Understanding Mean, Median, and Mode Equality in Number Sets

This question asks us to identify a specific set of numbers from the given options where the three main measures of central tendency – the mean, median, and mode – are all identical.

Defining Key Statistical Terms

Let's first clarify what each term means:

• Mean: This is the average of all the numbers in a set. It's calculated by summing up all the numbers and then dividing by the count of numbers in the set. The formula is represented as:

  $$ \text{Mean} = \frac{\sum x_i}{n} $$

  where $ \sum x_i $ represents the sum of all numbers in the set, and $ n $ is the total count of numbers.
• Median: This is the middle value in a list of numbers that has been arranged in ascending order. If there's an even number of values, the median is the average of the two middle numbers.
• Mode: This is the number that appears most frequently in the data set. A set can have one mode, more than one mode, or no mode at all.

Analyzing Each Number Set

We need to examine each option provided to see if the mean, median, and mode are the same.

Option 1: {2, 2, 2, 2, 4}

First, let's arrange the numbers in ascending order: 2, 2, 2, 2, 4.

In this set, the mean (2.4) is not equal to the median (2) or the mode (2).

Option 2: {1, 3, 3, 3, 5}

Arranging the numbers in ascending order: 1, 3, 3, 3, 5.

Here, the mean is 3, the median is 3, and the mode is 3. All three values are the same.

Option 3: {1, 1, 2, 5, 6}

The numbers are already in ascending order: 1, 1, 2, 5, 6.

In this set, the mean (3) is not equal to the median (2) or the mode (1).

Option 4: {1, 1, 1, 2, 5}

The numbers are already in ascending order: 1, 1, 1, 2, 5.

In this set, the mean (2) is not equal to the median (1) or the mode (1).

Conclusion

By analyzing all the options, we found that only the set {1, 3, 3, 3, 5} has the mean, median, and mode equal to the same value, which is 3.