What are the differences among the various measures of variation, such as the ra

Question

• What are the differences among the various measures of variation, such as the range, interquartile range, variance, and standard deviation?
• What are the advantages and disadvantages of each, using your own real-world example?
• How would you apply these concepts in a unique business scenario?

• What are the differences among the various measures of variation, such as the range, interquartile range, variance, and standard deviation?
• What are the advantages and disadvantages of each, using your own real-world example?
• How would you apply these concepts in a unique business scenario?

Explanation / Answer

Range : The range is the most obvious measure of dispersion and is the difference between the lowest and highest values in a dataset. In figure 1, the size of the largest semester 1 group is 6 students and the size of the smallest group is 4 students, resulting in a range of 2 (6-4). In semester 2, the largest group size is 7 students and the smallest group contains 3 students, therefore the range is 4 (7-3).

The inter-quartile range : is a measure that indicates the extent to which the central 50% of values within the dataset are dispersed. It is based upon, and related to, the median.

In the same way that the median divides a dataset into two halves, it can be further divided into quarters by identifying the upper and lower quartiles. The lower quartile is found one quarter of the way along a dataset when the values have been arranged in order of magnitude; the upper quartile is found three quarters along the dataset. Therefore, the upper quartile lies half way between the median and the highest value in the dataset whilst the lower quartile lies halfway between the median and the lowest value in the dataset. The inter-quartile range is found by subtracting the lower quartile from the upper quartile.

The standard deviation : is a measure that summarises the amount by which every value within a dataset varies from the mean. Effectively it indicates how tightly the values in the dataset are bunched around the mean value. It is the most robust and widely used measure of dispersion since, unlike the range and inter-quartile range, it takes into account every variable in the dataset. When the values in a dataset are pretty tightly bunched together the standard deviation is small. When the values are spread apart the standard deviation will be relatively large. The standard deviation is usually presented in conjunction with the mean and is measured in the same units.

Variance: measures how far a data set is spread out. The technical definition is “The average of the squared differences from the mean,” but all it really does is to give you a very general idea of the spread of your data. A value of zero means that there is no variability; All the numbers in the data set are the same.

The range, inter-quartile range and standard deviation are all measures that indicate the amount of variability within a dataset. The range is the simplest measure of variability to calculate but can be misleading if the dataset contains extreme values. The inter-quartile range reduces this problem by considering the variability within the middle 50% of the dataset. The standard deviation is the most robust measure of variability since it takes into account a measure of how every value in the dataset varies from the mean. However, care must be taken when calculating the standard deviation to consider whether the entire population or a sample is being examined and to use the appropriate formula.

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