Finding Mean, Median, and Mode

To find the set of numbers where the mean, median, and mode are all equal, we need to calculate these three measures of central tendency for each given set of numbers.

Understanding the Concepts

• Mean: The average of the numbers. It is calculated by summing all the numbers in the set and dividing by the count of numbers in the set.
• Median: The middle value in a set of numbers that are arranged in ascending or descending order. If the set has an odd number of elements, the median is the single middle value. If the set has an even number of elements, the median is the average of the two middle values.
• Mode: The number that appears most frequently in the set. A set can have one mode (unimodal), multiple modes (multimodal), or no mode if all numbers appear with the same frequency.

Analyzing Each Set of Numbers

Set 1: 12, 12, 12, 12, 24

• Numbers in ascending order: 12, 12, 12, 12, 24
• Mean: \( \text{Mean} = \frac{12 + 12 + 12 + 12 + 24}{5} = \frac{72}{5} = 14.4 \)
• Median: The middle value in the sorted list (5 numbers) is the 3rd number. Median = 12.
• Mode: The number 12 appears 4 times, which is more frequent than any other number. Mode = 12.

For this set, Mean (14.4), Median (12), and Mode (12) are not all equal.

Set 2: 6, 18, 18, 18, 30

• Numbers in ascending order: 6, 18, 18, 18, 30
• Mean: \( \text{Mean} = \frac{6 + 18 + 18 + 18 + 30}{5} = \frac{90}{5} = 18 \)
• Median: The middle value in the sorted list (5 numbers) is the 3rd number. Median = 18.
• Mode: The number 18 appears 3 times, which is more frequent than any other number. Mode = 18.

For this set, Mean (18), Median (18), and Mode (18) are all equal.

Set 3: 6, 6, 12, 30, 36

• Numbers in ascending order: 6, 6, 12, 30, 36
• Mean: \( \text{Mean} = \frac{6 + 6 + 12 + 30 + 36}{5} = \frac{90}{5} = 18 \)
• Median: The middle value in the sorted list (5 numbers) is the 3rd number. Median = 12.
• Mode: The number 6 appears 2 times, which is more frequent than any other number. Mode = 6.

For this set, Mean (18), Median (12), and Mode (6) are not all equal.

Set 4: 6, 6, 6, 12, 30

• Numbers in ascending order: 6, 6, 6, 12, 30
• Mean: \( \text{Mean} = \frac{6 + 6 + 6 + 12 + 30}{5} = \frac{60}{5} = 12 \)
• Median: The middle value in the sorted list (5 numbers) is the 3rd number. Median = 6.
• Mode: The number 6 appears 3 times, which is more frequent than any other number. Mode = 6.

For this set, Mean (12), Median (6), and Mode (6) are not all equal.

Conclusion

Comparing the results for all sets, the set where the mean, median, and mode have the same value is the set {6, 18, 18, 18, 30}, where all three measures are equal to 18.

Revision Table: Mean, Median, Mode Calculation

Additional Information: Measures of Central Tendency

Mean, median, and mode are fundamental measures used in statistics to find the center or typical value of a data set.

• The mean is sensitive to extreme values (outliers).
• The median is less affected by extreme values and is a good measure for skewed distributions.
• The mode is useful for categorical data or finding the most popular item.

In a perfectly symmetrical distribution, like a normal distribution, the mean, median, and mode are all located at the same point. However, in skewed distributions, they will typically be different.