To determine which statements are true about the weighted mean, let's go through each statement and evaluate its correctness.
1. It is used when there are several observations of the same value.
- True: The weighted mean is indeed useful in scenarios where we have several observations of the same value, each with different weights or frequencies. Assigning weights allows us to account for the varying significance or frequency of each observation.
2. The denominator of the weighted mean is always the sum of the weights.
- True: The formula for the weighted mean [tex]\(\bar{X}_w=\frac{\Sigma(w X)}{\Sigma w}\)[/tex] specifies that the denominator is the sum of the weights [tex]\(\Sigma w\)[/tex].
3. It gives a larger value for the mean than the formula for the arithmetic mean.
- False: The weighted mean can be either larger or smaller than the arithmetic mean, depending on the weights assigned to the values. Therefore, it is not necessarily larger.
4. It is a special case of the arithmetic mean.
- True: The weighted mean can be considered a special case of the arithmetic mean where each observation could have an equal weight of 1. In such a scenario, the weighted mean reduces to the arithmetic mean.
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.
