Quick Revision Notes: Mathematics (Class 10)

Chapter: Statistics (सांख्यिकी)

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1. Introduction & Basic Terms

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

• Data (आँकड़े): Facts and figures collected for a specific purpose.
• Raw Data: Data obtained in the original form.
• Grouped Data: Data organized into groups called class intervals.

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2. Mean of Grouped Data (माध्य)

The mean (or average) is the sum of all observations divided by the total number of observations. It is denoted by $\bar{x}$.

Method 1: Direct Method (प्रत्यक्ष विधि)

$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ Where:

• $x_i$ = Class mark (वर्ग चिह्न) of the $i^{th}$ interval
• $f_i$ = Frequency (बारंबारता) of the $i^{th}$ interval
• $\sum f_i x_i$ = Sum of the products of $f_i$ and $x_i$
• $\sum f_i$ = Total frequency ($N$)

Method 2: Assumed Mean Method (कल्पित माध्य विधि)

$$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$$ Where:

• $a$ = Assumed mean (कल्पित माध्य)
• $d_i = x_i - a$ = Deviation of each $x_i$ from $a$

Method 3: Step-Deviation Method (पद-विचलन विधि)

$$\bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h$$ Where:

• $u_i = \frac{x_i - a}{h}$
• $h$ = Class size / width (वर्ग माप)

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3. Mode of Grouped Data (बहुलक)

Mode is the value that appears most frequently in the data (maximum frequency).

$$\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$$

Where:

• $l$ = Lower limit of the modal class (बहुलक वर्ग की निचली सीमा)
• $h$ = Size of the class interval (वर्ग आमाप)
• $f_1$ = Frequency of the modal class (बहुलक वर्ग की बारंबारता)
• $f_0$ = Frequency of the class preceding the modal class (ठीक पहले वाले वर्ग की बारंबारता)
• $f_2$ = Frequency of the class succeeding the modal class (ठीक बाद वाले वर्ग की बारंबारता)

Note: Modal Class (बहुलक वर्ग): The class interval with the maximum frequency.

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4. Median of Grouped Data (माध्यक / माधिका)

Median is the middle-most value of a distribution when arranged in ascending or descending order.

$$\text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h$$

Where:

• $l$ = Lower limit of the median class (माध्यक वर्ग की निचली सीमा)
• $N$ = Total number of observations ($\sum f_i$)
• $cf$ = Cumulative frequency of the class preceding the median class (माध्यक वर्ग से ठीक पहले वाले वर्ग की संचयी बारंबारता)
• $f$ = Frequency of the median class (माध्यक वर्ग की बारंबारता)
• $h$ = Class size (वर्ग आमाप)

Note: Median Class (माध्यक वर्ग): The class interval whose cumulative frequency is closest to and greater than $\frac{N}{2}$.

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5. Empirical Relationship Between Mean, Median, and Mode

For a moderately skewed frequency distribution, the three measures of central tendency are connected by the famous empirical formula:

$$\text{3 Median} = \text{Mode} + \text{2 Mean}$$ (Or) $$\text{Mode} = 3(\text{Median}) - 2(\text{Mean})$$

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6. Quick Tips for MP Board Exams

1. Cumulative Frequency (संचयी बारंबारता - c.f.): Always remember that c.f. is calculated by adding frequencies progressively. It is essential for finding the Median.
2. Class Marks ($x_i$): $$x_i = \frac{\text{Upper Limit} + \text{Lower Limit}}{2}$$
3. Units: Always write appropriate units in your final answers (e.g., cm, kg, marks, etc., if given in the question).
4. Table Formatting: Always make clean, ruled columns for $x_i$, $f_i$, $f_i x_i$, or $c.f.$ during calculations to avoid silly arithmetic errors.