Statistics and Probability:
The chapter "Statistics and Probability" carrying the weightage of \(15\) marks in the board examination. It covers the concepts like Measures of dispersion, coefficient of variation, Probability, Algebra of events and Addition theorem of Probability.
| Total marks \(15\) |
Total marks \(= 15\)
|
To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Measures of dispersion - Finding Range, Mean deviation, Standard deviation, variance and coefficient of variation.
- Coefficient of variation - \(C.V = \frac{\sigma}{\overline x} \times 100\%\).
- Probability - Finding probability of an event.
- Algebra of events - Finding the probability of the events if they are mutually exclusive.
| Important Concept (Learning Outcomes) | Expected Question Type | Concept dealt with |
| Mean of the data Probability |
Sec A | |
| Range and coefficient of range Probability |
Sec A, Sec B and Sec C | |
| Probability | Sec C |
Let us recall the concepts in Statistics and Probability:
1. Measures of central tendency: The measures of central tendency are the value that tends to cluster around the middle value of the given set of data. The three measures of central tendency are mean, median and mode.
2. Range: The difference between the largest value and the smallest value in the given set of data is called Range.
3. Variance \(=\) Mean of squares of deviation
4. Standard deviation \(\sigma = \sqrt{\text{Variance}}\) \(= \sqrt{\frac{\sum_{i = 1}^n (x - \overline x)^2}{n}}\)
5. The formula for finding the arithmetic mean using the direct method is .
6. The formula for finding the arithmetic mean using the assumed mean method is where \(d_i = x_i - A\).
7. The formula for finding the arithmetic mean using the step deviation mean method is where \(d_i = \frac{x_i - A}{c}\).
8. The standard deviation of an ungrouped data can be calculated using one of the following methods:
Direct method : \(\sigma = \sqrt{\frac{\sum x_{i}^{2}}{n}- \left(\frac{\sum x_{i}}{n}\right)^2}\)
Assumed mean method : \(\sigma = \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}\) where \(d_{i} = x_{i} - A\).
Step deviation method : \(\sigma = c \times \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}\) where \(d_{i} = \frac{x_{i} - A}{c}\).
9. The standard deviation of an grouped data (either discrete or continuous) can be calculated using one of the following methods:
Mean method: \(\sigma\) \(=\) \(\sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}}\) where \(N = \sum_{i = 1}^{n} f_{i}\)
Assumed mean method: \(\sigma = \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right)^2}\) where \(N = \sum_{i = 1}^{n} f_{i}\)
Step deviation method: \(\sigma = c \times \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right) ^2}\) where \(N = \sum_{i = 1}^{n} f_{i}\).