Graphical Determination of Mode and Median

In statistics, graphical methods provide visual ways to understand and find measures of central tendency like mode and median. Different types of graphs are used for different purposes and different measures.

Graphical Determination of Mode using Histogram

The mode is the value that appears most frequently in a data set. For continuous data presented in frequency distributions, the mode can be graphically determined using a Histogram.

• First, construct the histogram for the given data. The highest rectangle in the histogram represents the modal class (the class interval with the highest frequency).
• To find the exact mode within this modal class, draw lines from the top corners of the modal rectangle to the opposite top corners of the adjacent rectangles (the rectangle preceding and succeeding the modal class).
• The point where these two lines intersect is located within the modal class rectangle.
• Draw a perpendicular line from this intersection point down to the x-axis.
• The value on the x-axis where this perpendicular line touches is the graphical mode.

This method is specifically used for finding the mode of a continuous frequency distribution.

Graphical Determination of Median using Ogive

The median is the middle value of a data set when it is arranged in order. For grouped data, the median can be graphically determined using an Ogive, which is a cumulative frequency curve.

• There are two types of Ogives: the 'less than' Ogive and the 'more than' Ogive.
• To find the median, you can use either one Ogive or both together.
• Using a single Ogive (e.g., 'less than' Ogive):
  • Plot the cumulative frequencies against the upper limits of the class intervals. Connect the points with a smooth curve to get the 'less than' Ogive.
  • Calculate the total frequency, \(N\). The median corresponds to the value at the \((N/2)\)-th position.
  • Locate \(N/2\) on the cumulative frequency (y-axis).
  • From this point on the y-axis, draw a horizontal line to intersect the Ogive curve.
  • From the intersection point on the curve, draw a perpendicular line down to the x-axis.
  • The value on the x-axis where this perpendicular line touches is the graphical median.
• Using both Ogives:
  • Plot both the 'less than' and the 'more than' Ogives on the same graph.
  • The point where the two Ogives intersect corresponds to the median.
  • Draw a perpendicular line from this intersection point down to the x-axis.
  • The value on the x-axis where this perpendicular line touches is the graphical median.

Ogive is the standard graphical tool for finding the median of a frequency distribution because it shows cumulative values, which are necessary to locate the middle value.

Conclusion

Based on these graphical methods:

• The mode of a continuous frequency distribution is determined graphically using a Histogram.
• The median of a frequency distribution is determined graphically using an Ogive.

Therefore, for the determination of mode and median graphically, one considers Histogram and Ogive.

Comparing this with the given options, the correct combination is Histogram and Ogive.

Revision Table: Graphical Methods for Central Tendency

Additional Information: Other Statistical Graphs

While Histogram and Ogive are crucial for finding mode and median graphically, other statistical graphs are used for different purposes:

• Bar Diagram: Used for discrete data or qualitative data. Shows the frequency of each category. Bars are separated. Not used for finding mode or median of continuous data this way.
• Line Diagram (Frequency Polygon): Often derived from a histogram by joining the midpoints of the tops of adjacent rectangles. Useful for showing the shape of the distribution and comparing distributions. Not directly used for finding median, though the area under it represents total frequency.
• Frequency Curve: A smoothed frequency polygon, assuming the data is continuous and the classes are very narrow.

Understanding which graph to use depends on the type of data and what statistical measure or feature you want to illustrate or determine.