Final answer:
The statement is true; a smaller variance and standard deviation indicate that the observations are less spread out, reflecting data points that are closer to the mean.
Step-by-step explanation:
True. The statement that the less spread out the observations, the smaller the variance and standard deviation (SD) is correct. Variance and standard deviation are measures of dispersion, indicating how much the data points in a set deviate from the mean. A smaller variance or standard deviation occurs when the data points are clustered more closely around the mean, meaning there is less spread. Conversely, when the observations are more dispersed, the variance and standard deviation are higher, showing greater spread in the data. This is supported by various statistical principles, including the fact that as the sample size increases, the standard deviation of the sampling distribution tends to decrease, assuming the samples are taken from the same population.
The standard deviation, which is always positive or zero, essentially describes how tightly the data is clustered around the mean. If there is no spread at all (all data points are identical), the standard deviation is zero. Conversely, a large standard deviation indicates that the data points are spread out over a wider range of values. Outliers in the data can also significantly increase the standard deviation, illustrating a greater spread.