To determine which statements are true about the weighted mean, we need to carefully consider each provided statement in reference to the weighted mean formula [tex]\(\bar{X}_w = \frac{\Sigma (w X)}{\Sigma w}\)[/tex].
1. "It is used when there are several observations of the same value."
This statement is true. The weighted mean is particularly useful when certain observations are repeated multiple times, each possibly having a different significance or frequency. Instead of computing the mean where each observation contributes equally, the weighted mean accounts for these repetitions by assigning a weight to each value.
2. "The denominator of the weighted mean is always the sum of the weights."
This statement is also true. According to the formula [tex]\(\bar{X}_w = \frac{\Sigma (w X)}{\Sigma w}\)[/tex], the denominator is [tex]\(\Sigma w\)[/tex], which is the sum of all the weights assigned to the observations. This sum ensures that the weights properly normalize the weighted contributions.
3. "It gives a larger value for the mean than the formula for the arithmetic mean."
This statement is false. The weighted mean does not necessarily give a larger value than the arithmetic mean. The value of the weighted mean depends on the weights assigned to each observation and their corresponding values. Therefore, depending on the context and the assigned weights, the weighted mean could be larger, smaller, or equal to the arithmetic mean.
4. "It is a special case of the arithmetic mean."
This statement is true. The arithmetic mean can be seen as a special case of the weighted mean where all weights are equal to 1. In other words, the arithmetic mean does not differentiate between the significance or frequency of different observations—each observation is equally weighted.
Therefore, the true statements about the weighted mean are:
- It is used when there are several observations of the same value.
- The denominator of the weighted mean is always the sum of the weights.
- It is a special case of the arithmetic mean.
