Find out the missing frequencies of the following data,...

Find out the missing frequencies of the following data, given that A.M. is $67.45$ inches. \begin{tabular}{|l|c|c|c|c|c|c|} \hline Height (inches) & $60-62$ & $63-65$ & $66-68$ & $69-71$ & $72-74$ & Total \\ \hline No. of Students & 5 & 18 & $f_{3}$ & $f_{4}$ & 8 & 100 \\ \hline \end{tabular}

Find out the missing frequencies of the following data, given that A.M. is $67.45$ inches. \begin{tabular}{|l|c|c|c|c|c|c|}
\hline Height (inches) & $60-62$ & $63-65$ & $66-68$ & $69-71$ & $72-74$ & Total \\
\hline No. of Students & 5 & 18 & $f_{3}$ & $f_{4}$ & 8 & 100 \\
\hline
\end{tabular}

Key Concepts

Total Frequency Constraint

The total frequency is the sum of all class frequencies in the distribution. This constraint is essential for ensuring consistency in the data and serves as one of the foundational equations when solving for unknown frequencies.

Grouped Frequency Distribution

A grouped frequency distribution organizes data into classes or intervals and records the frequency of observations within each class. This format helps in summarizing large datasets and facilitates analysis of the distribution's shape and central tendency.

Class Midpoints

Class midpoints are calculated by taking the average of the lower and upper boundaries of an interval. They serve as a representative value for the data in that interval, particularly when the exact data values are not known.

Arithmetic Mean for Grouped Data

When data is provided in grouped form, the arithmetic mean is estimated by multiplying each class midpoint by its corresponding frequency, summing these products, and dividing by the total number of observations. This method offers a way to approximate the central tendency of the data.

Setting Up Equations with Missing Frequencies

In situations where some class frequencies are unknown, equations can be formulated using known constraints such as the total sum of frequencies and the given arithmetic mean. By incorporating the class midpoints and known frequencies, these equations enable the determination of the missing values.

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