As the number of observations and classes increases, the shape of a frequency polygon
Tends to become increasingly smooth
Let's analyze how the shape of a frequency polygon changes when we increase the number of observations and the number of classes.
Understanding Frequency Polygons and Data
A frequency polygon is a way to visualize the distribution of data. It is constructed by plotting points at the midpoint of each class interval and connecting these points with straight lines. The height of each point corresponds to the frequency (or relative frequency) of observations within that class interval.
Let's consider the factors mentioned in the question:
- Number of Observations: This refers to the total amount of data collected. More observations generally provide a more complete picture of the underlying distribution.
- Number of Classes: This refers to how many intervals the entire range of data is divided into. Increasing the number of classes means each interval becomes narrower (assuming the range of data stays the same or doesn't increase proportionally).
Impact of Increasing Observations and Classes
When you increase the number of observations, you gather more data points. This gives a clearer idea of how frequencies are distributed across different values or ranges. With more data, the frequency counts for each class interval become more reliable representations of the population distribution.
When you increase the number of classes, you divide the data range into finer intervals. This means that the plotted points on the frequency polygon will represent frequencies over smaller segments of the data range and will be closer together horizontally.
Consider the combined effect:
- More observations provide more detail about the frequency within *each* class.
- More classes provide more points on the graph, spread over finer intervals.
As the number of observations increases, especially when combined with an increased number of classes (making class intervals narrower), the frequency polygon tends to have more points that are closer together. These points will more closely follow the true shape of the data distribution. Think about drawing a curve by connecting dots: if you have only a few dots widely spaced, the connecting lines might look jagged. If you have many dots very close together, the connecting lines will look much smoother and will better approximate a smooth curve.
Evaluating the Options
Let's look at the given options in light of this understanding:
- Tends to become jagged: This is incorrect. Jaggedness is typically associated with fewer data points or wider class intervals, leading to larger jumps in frequency between points.
- Tends to become increasingly smooth: This aligns with our explanation. More data points and finer class intervals lead to a more detailed and smoother representation of the underlying distribution.
- Stays the same: This is incorrect. The shape is directly influenced by the amount and grouping of data.
- Varies only if data become more reliable: While reliable data is crucial for accurate analysis, the question focuses on the impact of the *quantity* of observations and *number of classes* on the graphical shape, not the reliability of the data itself. Increasing reliability would improve the accuracy of the shape, but the *trend* towards smoothness with increased observations and classes is a separate phenomenon.
Therefore, as the number of observations and classes increases, the shape of a frequency polygon tends to become increasingly smooth, approximating the shape of the underlying theoretical distribution (like a normal curve) if the data follows one.
Revision Table: Frequency Polygon Shape
| Factor | Change | Impact on Frequency Polygon Shape |
|---|---|---|
| Number of Observations | Increases | More detailed frequencies, smoother polygon (especially with more classes) |
| Number of Classes | Increases | Narrower class intervals, more points on graph, smoother polygon |
| Number of Observations & Classes | Both Increase | Polygon becomes increasingly smooth, better approximates true distribution shape |
Additional Information on Frequency Polygons and Histograms
A frequency polygon is closely related to a histogram. A histogram uses bars to show the frequency of data within class intervals. A frequency polygon can be drawn by connecting the midpoints of the top of each bar in a histogram. Both graphs help visualize the shape, center, and spread of a data distribution.
When data is continuous and we have a very large number of observations divided into very small class intervals, the frequency polygon would approach the shape of a smooth curve, often called a frequency curve or density curve. This concept is important in understanding probability distributions.
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1. Bar diagram
2. Pie-diagram
3. Pictogram
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Important Questions from Classification of Data
Consider the following LPP.:
Max Z = 15x 1 + 10x 2
Subject to the constraints
4x 1 + 6x 2 ≤ 360
3x 1 + 0x 2 ≤ 180
0x 1 + 5x 2 ≤ 200
x 1, x 2 ≥ 0
The solution of the LPP using Graphical solution-technique is :
A graph of a cumulative frequency distribution is called :
Which of the following is not an example of compressed data?
A cumulative frequency distribution is given below
Class
60-62
63-65
66-68
69-71
72-74
Cumulative frequency
3
20
36
48
50
Which one of the following class has maximum frequency?
The measure of the central tendency is given by the X-coordinate of the point of intersection of the more than ogive and less than ogive is:
