Why is the standard deviation of the sample means less than the population standard deviation?
a. It is not less; taking a group of scores at a time introduces additional error.
b. The data is sampled twice, thereby reducing error.
c. It is not less; smaller samples will have greater variability.
d. The data is "condensed" by taking a group of scores at a time, so variability is reduced.
The correct answer is option d. The standard deviation of the sample means is generally less than the population standard deviation because the data is "condensed" by taking a group of scores at a time, which reduces variability.
Now, let's break down the other options to understand why they are not correct:
a. It is not less; taking a group of scores at a time introduces additional error.
This option suggests that taking a group of scores introduces additional error, which is incorrect. In fact, when you calculate the mean of a sample, any random errors tend to cancel each other out, resulting in a more accurate estimate.
b. The data is sampled twice, thereby reducing error.
This option incorrectly suggests that sampling the data twice reduces error. On the contrary, taking multiple samples can provide a more accurate estimate of the population mean, but it does not directly reduce the standard deviation of the sample means.
c. It is not less; smaller samples will have greater variability.
This option suggests that smaller samples have greater variability, which is partially true. Smaller samples tend to have more variability in their individual values, but when calculating the sample mean, the variability decreases due to the averaging effect.
In summary, option d is correct because when you calculate the sample means by taking a group of scores at a time, the variability is reduced due to the "condensing" effect of combining multiple values into a single mean.