Ungrouped Data

PYQ-2023-Statistics-Q1, PYQ-2023-Statistics-Q2, PYQ-2023-Statistics-Q5, PYQ-2023-Statistics-Q6, PYQ-2023-Statistics-Q8, PYQ-2023-Statistics-Q9, PYQ-2023-Statistics-Q10

• Mean (Average)

  $$ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} $$

  where (x_i) are the individual data points and (n) is the number of data points.

• Median

  • Arrange the data in ascending order.
  • If n is odd, the median is the middle value.
  • If n is even, the median is the average of the two middle values.

• Mode

  • The value that occurs most frequently in the data set.

• Standard Deviation

  $$\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$$

• Variance

  $$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$$ PYQ-2023-Statistics-Q4

• Mean Deviation

  $$\text{MD} = \frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n}$$

Grouped Data

$\quad$ For grouped data, we use class intervals and their corresponding frequencies. Let $f_i$ be the frequency and $x_i$ be the midpoint (or class mark) of the $i$-th class interval.

• Mean

  $$ \bar{x} = \frac{\sum_{i=1}^{k} f_i x_i}{\sum_{i=1}^{k} f_i}$$

$\quad$ where, $k$ is the number of class intervals.

• Median

  • Locate the class interval containing the median (the one whose cumulative frequency just exceeds or equals $n/2$).
  • Use the formula: $$ \text{Median} = L + \left(\frac{\frac{n}{2} - F}{f}\right) \times w $$

  where, $L$ is the lower boundary of the median class, $F$ is the cumulative frequency of the class before the median class, $f$ is the frequency of the median class, and $w$ is the width of the median class.

• Mode:

  • Identify the modal class (the class with the highest frequency).
  • Use the formula: $$ \text{Mode} = L + \left(\frac{f_m - f_{m-1}}{(f_m - f_{m-1}) + (f_m - f_{m+1})}\right) \times w $$ where $L$ is the lower boundary of the modal class, $f_m$ is the frequency of the modal class, $f_{m-1}$ is the frequency of the class before the modal class, $f_{m+1}$ is the frequency of the class after the modal class, and $w$ is the width of the modal class.

• Standard Deviation

  $$ \sigma = \sqrt{\frac{\sum_{i=1}^{k} f_i (x_i - \bar{x})^2}{\sum_{i=1}^{k} f_i}}$$

• Variance

  $$\sigma^2 = \frac{\sum_{i=1}^{k} f_i (x_i - \bar{x})^2}{\sum_{i=1}^{k} f_i}$$

• Mean Deviation

  $$\text{MD} = \frac{\sum_{i=1}^{k} f_i |x_i - \bar{x}|}{\sum_{i=1}^{k} f_i}$$