Yes, the arithmetic mean is dependent on changes in the origin (shifting the data by adding or subtracting a constant) and scale (multiplying or dividing by a constant). This is because the arithmetic mean is calculated by summing all the values and dividing by the number of values, so any transformation applied uniformly to all values affects the mean in a predictable way.

Here's how the arithmetic mean behaves with respect to these transformations:

1. Change in origin (Addition/Subtraction of a constant):

   • If each value of the variable $X$ is increased or decreased by a constant $k$, the arithmetic mean will also increase or decrease by $k$.
   • Example: If $X = \{ x_1, x_2, \ldots, x_n \}$ and a constant $k$ is added to each value, then: $\text{Mean of } (X + k) = \frac{(x_1 + k) + (x_2 + k) + \ldots + (x_n + k)}{n} = \text{Mean of } X + k$
   • The mean simply shifts by the constant.

2. Change in scale (Multiplication/Division by a constant):

   • If each value of the variable $X$ is multiplied or divided by a constant $k$, the arithmetic mean will also be multiplied or divided by $k$.
   • Example: If $X = \{ x_1, x_2, \ldots, x_n \}$ and each value is multiplied by $k$, then: $\text{Mean of } (kX) = \frac{kx_1 + kx_2 + \ldots + kx_n}{n} = k \times \text{Mean of } X$
   • The mean scales by the same factor.

In summary:

• Add/subtract a constant: The mean changes by the same amount.
• Multiply/divide by a constant: The mean is multiplied/divided by the same factor.

Would you like more details or further examples?

Related Questions:

1. How does the median change when adding a constant to all values?
2. What happens to the standard deviation if we multiply all values by a constant?
3. How does the variance respond to a change in scale?
4. Can you explain the effect of both addition and multiplication on the mean with a real-world example?
5. How does the geometric mean behave when each value is multiplied by a constant?

Tip:

The arithmetic mean is always sensitive to extreme values (outliers), while transformations like adding or multiplying affect the entire dataset evenly.