Quick Revision Notes & Formula Sheet

Class: 11 Mathematics Chapter: Statistics

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1. Introduction to Statistics

Statistics is the branch of mathematics that deals with the collection, presentation, analysis, and interpretation of numerical data.

• Raw Data: Data obtained in the original form.
• Arrayed Data: Data arranged in ascending or descending order.
• Frequency ($f$): The number of times an observation occurs in the given data.

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2. Measures of Dispersion

Measures of dispersion help us to study the variability or spread of the observations about the mean or any other central value. It is classified into two types:

1. Absolute Measures: Range, Quartile Deviation, Mean Deviation, Standard Deviation (expressed in the same units as data).
2. Relative Measures: Coefficient of Range, Coefficient of Quartile Deviation, Coefficient of Mean Deviation, Coefficient of Variation (unitless/pure ratios).

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3. Range

The range is the difference between the maximum and minimum observations in the data.

• Range = $x_{max} - x_{min}$
• Coefficient of Range = $\frac{x_{max} - x_{min}}{x_{max} + x_{min}}$

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4. Mean Deviation (MD)

Mean Deviation is the average of the absolute differences of the values from a central tendency (Mean or Median).

For Ungrouped Data / Discrete Frequency Distribution:

Let the observations be $x_1, x_2, \dots, x_n$ with respective frequencies $f_1, f_2, \dots, f_n$.

• Mean Deviation about Mean ($MD(\bar{x})$): $$MD(\bar{x}) = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - \bar{x}|$$ where $N = \sum f_i$ and $\bar{x}$ is the mean.

• Mean Deviation about Median ($MD(M)$): $$MD(M) = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - M|$$ where $M$ is the median.

For Continuous Frequency Distribution (Grouped Data):

• Use the mid-points ($x_i$) of each class interval in place of observations.
• $\bar{x} = \frac{\sum f_i x_i}{N}$
• Median $M = L + \frac{\frac{N}{2} - C}{f} \times h$ (where $L$ = lower limit of median class, $C$ = cumulative frequency of preceding class, $f$ = frequency of median class, $h$ = class size).

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5. Variance and Standard Deviation

Standard Deviation (SD) is the positive square root of the variance. It is denoted by Greek letter sigma ($\sigma$).

5.1 For Ungrouped Data

• Variance ($\sigma^2$): $$\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{n} \sum x_i^2 - (\frac{\sum x_i}{n})^2$$

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

5.2 For Discrete Frequency Distribution

• Variance ($\sigma^2$): $$\sigma^2 = \frac{1}{N} \sum_{i=1}^{n} f_i (x_i - \bar{x})^2 = \frac{1}{N} \sum_{i=1}^{n} f_i x_i^2 - \left(\frac{1}{N} \sum_{i=1}^{n} f_i x_i\right)^2$$ where $N = \sum f_i$.

• Standard Deviation ($\sigma$): $$\sigma = \sqrt{\frac{1}{N} \sum f_i (x_i - \bar{x})^2}$$

5.3 For Continuous Frequency Distribution (Grouped Data)

Use class mid-points as $x_i$: $$\sigma = \frac{1}{N} \sqrt{N \sum f_i x_i^2 - (\sum f_i x_i)^2}$$ (Shortcut method / Step-deviation method can also be used: $d_i = \frac{x_i - A}{h}$)

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6. Coefficient of Variation (CV)

The Coefficient of Variation is used to compare the variability of two or more series. The series with a greater CV is more variable (less consistent/less stable), while the series with a smaller CV is more consistent (more stable).

$$\text{CV} = \frac{\sigma}{\bar{x}} \times 100$$ (where $\sigma$ = Standard Deviation, $\bar{x}$ = Mean)

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Important Quick Tips for Exams:

1. Sum of absolute deviations is minimum when taken from the Median.
2. Sum of squares of deviations is minimum when taken from the Mean.
3. Standard deviation is independent of change of origin, but depends on change of scale. (If $y_i = \frac{x_i - a}{h}$, then $\sigma_y = \frac{1}{|h|} \sigma_x$).
4. Variance of first $n$ natural numbers: $\sigma^2 = \frac{n^2 - 1}{12}$.