Understanding the Cumulative Frequency Diagram for a Discrete Variable

The question asks about the graphical representation of the less than type cumulative frequency for a discrete variable. Let's break down the key concepts:

• Discrete Variable: A variable whose values can only take specific, distinct values, often integers (e.g., number of students, number of defects). There are gaps between possible values.
• Frequency Distribution: A table that shows the frequency (count) of each distinct value or range of values in a dataset.
• Cumulative Frequency (Less Than Type): For a given value of the variable, the cumulative frequency is the sum of the frequencies of all values that are less than or equal to that value.

Properties of Less Than Type Cumulative Frequency

Consider a discrete variable X with values \(x_1 < x_2 < x_3 < \dots < x_n\). Let the frequency of \(x_i\) be \(f_i\). The cumulative frequency (less than type) for \(x_i\) is given by:

\(CF(x_i) = f_1 + f_2 + \dots + f_i = \sum_{j=1}^{i} f_j\)

As we move from a smaller value \(x_i\) to a larger value \(x_{i+1}\), the cumulative frequency \(CF(x_{i+1})\) includes all the frequencies up to \(x_i\) plus the frequency of \(x_{i+1}\). Since frequencies (\(f_i\)) are non-negative, the cumulative frequency can either stay the same (if \(f_{i+1} = 0\)) or increase (if \(f_{i+1} > 0\)). It can never decrease.

Therefore, the cumulative frequency function for a less than type distribution is always non-decreasing.

Diagram of Cumulative Frequency for a Discrete Variable

When we plot the cumulative frequency against the value of the discrete variable, we plot points \((x_i, CF(x_i))\). Since the variable is discrete, the cumulative frequency only changes its value at the specific points \(x_1, x_2, \dots, x_n\). Between two consecutive distinct values \(x_i\) and \(x_{i+1}\), the cumulative frequency remains constant at \(CF(x_i)\) for any value greater than or equal to \(x_i\) and less than \(x_{i+1}\). This results in a graph that looks like a series of horizontal steps, with vertical jumps occurring at each value of the discrete variable where the frequency is non-zero.

A function whose graph consists of a series of horizontal steps is called a step function.

Because the cumulative frequency is non-decreasing and the changes occur in steps at discrete values, the diagram of the less than type cumulative frequency for a discrete variable is a non-decreasing step function.

A continuous function, in contrast, would have a graph that can be drawn without lifting the pen, implying smooth changes, which is not the case here due to the discrete nature of the variable.

Based on this analysis, the correct description for the diagram of the less than type cumulative frequency of a discrete variable is a non-decreasing step function.

Revision Table: Cumulative Frequency Diagrams

Additional Information: Ogive and Cumulative Frequency

The diagram representing cumulative frequencies is often called an Ogive. For discrete data, the ogive is typically a step function. For grouped continuous data, we usually plot the cumulative frequency against the upper class boundaries, and the points are joined by line segments, resulting in a curve (often smoothed) which is also a non-decreasing function, but continuous rather than a step function in its ideal representation.

Understanding whether a variable is discrete or continuous is crucial in determining the appropriate graphical representation for its cumulative frequency distribution.