Statistics
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General Learning Resources
Ungrouped Data
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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.
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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.
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Mode
- The value that occurs most frequently in the data set.
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Standard Deviation
$\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$
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Variance
$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$
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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.
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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.
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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.
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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.
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Standard Deviation
$ \sigma = \sqrt{\frac{\sum_{i=1}^{k} f_i (x_i - \bar{x})^2}{\sum_{i=1}^{k} f_i}}$
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Variance
$\sigma^2 = \frac{\sum_{i=1}^{k} f_i (x_i - \bar{x})^2}{\sum_{i=1}^{k} f_i}$
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Mean Deviation
$\text{MD} = \frac{\sum_{i=1}^{k} f_i |x_i - \bar{x}|}{\sum_{i=1}^{k} f_i}$
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