Final answer:
The measures of variability for grouped data include the range (A), variance (B and D), and standard deviation (C), with variance being the average of the squared differences from the mean and standard deviation being the square root of variance.
Step-by-step explanation:
When analyzing grouped data, one may need to find measures of variability such as the range, variance, and standard deviation. These measures help to understand the spread of the data relative to the mean.
The range of a data set is the difference between the highest and lowest values. It gives a basic estimate of the spread but does not account for how the data is distributed between these values (A).
The variance is a measure of the spread of the data and it represents the average of the squared differences from the mean. For a sample, this is calculated by summing the squares of the deviations and then dividing by the sample size minus one (B and D).
The standard deviation is the square root of the variance, which puts the spread in the same units as the data itself (C). It is particularly useful because it takes into account how each data point varies from the mean, giving a more comprehensive picture of the spread.
To calculate the variance and standard deviation of grouped data, we approximate the mean using the midpoints of the intervals multiplied by the corresponding frequencies. The variance is then the sum of the squared distances of these values from the approximated mean, divided by the total number of data points minus one (for a sample). The standard deviation is the square root of this variance.