Understanding the Student Marks Data

We are given a set of marks obtained by 20 students in mathematics out of 30. This is our dataset.

The marks are: 7, 9, 12, 12, 13, 12, 14, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 18, 20, 13.

We need to find the value of \( (2 \times \text{median} - \text{mode}) \) for this data. To do this, we first need to calculate the mode and the median of the dataset.

Calculating the Mode of the Marks Data

The mode of a dataset is the value that appears most frequently. To find the mode, we can count how many times each mark appears in the given data.

• Mark 7: appears 1 time
• Mark 9: appears 1 time
• Mark 12: appears 3 times
• Mark 13: appears 2 times
• Mark 14: appears 4 times
• Mark 15: appears 1 time
• Mark 16: appears 1 time
• Mark 17: appears 1 time
• Mark 18: appears 3 times
• Mark 19: appears 1 time
• Mark 20: appears 2 times

Comparing the frequencies, the mark '14' has the highest frequency, appearing 4 times.

Therefore, the mode of the data is 14.

Calculating the Median of the Marks Data

The median is the middle value in a dataset when the data is arranged in ascending or descending order. It represents the central value of the dataset.

First, let's arrange the given marks in ascending order:

7, 9, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 16, 17, 18, 18, 18, 19, 20, 20

There are a total of 20 data points (\(n=20\)) in this dataset. Since the number of data points is even, the median is calculated as the average of the two middle values.

The positions of the two middle values are given by \( \left(\frac{n}{2}\right)\text{th} \) and \( \left(\frac{n}{2} + 1\right)\text{th} \) terms.

\( \frac{n}{2} = \frac{20}{2} = 10 \)

\( \frac{n}{2} + 1 = \frac{20}{2} + 1 = 10 + 1 = 11 \)

So, we need to find the 10th and 11th values in our sorted list:

• The 10th value in the sorted list is 14.
• The 11th value in the sorted list is 14.

Now, we calculate the median by taking the average of these two values:

\( \text{Median} = \frac{\text{10th value} + \text{11th value}}{2} \)

\( \text{Median} = \frac{14 + 14}{2} = \frac{28}{2} = 14 \)

Therefore, the median of the data is 14.

Final Calculation: \( (2 \times \text{median} - \text{mode}) \)

We are asked to find the value of the expression \( (2 \times \text{median} - \text{mode}) \).

We have calculated the median to be 14 and the mode to be 14.

Substitute these values into the expression:

\( 2 \times \text{median} - \text{mode} = 2 \times 14 - 14 \)

First, perform the multiplication:

\( 2 \times 14 = 28 \)

Now, perform the subtraction:

\( 28 - 14 = 14 \)

Thus, the value of \( (2 \times \text{median} - \text{mode}) \) for the given data is 14.

Conclusion

Based on our calculations, the mode of the student marks data is 14 and the median is also 14. Using these values, we found that \( (2 \times \text{median} - \text{mode}) = (2 \times 14 - 14) = 28 - 14 = 14 \).

The value 14 corresponds to option 2.

Revision Table: Key Statistics Concepts

Additional Information: Measures of Central Tendency Explained

Measures of central tendency are single values that attempt to describe a set of data by identifying the central position within that set of data. Mean, median, and mode are the most common measures.

• The mean is what most people call the "average". It's useful but can be heavily influenced by outliers (extremely high or low values).
• The median is the middle value, dividing the dataset into two equal halves. It is less affected by outliers and is a good measure for skewed distributions.
• The mode tells us which value is the most common. It is particularly useful for categorical data where calculating a mean or median is not possible. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode.

These measures provide different perspectives on the 'center' of the data and are used depending on the nature of the data and the question being asked.