Statistics — CDS Elementary Mathematics

Formulas (1)

• Classes, histograms and polygons · Frequency density
  $$\text{frequency density} = \dfrac{\text{class frequency}}{\text{class width}}$$

Reference tables (1)

Watch out for (2)

• Ranked is not measured
  → Kinds of data and scales of measurement
• Inclusive and exclusive are easy to swap
  → Classes, histograms and polygons

Formulas (2)

• Cumulative frequency tables · Frequency from a cumulative table
  $$f(a\text{–}b) = F(b) - F(a)$$
• Mean, median and mode of an x–f table · Mean of a frequency table
  $$\bar x = \dfrac{\sum f x}{\sum f}$$

Watch out for (2)

• Percent of the maximum, not marks
  → Cumulative frequency tables
• Sort the x values first
  → Mean, median and mode of an x–f table

Formulas (3)

• Shifting and scaling the data · Linear change
  $$y = ax + b \;\Rightarrow\; \bar y = a\bar x + b$$
• Deviations from the mean and from other points · Deviations
  $$\sum (x_i - \bar x) = 0, \qquad \sum (x_i - A) = n(\bar x - A)$$
• Combined and weighted means · Combined mean
  $$\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}$$

Watch out for (3)

• The added constant is added n times to the total
  → Shifting and scaling the data
• The zero sum needs no arithmetic
  → Deviations from the mean and from other points
• Weight by the counts
  → Combined and weighted means

Formulas (2)

• Finding the median · Median position
  $$n \text{ odd: } \left(\tfrac{n + 1}{2}\right)\text{th}; \quad n \text{ even: mean of the } \tfrac n2\text{th and } \left(\tfrac n2 + 1\right)\text{th}$$
• How the median responds to changes · Middle of ten values
  $$\text{median} = \tfrac12\left(x_{(5)} + x_{(6)}\right)$$

Watch out for (2)

• Sort before you pick the middle
  → Finding the median
• Half the change when two values are averaged
  → How the median responds to changes

Formulas (3)

• Mean of grouped data and missing frequencies · Grouped mean
  $$\bar x = \dfrac{\sum f m}{\sum f}, \quad m = \text{class mid-point}$$
• Median of grouped data · Grouped median
  $$\text{Median} = l + \dfrac{\tfrac N2 - cf}{f}\,h$$
• Mode of grouped data · Grouped mode
  $$\text{Mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h$$

Watch out for (3)

• An open last class
  → Mean of grouped data and missing frequencies
• Use boundaries, not printed limits
  → Median of grouped data
• Two times f₁ in the denominator
  → Mode of grouped data

Formulas (1)

• The empirical relation · Empirical relation
  $$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$$

Reference tables (1)

Watch out for (2)

• Positional does not mean 'the mode'
  → Which average to use
• Three median, two mean
  → The empirical relation