DATA HANDLING

Data is a collection of facts, figures, or information collected for a specific purpose. Raw data is data collected in its original form, which is often unorganized.

Types of Data

• Qualitative Data: Describes qualities or characteristics (e.g., colours, types of cars). Cannot be measured numerically.
• Quantitative Data: Deals with numbers and can be measured or counted.
• Discrete Data: Can only take specific values (e.g., number of students, number of cars). Usually obtained by counting.
• Continuous Data: Can take any value within a given range (e.g., height, weight, temperature). Usually obtained by measurement.

Organizing Raw Data

1. Array Data: Arranging raw data in ascending or descending order. Helps in quickly finding minimum, maximum, and range.
2. Frequency: The number of times a particular observation occurs in a data set.
3. Frequency Distribution Table: A table that shows the frequency of each observation or class interval.

Tally Marks

• Used to count frequencies efficiently.
• Each occurrence is marked by a vertical line (|).
• Every fifth mark is drawn diagonally across the previous four (\), making a bundle of five. This helps in quick counting.

Example: If the data is 2, 3, 5, 2, 4, 3, 2, 5, 1, 3

| Observation | Tally Marks | Frequency | |-------------|-------------|-----------| | 1 | | | 1 | | 2 | ||| | 3 | | 3 | ||| | 3 | | 4 | | | 1 | | 5 | || | 2 | | Total | | 10 |

When the number of observations is large, it's impractical to make a frequency distribution for each individual value. In such cases, we group the data into class intervals.

Key Terms for Grouped Data

• Class Interval: A range of values within which observations fall (e.g., 0-10, 10-20).
• Lower Limit: The smallest value in a class interval (e.g., 0 in 0-10).
• Upper Limit: The largest value in a class interval (e.g., 10 in 0-10).
• Note: In exclusive class intervals (e.g., 0-10, 10-20), the upper limit of one class is the lower limit of the next. The upper limit is not included in that class, but in the next one. So, 10 belongs to 10-20, not 0-10.
• Class Size (or Width): The difference between the upper limit and the lower limit of a class interval. Class Size = Upper Limit - Lower Limit.
• For 0-10, class size is 10 - 0 = 10.
• Range: The difference between the highest and lowest values in the entire data set. Range = Maximum Value - Minimum Value.

Steps to Create a Grouped Frequency Distribution Table

1. Find the range of the data.
2. Decide on the number of class intervals (usually 5-10) and an appropriate class size. Ensure all data points are covered.
3. Set up the class intervals. Use the exclusive method (e.g., 0-10, 10-20) to avoid ambiguity.
4. Use tally marks to count the frequency of observations falling into each class interval.
5. Write down the frequency for each class interval.

Example: Marks of 20 students: 12, 25, 30, 18, 22, 35, 15, 28, 32, 10, 20, 24, 38, 16, 29, 31, 14, 26, 33, 19.

• Minimum = 10, Maximum = 38. Range = 38 - 10 = 28.
• Let's choose class size = 10. Intervals: 10-20, 20-30, 30-40.

| Class Interval (Marks) | Tally Marks | Frequency | |------------------------|-------------|-----------| | 10-20 | |||| || | 7 | | 20-30 | |||| ||| | 8 | | 30-40 | |||| || | 5 | | Total | | 20 |

Graphical representation makes data easier to understand and compare.

Bar Graph

• Used for discrete data or categorical data.
• Bars are of uniform width.
• Bars are drawn vertically or horizontally with equal spacing between them.
• The height (or length) of each bar is proportional to the frequency of the category it represents.
• Each bar represents a single category.

Example: Number of students in different classes.

| Class | Number of Students | |-------|--------------------| | 6th | 40 | | 7th | 35 |\ | 8th | 45 | | 9th | 30 |

[IMAGE: TODO: Bar graph showing number of students per class]

Histogram

• Used for continuous data, specifically for grouped frequency distributions.
• Similar to a bar graph, but there are no gaps between the bars because the class intervals are continuous.
• The width of each bar represents the class size.
• The height of each bar represents the frequency of the class interval.
• If the class intervals are not uniform, the area of the rectangle should be proportional to the frequency.
• The horizontal axis represents the class intervals, and the vertical axis represents the frequency.

Steps to Draw a Histogram:

1. Draw horizontal (x-axis) and vertical (y-axis) axes.
2. Mark class intervals on the x-axis. Ensure the scale is uniform.
3. Mark frequencies on the y-axis. Choose an appropriate scale.
4. Draw rectangles for each class interval. The width of the rectangle corresponds to the class size, and the height corresponds to the frequency.
5. If the first class interval does not start from 0, use a kink or zig-zag line (broken line) on the x-axis to indicate that the scale has been broken.

Example: Weekly wages of 30 workers (from NCERT Q4)

| Class Interval (Wages in Rs.) | Frequency | |-------------------------------|-----------| | 800-810 | 3 |\ | 810-820 | 2 |\ | 820-830 | 1 |\ | 830-840 | 9 |\ | 840-850 | 5 |\ | 850-860 | 1 |\ | 860-870 | 3 |\ | 870-880 | 1 |\ | 880-890 | 1 |\ | 890-900 | 4 |\

[IMAGE: TODO: Histogram for weekly wages of workers]

A pie chart or circle graph shows the relationship between a whole and its parts. The circle represents the whole, and its sectors (parts) represent the different components.

Key Principles

• The entire circle represents 360°.
• The size of each sector (its central angle) is proportional to the component it represents.
• The sum of all central angles in a pie chart must be 360°.

Steps to Construct a Pie Chart

1. Calculate the fraction of each component relative to the total.

Fraction = (Component Value / Total Value)

1. Calculate the central angle for each component.

Central Angle = (Fraction) 360° Or, Central Angle = (Component Value / Total Value) 360°

1. Draw a circle using a compass.
2. Draw a radius.
3. Using a protractor, draw the central angles calculated for each component, starting from the radius.
4. Label each sector clearly with its component name and/or percentage.

Example: (NCERT Q2) Languages spoken by students in a hostel. Total students = 72.

| Language | Number of Students | Fraction of Total | Central Angle Calculation | Central Angle | |----------|--------------------|-------------------|---------------------------|---------------| | Hindi | 40 | 40/72 | (40/72) 360° | 200° |\ | English | 12 | 12/72 | (12/72) 360° | 60° |\ | Marathi | 9 | 9/72 | (9/72) 360° | 45° |\ | Tamil | 7 | 7/72 | (7/72) 360° | 35° |\ | Bengali | 4 | 4/72 | (4/72) * 360° | 20° |\ | Total| 72 | 1 | | 360° |\

[IMAGE: TODO: Pie chart for languages spoken by students]

Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Key Terms

• Experiment: An action or process that results in well-defined outcomes (e.g., tossing a coin, rolling a die).
• Outcome: A possible result of an experiment (e.g., Head, Tail when tossing a coin).
• Event: One or more outcomes of an experiment (e.g., getting an even number when rolling a die).
• Random Experiment: An experiment where the outcome cannot be predicted with certainty, but all possible outcomes are known (e.g., tossing a fair coin).
• Equally Likely Outcomes: Outcomes that have the same chance of occurring (e.g., getting any number from 1 to 6 when rolling a fair die).
• Favourable Outcomes: The outcomes that result in the occurrence of a specific event.

Formula for Probability

For an event E:

$$P(E) = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

Properties of Probability

• The probability of any event E is always between 0 and 1