Some measure central tendency for a discrete observations are given below
I. Arithmetic mean
II Geometric mean
III Harmonic mean
IV. Median
A desirable property of a measure of central tendency is if every observation is multiplied by c. then the measure of central tendency is abo multiplied by c. where c<0. Which of the above measures satisfy the property

Options

A. I, II and III
B. I, II and IV
C. III and IV
D. I, II, III and IV

Answer

To determine which measures of central tendency satisfy the property that if every observation is multiplied by a constant \( c \) (where \( c < 0 \)), then the measure of central tendency is also multiplied by \( c \), we will analyze each measure step by step.

### Step 1: Arithmetic Mean
1. **Definition**: The arithmetic mean of two observations \( a \) and \( b \) is given by:
\[ \text{Arithmetic Mean} = \frac{a + b}{2} \]
2. **Multiplying by \( c \)**: If we multiply both observations by \( c \), we get:
\[ \text{New Observations} = ca \quad \text{and} \quad cb \]
3. **New Arithmetic Mean**:
\[ \text{New Arithmetic Mean} = \frac{ca + cb}{2} = c \cdot \frac{a + b}{2} \]
4. **Conclusion**: The arithmetic mean is multiplied by \( c \). Thus, it satisfies the property.

### Step 2: Geometric Mean
1. **Definition**: The geometric mean of two observations \( a \) and \( b \) is given by:
\[ \text{Geometric Mean} = \sqrt{ab} \]
2. **Multiplying by \( c \)**: If we multiply both observations by \( c \), we get:
\[ \text{New Observations} = ca \quad \text{and} \quad cb \]
3. **New Geometric Mean**:
\[ \text{New Geometric Mean} = \sqrt{(ca)(cb)} = \sqrt{c^2 ab} = |c| \sqrt{ab} \]
Since \( c < 0 \), we have:
\[ \text{New Geometric Mean} = c \sqrt{ab} \]
4. **Conclusion**: The geometric mean is multiplied by \( c \). Thus, it satisfies the property.

### Step 3: Harmonic Mean
1. **Definition**: The harmonic mean of two observations \( a \) and \( b \) is given by:
\[ \text{Harmonic Mean} = \frac{2ab}{a + b} \]
2. **Multiplying by \( c \)**: If we multiply both observations by \( c \), we get:
\[ \text{New Observations} = ca \quad \text{and} \quad cb \]
3. **New Harmonic Mean**:
\[ \text{New Harmonic Mean} = \frac{2(ca)(cb)}{ca + cb} = \frac{2c^2ab}{c(a + b)} = c \cdot \frac{2ab}{a + b} \]
4. **Conclusion**: The harmonic mean is multiplied by \( c \). Thus, it satisfies the property.

### Step 4: Median
1. **Definition**: The median of two observations \( a \) and \( b \) is given by:
\[ \text{Median} = \frac{a + b}{2} \]
2. **Multiplying by \( c \)**: If we multiply both observations by \( c \), we get:
\[ \text{New Observations} = ca \quad \text{and} \quad cb \]
3. **New Median**:
\[ \text{New Median} = \frac{ca + cb}{2} = c \cdot \frac{a + b}{2} \]
4. **Conclusion**: The median is multiplied by \( c \). Thus, it satisfies the property.

### Final Conclusion
All four measures of central tendency (Arithmetic Mean, Geometric Mean, Harmonic Mean, and Median) satisfy the property that if every observation is multiplied by \( c \) (where \( c < 0 \)), then the measure of central tendency is also multiplied by \( c \).

### Answer
All of the above measures satisfy the property: **I, II, III, IV**.

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