Question

Short Response Add two numbers to the data below so that the median does not change. (Statistics Review) \(11,13,16,12,25,8,25,33,51\)

Short answer

Add numbers 7 and 10; the median remains 16.

Step-by-step solution

Identify the Current Median

To find the median, we first need to order the numbers from smallest to largest. The ordered list is \(8, 11, 12, 13, 16, 25, 25, 33, 51\). Since there are 9 numbers, the median will be the 5th number in this list. Thus, the median is \(16\).

Determine Positions for New Numbers

To ensure the median remains \(16\) after adding two new numbers, both numbers must be positioned such that the central number (now the 6th position in an 11-element list) remains unchanged.

Add Numbers Below the Median

We can add numbers that are less than or equal to \(11\) to maintain the position of the current 5th number in the list as the median. Let's add \(7\) and \(10\) as our two numbers.

Verify the Solution

Reorder the list with the two new numbers: \(7, 8, 10, 11, 12, 13, 16, 25, 25, 33, 51\). The list still has \(16\) as the median because it is the 6th number in the 11-element ordered list.

Key concepts

Understanding Statistics

Statistics is an exciting field that helps us make sense of data. At its core, statistics involves collecting, organizing, analyzing, and interpreting numerical data. In many cases, it provides insights or highlights a trend that can be vital for decision-making. Let’s consider some fundamental elements that you might come across:

• Data Collection: Gathering information is the first crucial step.
• Organization: Structuring data to derive meaningful insights.
• Analysis: Using different methods to extract information.
• Interpretation: Making sense of the results for practical conclusions.

In this context, understanding the median helps us grasp how numbers can tell a story about a dataset's central tendency.

The Importance of an Ordered List

An ordered list is simply a sequence of numbers sorted from the smallest to the largest. It is crucial when finding the median, as an ordered list allows you to identify the middle number clearly.
To transform a list into its ordered form, you compare each pair of numbers and rearrange them accordingly. For example, let’s look at how you can order the numbers from the original problem:

• List before ordering: 11, 13, 16, 12, 25, 8, 25, 33, 51
• Ordered list: 8, 11, 12, 13, 16, 25, 25, 33, 51

Having an ordered list ensures accuracy in identifying measures like the median, and it makes further statistical analysis straightforward.

Adding Numbers While Maintaining Balance

Adding numbers to a dataset is a careful process, especially if you want to maintain a balance, like keeping the median unchanged. The goal is to ensure any additions do not shift the central point of the data too much.
When adding two numbers to the list, as in the exercise, you should think about where these numbers will fit:

• Insert numbers below or equal to the lower end of the median value until you reach strategic positions.
• This keeps the original median in the balanced center of the newly extended list.

In our example, adding 7 and 10 to the initial sequence was a strategic decision since these values appear before the existing median of 16 when reordered.

Understanding the Median Position

The median is a vital statistical measure, representing the middle value in an ordered list. If the list has an odd number of elements, the median is simply the middle number. For even numbers of elements, the median is the average of the two central numbers.
In our problem, the ordered list initially had 9 numbers, so the median was the 5th element. When planning to add two new numbers:

• The list extends to 11 elements.
• The new median becomes the 6th position in the ordered list.

Ensuring that the new additions don't alter the central (median) position is key to keeping statistical consistency. Therefore, knowing this position allows for strategic decisions when expanding or modifying data sets.