Draw the less than Ogive and estimate the value of median on...

Draw the less than Ogive and estimate the value of median on the basis of the data given below:
\begin{tabular}{|l|r|r|r|r|r|r|r|r|}
\hline \mathrm{Mid-Point } & 18 & 25 & 32 & 39 & 46 & 53 & 60 & \\
\hline Frequency & 10 & 15 & 32 & 42 & 26 & 12 & 9 & $N=146$ \\
\hline
\end{tabular}

Key Concepts

Less than Ogive

An Ogive is a graphical representation of the cumulative frequency distribution of a data set. In a less than Ogive, cumulative frequencies are plotted at the upper boundary of the class intervals. This curve is used to visualize how frequencies accumulate over increasing values and facilitates the estimation of percentiles, medians, and other quantiles by identifying the corresponding value on the horizontal axis for a given cumulative frequency.

Cumulative Frequency Distribution

A cumulative frequency distribution is a running total of frequencies through the classes in an ordered data set. It shows how many observations lie below the upper boundary of each class interval. This concept is central to constructing an Ogive and is also used in statistical analysis to determine the position of various quantiles and percentiles in the data.

Median Estimation in Grouped Data

For grouped data, the median is estimated by identifying the point on the Ogive corresponding to half of the total frequency. This involves locating the cumulative frequency that equals half of the total observations and then determining the corresponding value on the horizontal axis. Interpolation may be used between class boundaries if the exact median value does not fall exactly on one of the plotted points, making the median estimation a graphical approximation in the context of grouped data.

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