Calculate the quartile deviation and its coefficient from...

Calculate the quartile deviation and its coefficient from the following: $$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|} \hline \text { Cl. Interval } & 10-15 & 15-20 & 20-25 & 25-30 & 30-40 & 40-50 & 50-60 & 60-70 & \text { Total } \\ \hline \text { Frequency } & 4 & 12 & 16 & 22 & 10 & 8 & 6 & 4 & 82 \\ \hline \end{array} $$

Calculate the quartile deviation and its coefficient from the following:
$$
\begin{array}{|l|c|c|c|c|c|c|c|c|c|}
\hline \text { Cl. Interval } & 10-15 & 15-20 & 20-25 & 25-30 & 30-40 & 40-50 & 50-60 & 60-70 & \text { Total } \\
\hline \text { Frequency } & 4 & 12 & 16 & 22 & 10 & 8 & 6 & 4 & 82 \\
\hline
\end{array}
$$

Key Concepts

Quartile Deviation

The quartile deviation, also known as the semi-interquartile range, measures the spread of the middle 50% of the data and is calculated as half the difference between the third quartile (Q3) and the first quartile (Q1). It is a robust measure of dispersion that reduces the influence of extreme values.

Interpolation in Grouped Data

Interpolation is a technique used to estimate precise data values within a class interval when only grouped data is available. When calculating quartiles from grouped data, interpolation assumes that data are uniformly distributed within each interval to determine the exact point corresponding to a particular cumulative frequency.

Coefficient of Quartile Deviation

The coefficient of quartile deviation is a relative measure of dispersion that standardizes the quartile deviation by comparing it to the central tendency of the data. It is often calculated as the ratio of the quartile deviation to the sum of the first and third quartiles (or their average), expressed as a percentage, which allows for comparison of variability across different datasets.

Grouped Frequency Distribution

A grouped frequency distribution organizes data into intervals (or classes) and lists the number of observations (frequencies) that fall into each interval. This method is used when dealing with large datasets, allowing for a summarized view of the distribution and facilitating the calculation of various descriptive statistics.

Quartiles

Quartiles are statistical measures that divide a dataset into four equal parts. In the context of grouped data, quartiles (especially the first quartile Q1 and the third quartile Q3) are estimated using the cumulative frequency distribution and interpolation methods to identify the points below which 25% and 75% of the data fall, respectively.

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