Finding the Mode for Grouped Frequency Data

The mode is the value that appears most frequently in a data set. For grouped data, we cannot find the exact mode value because the individual values within each class are unknown. Instead, we estimate the mode using a formula based on the frequencies of the classes.

Understanding the Data Set

The provided data is a frequency distribution showing the number of observations (frequency) falling within specified class limits.

Identifying the Modal Class

The modal class is the class interval with the highest frequency. Looking at the frequency column in the table:

• The highest frequency is 40.
• This frequency corresponds to the class limit 401-500.

Therefore, the modal class is 401-500.

Formula for Estimating Mode in Grouped Data

The formula used to estimate the mode for grouped frequency distribution is:

\(\text{Mode} = \text{L} + \left(\frac{\text{f}_1 - \text{f}_0}{2\text{f}_1 - \text{f}_0 - \text{f}_2}\right) \times \text{h}\)

Where:

• L: The lower limit of the modal class.
• f\(_1\): The frequency of the modal class.
• f\(_0\): The frequency of the class preceding the modal class.
• f\(_2\): The frequency of the class succeeding the modal class.
• h: The class width of the modal class.

Extracting Values from the Data

From the identified modal class (401-500) and the frequency table, we can extract the necessary values:

• Modal class: 401-500
• L (Lower limit of the modal class): 401
• f\(_1\) (Frequency of the modal class): 40
• f\(_0\) (Frequency of the preceding class): The class before 401-500 is 301-400, with frequency 24. So, f\(_0\) = 24.
• f\(_2\) (Frequency of the succeeding class): The class after 401-500 is 501-600, with frequency 16. So, f\(_2\) = 16.
• h (Class width): For the class limit 401-500, the width is the difference between the upper limit and lower limit plus one (500 - 401 + 1), which is 100. Alternatively, the difference between consecutive lower limits (501 - 401) or upper limits (500 - 400) is 100. So, h = 100.

Calculating the Mode Value

Now, substitute these values into the mode formula:

\(\text{Mode} = \text{L} + \left(\frac{\text{f}_1 - \text{f}_0}{2\text{f}_1 - \text{f}_0 - \text{f}_2}\right) \times \text{h}\)

\(\text{Mode} = 401 + \left(\frac{40 - 24}{2 \times 40 - 24 - 16}\right) \times 100\)

\(\text{Mode} = 401 + \left(\frac{16}{80 - 24 - 16}\right) \times 100\)

\(\text{Mode} = 401 + \left(\frac{16}{80 - 40}\right) \times 100\)

\(\text{Mode} = 401 + \left(\frac{16}{40}\right) \times 100\)

\(\text{Mode} = 401 + (0.4) \times 100\)

\(\text{Mode} = 401 + 40\)

\(\text{Mode} = 441\)

The calculated mode value for the given data set is 441.

Conclusion

Based on the calculations using the mode formula for grouped data, the estimated mode value is 441. This value represents the peak frequency within the distribution.

Revision Table: Key Concepts for Mode Calculation

Review the main steps and components involved in calculating the mode for grouped data.

• Identify the class with the highest frequency (Modal Class).
• Determine the lower limit (L), modal class frequency (f\(_1\)), preceding class frequency (f\(_0\)), succeeding class frequency (f\(_2\)), and class width (h).
• Apply the mode formula: \(\text{Mode} = \text{L} + \left(\frac{\text{f}_1 - \text{f}_0}{2\text{f}_1 - \text{f}_0 - \text{f}_2}\right) \times \text{h}\).
• Calculate the result step-by-step.

Additional Information on Measures of Central Tendency

The mode is one of the three main measures of central tendency, along with the mean and the median. Each measure provides a different perspective on the 'center' of a data set.

• Mean: The average value (sum of all values divided by the number of values). For grouped data, it's estimated using class midpoints.
• Median: The middle value when the data is arranged in order. For grouped data, it's estimated using a formula involving cumulative frequency.
• Mode: The most frequent value. Useful for identifying the most common category or observation. For grouped data, it indicates the class interval with the highest density of data points.

The choice of which measure of central tendency to use depends on the type of data and the shape of the distribution.