Finding Mode for Grouped Frequency Distribution

The question asks us to find the mode for a given grouped frequency distribution. The mode is the value that appears most frequently in a data set. For grouped data, the mode is found within the class interval that has the highest frequency, known as the modal class.

Identifying the Modal Class

Let's look at the provided distribution table:

We need to find the class interval with the highest frequency. Looking at the 'Frequency' column, the highest frequency is 11, which corresponds to the class interval 25-30.

So, the modal class is 25-30.

Mode Calculation Formula for Grouped Data

The formula to calculate the mode for grouped data is:

$\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$

Where:

• $l$ = lower limit of the modal class
• $f_1$ = frequency of the modal class
• $f_0$ = frequency of the class preceding the modal class
• $f_2$ = frequency of the class succeeding the modal class
• $h$ = class size (width of the class interval)

Extracting Values from the Distribution

From our identified modal class (25-30) and the table, we can extract the necessary values:

• Modal class = 25-30
• Lower limit of modal class, $l = 25$
• Frequency of modal class, $f_1 = 11$
• Class preceding the modal class is 20-25. Its frequency, $f_0 = 9$
• Class succeeding the modal class is 30-35. Its frequency, $f_2 = 10$
• Class size, $h = 10 - 5 = 5$ (The difference between the upper and lower limit of any class, assuming uniform class size).

Step-by-Step Mode Calculation

Now, substitute these values into the mode formula:

$\text{Mode} = 25 + \frac{11 - 9}{2(11) - 9 - 10} \times 5$

First, calculate the differences and the denominator:

• $f_1 - f_0 = 11 - 9 = 2$
• $2f_1 = 2 \times 11 = 22$
• $f_0 + f_2 = 9 + 10 = 19$
• $2f_1 - f_0 - f_2 = 22 - 9 - 10 = 22 - 19 = 3$

Now substitute these back into the formula:

$\text{Mode} = 25 + \frac{2}{3} \times 5$

$\text{Mode} = 25 + \frac{10}{3}$

Calculate the fraction $\frac{10}{3}$:

$\frac{10}{3} \approx 3.3333...$

Finally, add this to the lower limit:

$\text{Mode} = 25 + 3.3333...$

$\text{Mode} \approx 28.3333...$

Rounding the Result

The question asks for the mode rounded off to two decimal places. Rounding 28.3333... to two decimal places gives 28.33.

Therefore, the mode for the given distribution is approximately 28.33.

Revision Table: Key Concepts

Additional Information on Measures of Central Tendency

Mode is one of the three main measures of central tendency used in statistics, along with Mean and Median. Each measure provides a different way to describe the "center" or "typical" value of a dataset.

• Mean: The average value, calculated by summing all values and dividing by the number of values. For grouped data, a specific formula using class marks and frequencies is used. The mean is affected by extreme values.
• Median: The middle value in a dataset that is ordered from least to greatest. For grouped data, it is calculated using a formula that involves the median class (the class containing the cumulative frequency midpoint). The median is less affected by extreme values than the mean.
• Mode: The most frequent value. It is the only measure of central tendency that can be used for qualitative data (like colors or types). For grouped data, it is calculated using the formula demonstrated above. The mode is useful for identifying the most popular category or score.

Choosing which measure of central tendency to use depends on the type of data and what aspect of the data's center you want to highlight. For example, if you want to know the most common salary in a company, you would look at the mode. If you want to know the average salary, you would calculate the mean. If you want to know the middle salary after ranking them, you would find the median.