirical relation
  $$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$$

Watch out for (10)

• Order the data first
  → Median and mode of a list
• Equal counts mean no mode
  → Median and mode of a list
• F is the count before the class
  → Median of grouped data
• Grouped data use N over 2
  → Median of grouped data
• Both neighbours go in the denominator
  → Mode of grouped data
• The tallest class, not the middle one
  → Mode of grouped data
• Multiply, do not add
  → Geometric mean, and numbers from their averages
• Adding to every value does not add to the geometric mean
  → Geometric mean, and numbers from their averages
• 3 times the median, 2 times the mean
  → Mean, median and mode compared
• The mean is not always above the median
  → Mean, median and mode compared

Formulas (1)

• The ogive and reading the median · Reading the median
  $$\text{cumulative frequency at } M = \dfrac N2$$

Reference tables (1)

Watch out for (4)

• Area, not height, follows the frequency
  → The graphs of a frequency distribution
• A histogram is not a bar chart
  → The graphs of a frequency distribution
• Less than goes at the upper boundary
  → The ogive and reading the median
• Half the total frequency, not half the axis
  → The ogive and reading the median