Statistics - Presentation of Data (Frequency Distribution)

Data and Variables: Data is a collection of observations. In statistics, variables are classified as Discrete (values that can be counted, like the number of children in a family) or Continuous (values that can be measured, like height or temperature, which can take any value within a range).

Frequency and Tally Marks: Frequency refers to the number of times a particular observation occurs in a data set. To organize raw data visually, we use tally marks where vertical bars $|, ||, |||, ||||$ represent counts of 1 to 4, and the fifth count is shown by a diagonal line crossing the four bars to form a bundle of five.

Class Intervals (Grouped Data): When dealing with large datasets, data is organized into groups called class intervals. There are two types: Exclusive (Continuous) method, where the upper limit of one class is the lower limit of the next (e.g., $0-10, 10-20$), and Inclusive (Discontinuous) method, where the limits are inclusive (e.g., $0-9, 10-19$).

Class Limits and Class Boundaries: In a class interval like $10-20$, 10 is the Lower Limit and 20 is the Upper Limit. For inclusive classes, we convert limits into boundaries to ensure continuity; this is visualized by closing the gap between the end of one class and the start of the next.

Class Mark (Mid-value): The class mark is the central value of a class interval. It represents the entire class in various calculations and is the point that would be plotted in the middle of a class interval on the horizontal axis of a frequency polygon.

Class Size (Width): The class size is the difference between the actual upper boundary and the lower boundary of a class. In a visual distribution like a histogram, this represents the uniform width of the rectangular bars.

Cumulative Frequency: This is the sum of the frequency of a class and all previous classes. A 'Less than' cumulative frequency table shows the total number of observations below the upper limit of each class, representing a running total that grows towards the total number of observations ($N$).