Understanding Frequency Distribution and Cumulative Frequency

A frequency distribution table organizes data by showing the frequency (count) of each distinct value or category. The cumulative frequency for a particular value is the sum of the frequencies of all values up to and including that value.

In the given table, we have the values of \(x\), their corresponding frequencies, and their cumulative frequencies. The cumulative frequency for a given row is obtained by adding the frequency of that row to the cumulative frequency of the previous row.

Calculating Missing Frequency \(f_1\)

The cumulative frequency for \(x=3\) is given as 29. According to the definition, the cumulative frequency for \(x=3\) is the sum of the frequencies for \(x=1\), \(x=2\), and \(x=3\).

Cumulative Frequency at \(x=3\) = Frequency at \(x=1\) + Frequency at \(x=2\) + Frequency at \(x=3\)

We are given:

• Frequency at \(x=1\) = 8
• Frequency at \(x=2\) = 10
• Frequency at \(x=3\) = \(f_1\)
• Cumulative Frequency at \(x=2\) = 18 (which is 8 + 10)
• Cumulative Frequency at \(x=3\) = 29

Using the relationship between cumulative frequency and frequency:

Cumulative Frequency at \(x=3\) = Cumulative Frequency at \(x=2\) + Frequency at \(x=3\)

Substituting the given values:

\(29 = 18 + f_1\)

To find \(f_1\), we subtract 18 from 29:

\(f_1 = 29 - 18\)

\(f_1 = 11\)

So, the missing frequency \(f_1\) is 11.

Calculating Missing Frequency \(f_2\)

Now that we know \(f_1 = 11\), we can find \(f_2\). The cumulative frequency for \(x=4\) is given as 45. This is the sum of frequencies up to \(x=4\).

Cumulative Frequency at \(x=4\) = Frequency at \(x=1\) + Frequency at \(x=2\) + Frequency at \(x=3\) + Frequency at \(x=4\)

We know the sum of frequencies up to \(x=3\) is the Cumulative Frequency at \(x=3\), which is 29.

So, we can write:

Cumulative Frequency at \(x=4\) = Cumulative Frequency at \(x=3\) + Frequency at \(x=4\)

Substituting the values:

• Cumulative Frequency at \(x=4\) = 45
• Cumulative Frequency at \(x=3\) = 29
• Frequency at \(x=4\) = \(f_2\)

So the equation is:

\(45 = 29 + f_2\)

To find \(f_2\), we subtract 29 from 45:

\(f_2 = 45 - 29\)

\(f_2 = 16\)

So, the missing frequency \(f_2\) is 16.

Final Values of f1 and f2

Based on the calculations, the value of \(f_1\) is 11 and the value of \(f_2\) is 16.

Therefore, the values of \(f_1\) and \(f_2\) are 11 and 16 respectively.

Revision Table: Frequency Distribution Concepts

Additional Information: Applications of Frequency Distribution

Frequency distributions are fundamental in statistics and data analysis. They help in summarizing datasets and understanding the pattern of data. Some applications include:

• Easily visualizing data patterns using histograms or frequency polygons.
• Calculating measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation).
• Identifying the most common values or ranges in the data.
• Comparing different datasets.

Understanding how frequency and cumulative frequency relate is key to interpreting statistical data presented in tabular form.