Statistics
Ungrouped Data
PYQ2023StatisticsQ1, PYQ2023StatisticsQ2, PYQ2023StatisticsQ5, PYQ2023StatisticsQ6, PYQ2023StatisticsQ8, PYQ2023StatisticsQ9, PYQ2023StatisticsQ10

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
$$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i  \bar{x})^2}{n}$$ PYQ2023StatisticsQ4

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: $$ \text{Mode} = L + \left(\frac{f_m  f_{m1}}{(f_m  f_{m1}) + (f_m  f_{m+1})}\right) \times w $$ where $L$ is the lower boundary of the modal class, $f_m$ is the frequency of the modal class, $f_{m1}$ 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
$$\text{MD} = \frac{\sum_{i=1}^{k} f_i x_i  \bar{x}}{\sum_{i=1}^{k} f_i}$$