Finding the New Median After Data Changes

The question asks us to find the median of a dataset after two specific observations have been changed. We are given the original set of observations, its median, and the changes made to two of the observations. The median is the middle value in a dataset that is ordered from least to greatest.

To find the median, the first step is always to arrange the data in ascending order.

Original Observations and Median Calculation

The original observations are:

46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33

Let's count the number of observations. There are 11 observations. For an odd number of observations (n), the median is the value at the \( \frac{n+1}{2} \) position after sorting.

Arranging the original observations in ascending order:

33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92

The number of observations is 11. The median position is \( \frac{11+1}{2} = \frac{12}{2} = 6 \).

The 6th observation in the sorted list is 58. This matches the given median in the question.

New Observations After Replacement

Now, two observations are replaced in the original data:

• 92 is replaced by 99.
• 41 is replaced by 43.

The new set of observations is formed by taking the original list and making these replacements:

Original: 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33

After replacing 92 with 99 and 41 with 43:

46, 64, 87, 43, 58, 77, 35, 90, 55, 99, 33

The number of observations is still 11.

Calculating the New Median

To find the new median, we must arrange the new observations in ascending order:

33, 35, 43, 46, 55, 58, 64, 77, 87, 90, 99

Again, with 11 observations, the median is the value at the 6th position.

Let's find the 6th observation in this new sorted list:

33 (1st), 35 (2nd), 43 (3rd), 46 (4th), 55 (5th), 58 (6th), 64 (7th), 77 (8th), 87 (9th), 90 (10th), 99 (11th)

The 6th observation in the new sorted list is 58.

Even though two observations were changed (41 to 43 and 92 to 99), these changes did not affect the value at the 6th position in the sorted list. Both 41 and 43 are smaller than 58 and come before it in the sorted list. Both 92 and 99 are larger than 58 and come after it in the sorted list. The middle value remained 58.

Therefore, the new median is 58.

Revision Table: Key Concepts

Additional Information: Median Properties and Calculations

The median is a measure of central tendency. It is often preferred over the mean when the data contains outliers because it is less affected by extremely large or small values.

Here's a quick summary of finding the median:

1. Arrange the data in ascending order.
2. Count the number of observations, n.
3. If n is odd, the median is the value at the position \( \frac{n+1}{2} \).
4. If n is even, the median is the average of the values at positions \( \frac{n}{2} \) and \( \frac{n}{2}+1 \).

In this problem, replacing 41 with 43 (a small change that keeps it below the median) and 92 with 99 (a larger change that keeps it above the median) did not shift the middle position's value, resulting in the same median.