Statistics — JEE Mains Mathematics

Formulas (2)

• Mean and variance from sums · Variance from sums
  $$\sigma^2=\frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2$$
• Shifts, scales and arithmetic progressions · Linear change of data
  $$y_i=ax_i+b\ \Rightarrow\ \bar y=a\bar x+b,\quad \sigma_y^2=a^2\sigma_x^2$$

Watch out for (2)

• Squares about a is not the variance
  → Mean and variance from sums
• The sign of a is lost
  → Shifts, scales and arithmetic progressions

Formulas (2)

• Two unknowns from the mean and variance · The pair from its sum and sum of squares
  $$(a-b)^2=2\left(a^2+b^2\right)-(a+b)^2$$
• A part of the data · Totals of the whole
  $$\sum x_i=n\bar x,\qquad \sum x_i^2=n\left(\sigma^2+\bar x^2\right)$$

Watch out for (2)

• Keep the square of the mean
  → Two unknowns from the mean and variance
• Divide by the new count
  → A part of the data

Formulas (2)

• Correcting a wrong value · Replace in both totals
  $$\sum x\to\sum x-w+c,\qquad \sum x^2\to\sum x^2-w^2+c^2$$
• Combining two groups · Combined variance
  $$\sigma^2=\frac{n_1\sigma_1^2+n_2\sigma_2^2}{n_1+n_2}+\frac{n_1n_2\left(\bar x_1-\bar x_2\right)^2}{\left(n_1+n_2\right)^2}$$

Watch out for (2)

• Rebuild the totals from the wrong data
  → Correcting a wrong value
• Group variances do not simply average
  → Combining two groups

Formulas (2)

• Mean deviation about the mean · Mean deviation about the mean
  $$\text{MD}(\bar x)=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-\bar x\right|$$
• Mean deviation about the median or the mode · Mean deviation about the median
  $$\text{MD}(M)=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-M\right|$$

Watch out for (2)

• The mean deviation is not the SD
  → Mean deviation about the mean
• Sort before you pick the median
  → Mean deviation about the median or the mode

Formulas (2)

• Mean and variance of a frequency table · Frequency table
  $$\bar x=\frac{\sum f_ix_i}{\sum f_i},\qquad \sigma^2=\frac{\sum f_ix_i^2}{\sum f_i}-\bar x^2$$
• Median and mean deviation of a table · Grouped median
  $$M=l+\frac{\frac{N}{2}-F}{f}\times h$$

Watch out for (2)

• The unknown frequency is in N too
  → Mean and variance of a frequency table
• F stops before the median class
  → Median and mean deviation of a table