Mathematics: Class 8 (Part 2) — Chapter 5: Tales by Dots and Lines

5.1 The Balancing Act

Arithmetic Mean: The sum of all values divided by the total number of values. It represents the "centre of gravity" or balance point of the data.
Median: The middle value when the data is sorted in ascending or descending order.
Balancing Property of Mean: The sum of the distances of all data points to the left of the mean is exactly equal to the sum of the distances of all data points to the right of the mean (LHS Distance = RHS Distance).

Adding/Removing Values:
- Including a value greater than the mean increases the mean.
- Including a value less than the mean decreases the mean.
- Including a value equal to the mean leaves the mean unchanged.
Shifting Data (Addition/Subtraction): If a constant $k$ is added to (or subtracted from) every value in a dataset, the new mean increases (or decreases) by $k$.
Scaling Data (Multiplication/Division): If every value in a dataset is multiplied by a constant $k$, the new mean is also multiplied by $k$.

Weighted Mean: When values have frequencies, the mean is calculated as:
$$\text{Mean} = \frac{\sum (x_i \times f_i)}{\sum f_i}$$ where $x_i$ represents the values and $f_i$ represents their respective frequencies.
Cumulative Frequency for Median: To find the median from a frequency table without listing all numbers, successively add frequencies from the smallest value until reaching the middle position(s).

Cells: Individual boxes in a spreadsheet identified by column letters and row numbers (e.g., E5).
Formulas: Used to perform quick calculations over a range of cells:
- =SUM(Start:End): Calculates the total sum of the specified range.
- =AVERAGE(Start:End): Calculates the arithmetic mean of the specified range.

5.2 Visualising and Interpreting Data

Line Graph: A visual representation consisting of data points connected by straight line segments. It is primarily used to display and track changes over time.
Interpretation: The steepness of a line segment indicates the rate of change; steeper lines represent more rapid increases or decreases.

Infographics: Visual representations of information, data, or knowledge designed to communicate complex patterns quickly and clearly (e.g., geographic preference maps).
Activity Strips: A method of tracking daily time allocation by dividing a strip into proportional, color-coded blocks representing different activities over 24 hours.