For which of the following sets of numbers do the mean, median and mode have the same value?

Options

A. 12, 12, 12, 12, 24
B. 6, 18, 18, 18, 30
C. 6, 6, 12, 30, 36
D. 6, 6, 6, 12, 30

Answer

To determine which set of numbers has the same mean, median, and mode, we will analyze each option step by step.

### Step-by-Step Solution:

**Option 1: {12, 12, 12, 12, 24}**

1. **Calculate the Mean:**
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{12 + 12 + 12 + 12 + 24}{5} = \frac{72}{5} = 14.4 \]

2. **Calculate the Median:**
- The numbers in ascending order: {12, 12, 12, 12, 24}
- The median is the middle number (3rd number in this case):
\[ \text{Median} = 12 \]

3. **Calculate the Mode:**
- The mode is the number that appears most frequently:
\[ \text{Mode} = 12 \]

4. **Comparison:**
- Mean = 14.4, Median = 12, Mode = 12
- Not all values are the same, so **Option 1 is rejected**.

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**Option 2: {6, 18, 18, 18, 30}**

1. **Calculate the Mean:**
\[ \text{Mean} = \frac{6 + 18 + 18 + 18 + 30}{5} = \frac{90}{5} = 18 \]

2. **Calculate the Median:**
- The numbers in ascending order: {6, 18, 18, 18, 30}
- The median is the middle number (3rd number in this case):
\[ \text{Median} = 18 \]

3. **Calculate the Mode:**
- The mode is the number that appears most frequently:
\[ \text{Mode} = 18 \]

4. **Comparison:**
- Mean = 18, Median = 18, Mode = 18
- All values are the same, so **Option 2 is the correct answer**.

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**Option 3: {5, 10, 15, 20, 25}**

1. **Calculate the Mean:**
\[ \text{Mean} = \frac{5 + 10 + 15 + 20 + 25}{5} = \frac{75}{5} = 15 \]

2. **Calculate the Median:**
- The numbers in ascending order: {5, 10, 15, 20, 25}
- The median is the middle number (3rd number):
\[ \text{Median} = 15 \]

3. **Calculate the Mode:**
- All numbers appear only once, so there is no mode:
\[ \text{Mode} = \text{None} \]

4. **Comparison:**
- Mean = 15, Median = 15, Mode = None
- Not all values are the same, so **Option 3 is rejected**.

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**Option 4: {1, 2, 2, 3, 4}**

1. **Calculate the Mean:**
\[ \text{Mean} = \frac{1 + 2 + 2 + 3 + 4}{5} = \frac{12}{5} = 2.4 \]

2. **Calculate the Median:**
- The numbers in ascending order: {1, 2, 2, 3, 4}
- The median is the middle number (3rd number):
\[ \text{Median} = 2 \]

3. **Calculate the Mode:**
- The mode is the number that appears most frequently:
\[ \text{Mode} = 2 \]

4. **Comparison:**
- Mean = 2.4, Median = 2, Mode = 2
- Not all values are the same, so **Option 4 is rejected**.

### Conclusion:
The only option where the mean, median, and mode are the same is **Option 2**: {6, 18, 18, 18, 30}.

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