Quick Revision Notes: Class 9 Mathematics

Chapter: Statistics (सांख्यिकी)

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1. Introduction to Statistics

• Statistics (सांख्यिकी): The branch of mathematics that deals with the collection, presentation, analysis, and interpretation of numerical data.
• Data (आँकड़े): Facts or figures collected with a specific purpose.
• Types of Data:
  1. Primary Data (प्राथमिक आँकड़े): Data collected by the investigator herself/himself with a definite objective in mind.
  2. Secondary Data (गौण आँकड़े): Data collected from a source that already had the information stored (e.g., from books, newspapers, or official records).

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2. Presentation of Data

• Raw Data (अवर्गीकृत आँकड़े): Data obtained in the original form (as initial measurements).
• Frequency (बारंबारता - $f$): The number of times a particular observation occurs in the given data.
• Grouped Frequency Distribution (वर्गीकृत बारंबारता बंटन): When data is organized into groups (called Class Intervals like $0-10$, $10-20$).

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3. Terms Related to Grouped Data

• Class Interval (वर्ग अंतराल): The range of values in a group (e.g., in $10-20$, $10$ is the lower limit and $20$ is the upper limit).
• Lower Limit (निम्न सीमा): The lowest value in a class interval.
• Upper Limit (ऊपरी सीमा): The highest value in a class interval.
• Class Size / Width (वर्ग आमाप): The difference between the upper limit and the lower limit of a class. $$\text{Class Size} = \text{Upper Limit} - \text{Lower Limit}$$
• Class Mark (वर्ग चिन्ह): The mid-point of a class interval. $$\text{Class Mark} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$$

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4. Graphical Representation of Data

The three main graphical representations studied in Class 9 are:

1. Bar Graph (दण्ड आरेख): A display of bars with uniform width and heights proportional to the respective frequencies.
2. Histogram (आयतचित्र): A type of bar graph for continuous class intervals where bars are drawn adjacent to each other with no gaps.
   • Note: If the first class interval does not start at origin (0), a kink (zig-zag mark) is used on the x-axis.
3. Frequency Polygon (बारंबारता बहुभुज): A polygon constructed by plotting class marks against frequencies and joining the points with straight lines. It can be drawn with or without a histogram.

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5. Measures of Central Tendency (केंद्रीय प्रवृत्ति की माप)

Measures of central tendency help to represent the entire data with a single value.

I. Arithmetic Mean (माध्य - $\bar{x}$)

For raw data observations $x_1, x_2, \dots, x_n$: $$\text{Mean } (\bar{x}) = \frac{\text{Sum of all observations}}{\text{Total number of observations}} = \frac{x_1 + x_2 + \dots + x_n}{n}$$

Using summation notation ($\sum$): $$\bar{x} = \frac{\sum x_i}{n}$$

For a frequency distribution where observations $x_i$ have frequencies $f_i$: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$

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II. Median (माध्यिका / मध्यक)

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

1. Step 1: Arrange the data in increasing order.
2. Step 2: Let $n$ be the total number of observations.
   • If $n$ is Odd (विषम): $$\text{Median} = \left(\frac{n + 1}{2}\right)^{\text{th}} \text{ term}$$
   • If $n$ is Even (सम): $$\text{Median} = \frac{\left(\frac{n}{2}\right)^{\text{th}} \text{ term} + \left(\frac{n}{2} + 1\right)^{\text{th}} \text{ term}}{2}$$

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III. Mode (बहुलक)

• Definition: The value that occurs most frequently in the data (the observation with the highest frequency).
• For Raw Data: Simply count the frequency of each observation and identify the one with the maximum count. A data set can have one mode (unimodal), two modes (bimodal), or more.