In general, for samples with an odd number of observations, there will always be values both smaller and larger than the median.

You claim it isn't possible. I claim I can come up with a simple counter example to what you just stated. Try to come up with a simple counter example to your statement on your own.

Chat GPT4:
You are correct to challenge my initial response. Upon reconsideration, there can indeed be a counterexample to my initial claim. Let's consider a simple set of numbers where not all are identical, yet there is no number smaller than the median:

Suppose we have a set of numbers: [4, 4, 4, 5]

In this set, the numbers are not all identical (there's a 5 which is different from the 4s).
The median in this case, for an even number of observations, is the average of the two middle numbers. Here, the two middle numbers are both 4, so the median is 4.
In this set, there is no number smaller than 4.
Hence, this serves as a counterexample where not all numbers are identical, yet there is no number smaller than the median.

Let's reexamine the question and my response to identify where the misunderstanding may have occurred.

Revisiting the Question
The question asks if it's possible, in a group of numbers where not all numbers are identical, for there to be no number smaller than the median.

Analyzing the Initial Explanation
In my previous response, I stated that the median divides the numbers into two halves, where at least half of the numbers are equal to or less than the median. This explanation assumes a standard scenario where the numbers are distinct.

Identifying the Mistake
Upon further reflection, the key lies in considering the presence of multiple identical numbers in the dataset. If the set contains multiple instances of the same number, and this number happens to be the median, it's possible that no number is smaller than the median. For example, in a dataset like [4, 4, 4, 4, 5, 6, 7], the median is 4, but there is no number smaller than 4 in this dataset.

Corrected Solution By Steps

Step 1: Reconsidering the Definition of Median with Identical Numbers
In a dataset with identical numbers, the median is still the middle value when the dataset is ordered. If the median happens to be the most frequently occurring number (mode), it is possible that all numbers in one half of the dataset are equal to the median.

Step 2: Applying this to the Given Scenario
In a scenario where the dataset consists mostly or entirely of the same number, which is also the median, there can indeed be no number smaller than the median. This is a special case that occurs when the median is equal to the smallest number in the dataset, especially in datasets with high repetition of a single number.

Final Answer
Yes, it is possible in a special case for there to be no number smaller than the median in a group of numbers where not all are identical, particularly when the dataset contains multiple instances of the median value, and this value is equal to the smallest number in the set.

Source: Conversation with Bing, 25/01/2024
(1) [1706.03762] Attention Is All You Need - arXiv.org. https://arxiv.org/abs/1706.03762.
(2) Attention Is All You Need - arXiv.org. https://arxiv.org/pdf/1706.03762v5.pdf.
(3) Understanding Google’s “Attention Is All You Need” Paper and Its .... https://alok-shankar.medium.com/understanding-googles...
(4) Transformer (machine learning model) - Wikipedia. Transformer_%28machine_learning_model%29. [Wikipedia]
(5) undefined. https://doi.org/10.48550/arXiv.1706.03762.
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‏¹ To Zummarize?