Question:
If all observations are increased by 5 their mean ____.
If all observations are increased by 5 their mean ____.
Show Hint
This is a property of "Change of Origin." Adding or subtracting a constant shifted the entire distribution, including the center point (mean), by that exact amount.
Updated On: May 19, 2026
- Decreases
- Increase by 5
- Remain same
- Can not be Predicated
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
The arithmetic mean is sensitive to changes in the data. If a constant value is added to, subtracted from, multiplied by, or divided from every observation in a set, the mean will change by that same constant.
Step 2: Detailed Explanation:
Let the original observations be $x_1, x_2, \dots, x_n$ with mean $\bar{x} = \frac{\sum x_i}{n}$. If every observation is increased by 5, the new observations become $(x_1 + 5), (x_2 + 5), \dots, (x_n + 5)$. The new mean $\bar{x}'$ is: \[ \bar{x}' = \frac{(x_1 + 5) + (x_2 + 5) + \dots + (x_n + 5)}{n} \] \[ \bar{x}' = \frac{(x_1 + x_2 + \dots + x_n) + (5 \times n)}{n} \] \[ \bar{x}' = \frac{\sum x_i}{n} + \frac{5n}{n} = \bar{x} + 5 \]
Step 3: Final Answer:
The new mean will increase by 5.
The arithmetic mean is sensitive to changes in the data. If a constant value is added to, subtracted from, multiplied by, or divided from every observation in a set, the mean will change by that same constant.
Step 2: Detailed Explanation:
Let the original observations be $x_1, x_2, \dots, x_n$ with mean $\bar{x} = \frac{\sum x_i}{n}$. If every observation is increased by 5, the new observations become $(x_1 + 5), (x_2 + 5), \dots, (x_n + 5)$. The new mean $\bar{x}'$ is: \[ \bar{x}' = \frac{(x_1 + 5) + (x_2 + 5) + \dots + (x_n + 5)}{n} \] \[ \bar{x}' = \frac{(x_1 + x_2 + \dots + x_n) + (5 \times n)}{n} \] \[ \bar{x}' = \frac{\sum x_i}{n} + \frac{5n}{n} = \bar{x} + 5 \]
Step 3: Final Answer:
The new mean will increase by 5.
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