The arithmetic mean of a frequency distribution is 5. If each frequency is increased to 5 times, then the arithmetic mean will be:

Question

Accepted Answer

The correct option is 3:
The arithmetic mean ($\bar{X}$) of a frequency distribution is calculated using the formula: $$\bar{X} = \frac{\sum f\_i x\_i}{\sum f\_i}$$, where $f\_i$ represents the frequency of a class and $x\_i$ represents the mid-value of that class. The arithmetic mean is a measure of central tendency which depends on the actual values ($x\_i$) and the distribution of their weights ($f\_i$).

When every single frequency in the distribution is increased by the same constant factor (in this case, multiplied by 5), the impact on the mean is mathematically nullified. If the new frequencies are $f'\_i = 5 f\_i$, the new mean ($\bar{X}'$) will be:
$$\bar{X}' = \frac{\sum f'\_i x\_i}{\sum f'\_i} = \frac{\sum (5 f\_i) x\_i}{\sum (5 f\_i)} = \frac{5 \sum f\_i x\_i}{5 \sum f\_i}$$
The common factor of 5 cancels out from both the numerator and the denominator. This demonstrates that scaling all frequencies by a uniform constant does not change the ratio of the total weighted sum to the total frequency count. Since the original mean was 5, the new arithmetic mean will also remain 5. The mean is invariant under the scaling of all frequencies.

The incorrect option is 1 (25):
This result would only occur if the values of the items ($x\_i$) were also multiplied by 5, or if the original mean (5) was multiplied by the constant (5). Since only the frequencies are changed, this option is incorrect.

The incorrect option is 2 (1):
This value is the original mean (5) divided by the scaling factor (5). The inverse relationship implied by this option does not apply because the scaling factor is present in both the numerator and the denominator of the mean formula, leading to cancellation.

The incorrect option is 4 (6.33):
There is no statistical principle or relationship that would lead to a non-integer or slightly increased mean simply by uniformly multiplying all frequencies. This value is mathematically irrelevant to the operation performed on the data set.