Download RS Aggarwal Solutions Class 10 Chapter 9 - Mean, Median & Mode Exercise 9.2 Free PDF
FAQs on Quick Study with RS Aggarwal Solutions Class 10 Chapter 9 - Mean, Median & Mode (Ex 9B) Exercise 9.2
1. How do you construct a 'less than' type ogive for the problems in RS Aggarwal Class 10 Maths Chapter 9, Exercise 9.2?
To construct a 'less than' type ogive, you must first convert the given frequency distribution into a 'less than' cumulative frequency distribution. The steps are as follows:
- Take the upper class limits on the x-axis and the corresponding cumulative frequencies on the y-axis.
- Plot the points (upper class limit, cumulative frequency).
- Join these points with a smooth, freehand curve. The resulting curve is the 'less than' ogive. It will always be an upward-sloping curve.
2. What is the step-by-step method to find the median from a 'less than' ogive?
Once you have drawn the 'less than' ogive, you can find the median graphically by following these steps:
- Calculate the value of N/2, where N is the sum of all frequencies.
- Locate this N/2 value on the y-axis (the cumulative frequency axis).
- From this point on the y-axis, draw a horizontal line parallel to the x-axis until it intersects the ogive curve.
- From the point of intersection, draw a vertical line downwards to the x-axis.
- The point where this vertical line meets the x-axis is the median of the data.
3. Why must we use upper class limits for a 'less than' ogive and lower class limits for a 'more than' ogive?
This is based on the definition of cumulative frequency. A 'less than' ogive represents the cumulative count of observations that are less than a certain value. This value is naturally the upper boundary of a class interval. For example, the cumulative frequency for the class 10-20 shows the total number of items with a value 'less than 20'. Conversely, a 'more than' ogive shows the cumulative count of observations 'more than or equal to' a certain value, which corresponds to the lower boundary of the class interval.
4. How can the median be found if both 'less than' and 'more than' ogives are drawn on the same graph?
When both the 'less than' and 'more than' type ogives are plotted on the same graph paper, they will intersect at a single point. To find the median using this method:
- Draw both the 'less than' and 'more than' ogives accurately.
- Locate their point of intersection.
- Draw a perpendicular line from this intersection point down to the x-axis.
- The value of the x-coordinate at this point gives the median of the distribution. This is because the intersection represents the value for which the number of observations above it and below it are equal.
5. In RS Aggarwal Ex 9.2, why is it necessary to convert a standard frequency distribution to a cumulative one before plotting an ogive?
An ogive is, by definition, a cumulative frequency curve. Its purpose is to show the running total of frequencies, which helps in graphically locating partition values like the median and quartiles. A standard frequency distribution only shows the number of observations within each specific class interval. An ogive cannot be plotted from this directly. Therefore, the first and most crucial step is to calculate the cumulative frequencies (either 'less than' or 'more than') to get the data points needed for the y-axis of the graph.
6. Can an ogive be used to determine the mean or mode of a dataset? Explain your answer.
No, an ogive is not the correct graphical tool for finding the mean or mode.
- Median: An ogive is the specific graphical method used to find the median and other partition values (like quartiles and percentiles).
- Mode: The mode is found graphically using a histogram. It corresponds to the x-value of the tallest bar in the histogram.
- Mean: The mean cannot be determined graphically. It must be calculated using a formula (Direct, Assumed Mean, or Step-Deviation Method) as it depends on the actual values and frequencies of the data, not just their cumulative count.
7. What is the key difference between a frequency polygon and an ogive in statistics?
The primary difference lies in what is plotted on each axis. A frequency polygon is plotted by joining the points formed by the class marks (mid-points) on the x-axis and their corresponding individual class frequencies on the y-axis. In contrast, an ogive is plotted using the class boundaries (upper or lower limits) on the x-axis and the cumulative frequencies on the y-axis. Consequently, an ogive is a curve that never decreases, while a frequency polygon can go up and down.
