Question

If all observations are increased by 5 their mean ____.

• A. Decreases
• B. Increase by 5
• C. Remain same
• D. Can not be Predicated

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.

Answer 1

The Correct Option is B

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.

Answer 2

This question checks whether you understand how a mean responds when a fixed constant is added to every value in a data set. Let's test this with a small worked example alongside checking each option.

1. Decreases: If every value in a set is made larger by adding 5, the total sum of the data can only increase, never decrease, so the average cannot decrease either.
2. Increase by 5: Take a simple example: the set \( \{1, 2, 3\} \) has mean \( \frac{1+2+3}{3} = 2 \). Add 5 to each value to get \( \{6, 7, 8\} \), whose mean is \( \frac{6+7+8}{3} = \frac{21}{3} = 7 \). The mean rose from 2 to 7, an increase of exactly 5, matching the constant added to each observation.
3. Remain same: The mean only stays unchanged if the data itself is unchanged, or if some values increase while others decrease by compensating amounts. Since every value here increases by the same fixed amount, the mean must shift too.
4. Cannot be predicted: The outcome is actually fully determined and doesn't depend on what the original numbers were, adding the same constant to every observation always shifts the mean by that exact constant, regardless of the data set's size or values.

The worked example confirms that adding 5 to every observation increases the mean by exactly 5, a result that holds for any data set because of how the mean is defined as a sum divided by a fixed count.

Therefore, the correct answer is increase by 5.