Understanding the Median and Data Changes

The question asks how the median of a set of 11 distinct observations changes if the largest five observations are increased by 3. Let's first understand what the median is.

The median is the middle value in a dataset that is ordered from least to greatest. When you have an odd number of observations, the median is the single middle value. For a set of 'n' observations, the position of the median is given by the formula:

\(\text{Median Position} = \frac{n+1}{2}\)

Calculating the Median Position

In this problem, we have 11 distinct observations. Let the observations be \(x_1, x_2, \dots, x_{11}\) sorted in ascending order:

\(x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10} < x_{11}\)

The number of observations, \(n\), is 11. Using the formula for the median position:

\(\text{Median Position} = \frac{11+1}{2} = \frac{12}{2} = 6\)

So, the median of the original set is the 6th observation in the sorted list, which is \(x_6\). We are given that the original median is 73.2, so \(x_6 = 73.2\).

Analyzing the Change in Observations

The problem states that "each of the largest five observations of the set is increased by 3". In our sorted list, the largest five observations are the last five values.

The observations are \(x_1, x_2, x_3, x_4, x_5, \textbf{x_6}, x_7, x_8, x_9, x_{10}, x_{11}\).

The largest five observations are \(x_7, x_8, x_9, x_{10}, x_{11}\). These are the 7th, 8th, 9th, 10th, and 11th observations.

These five observations are increased by 3. The new values will be \(x_7+3, x_8+3, x_9+3, x_{10}+3, x_{11}+3\).

Determining the New Median

The original sorted list is:

\(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}, x_{11}\)

The new list, after increasing the largest five observations by 3, is:

\(x_1, x_2, x_3, x_4, x_5, x_6, x_7+3, x_8+3, x_9+3, x_{10}+3, x_{11}+3\)

Since the original observations were distinct and sorted (\(x_1 < \dots < x_6 < \dots < x_{11}\)), increasing the largest five observations by the same positive value (3) will maintain their relative order and also ensure they remain larger than the 6th observation, \(x_6\).

For example, since \(x_6 < x_7\), it follows that \(x_6 < x_7+3\). This applies to all observations from \(x_7\) to \(x_{11}\).

The new sorted list is still:

\(x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7+3 < x_8+3 < x_9+3 < x_{10}+3 < x_{11}+3\)

There are still 11 observations in the new set. The median position is still the 6th observation.

The 6th observation in the new set is \(x_6\). This observation was not among the largest five observations that were increased. Therefore, its value remains unchanged.

The median of the new set is \(x_6\), which is equal to the original median, 73.2.

Conclusion on Median Change

Increasing values in the upper half of a sorted dataset (specifically, values greater than the median) does not change the median itself, as long as the relative order of the elements is preserved and the median value itself is not affected by the change. In this case, the median value is the 6th observation, and only the 7th through 11th observations were changed. Thus, the median remains the same.

Let's look at the options:

• Option 1: is 3 times that of the original set. This is incorrect, as the median value did not change.
• Option 2: is increased by 3. This would happen if the median observation itself was increased by 3, or if the change caused a different observation to become the median which was 3 greater than the original median. Neither is the case here.
• Option 3: remains the same as that of the original set. This matches our conclusion.
• Option 4: is decreased by 3. This is incorrect, as the median value did not change and values were increased, not decreased.

Therefore, the median of the new set remains the same as that of the original set.

Revision Table: Understanding Median Changes

Additional Information: Properties of the Median

The median is a measure of central tendency that is less affected by extreme values (outliers) compared to the mean. Here's why:

• The median only considers the position of the values when the data is ordered.
• Its value is determined by one or two specific values in the middle of the sorted dataset.
• Adding or removing extreme values usually shifts the median position slightly but doesn't drastically change the median value unless the extreme values are very numerous or the dataset is small.
• In this problem, increasing the largest five observations are essentially making the upper values "more extreme", but since these changes occur above the median position, the median value itself, which is the 6th value, remains unchanged.

Contrast this with the mean (average), which is calculated using all values. If you increased the largest five observations by 3, the sum of all observations would increase by \(5 \times 3 = 15\), and the mean would definitely increase.