Statistics: SPI Mathematics

Formulas (3)

• Mean, sum and count · Mean of n values
  $$\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} \qquad \sum x = n\bar{x}$$
• Mean of a number pattern · Means of common patterns
  $$\text{mean of } 1, 2, \dots, n = \frac{n + 1}{2} \qquad \text{mean of first } n \text{ odd numbers} = n \qquad \text{mean of first } n \text{ multiples of } k = \frac{k(n + 1)}{2}$$
• When every value changes the same way · Changing every value
  $$\text{new values } kx + c \;\Rightarrow\; \text{new mean} = k\bar{x} + c \qquad \sum (x - \bar{x}) = 0$$

Watch out for (7)

• Dividing by the number of known values
  → Mean, sum and count
• Looking for x and y separately
  → Mean, sum and count
• Starting whole numbers at 1
  → Mean of a number pattern
• Taking the sum for the mean
  → Mean of a number pattern
• Using first plus last over two for squares
  → Mean of a number pattern
• Adding n times the change to the mean
  → When every value changes the same way
• Doing the steps in the wrong order
  → When every value changes the same way

Formulas (1)

• Median of a list · Position of the median
  $$n \text{ odd: } \left(\frac{n + 1}{2}\right)\text{th value} \qquad n \text{ even: mean of the } \left(\frac{n}{2}\right)\text{th and } \left(\frac{n}{2} + 1\right)\text{th values}$$

Reference tables (1)

Watch out for (4)

• Calling the mode the middle value
  → Mean, median and mode side by side
• Taking the middle of an unordered list
  → Median of a list
• Giving the position for the value
  → Median of a list
• Losing count in a long list
  → Mode of a list

Formulas (3)

• Class marks and the direct method · Direct method
  $$\text{class mark} = \frac{\text{lower} + \text{upper}}{2} \qquad \bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$
• Assumed mean method · Assumed mean method
  $$d_i = x_i - a \qquad \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$$
• Step deviation method · Step deviation method
  $$u_i = \frac{x_i - a}{h} \qquad \bar{x} = a + h \times \frac{\sum f_i u_i}{\sum f_i}$$

Watch out for (5)

• Multiplying by a class limit
  → Class marks and the direct method
• Dividing by the number of classes
  → Class marks and the direct method
• Using only one equation for two unknown frequencies
  → Class marks and the direct method
• Dropping the minus signs
  → Assumed mean method
• Forgetting to multiply by h
  → Step deviation method

Formulas (3)

• Cumulative frequency and the median class · Median class
  $$\text{median class: first class with } cf \ge \frac{N}{2}$$
• The median formula · Median of grouped data
  $$\text{Median} = l + \frac{\frac{N}{2} - cf}{f} \times h$$
• The mode formula · Mode of grouped data
  $$\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$

Watch out for (5)

• Picking the tallest class as the median class
  → Cumulative frequency and the median class
• Using the median class's own cumulative frequency
  → The median formula
• Plugging mid-values in as limits
  → The median formula
• Swapping f0 and f1 in the top line
  → The mode formula
• Using classes with gaps as they are
  → The mode formula

Formulas (1)

• Mode equals 3 median minus 2 mean · Empirical relation
  $$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$$

Reference tables (1)

Watch out for (4)

• Swapping mean and median
  → Mode equals 3 median minus 2 mean
• Losing the minus sign when matching a pattern
  → Mode equals 3 median minus 2 mean
• Putting the classes on the y-axis
  → Which graph shows what
• Thinking every average has a graph
  → Which graph shows what