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Which measure of variability—range or IQR—best describes the spread of the dataset?

Social studies group project grades (out of 100 points) for Mr. Chang’s first period class

85 88 90 90 92 92 95 96 99

(1 point)
Responses

IQR; the distribution is skewed, and there are outliers in the dataset.
IQR; the distribution is skewed, and there are outliers in the dataset.

IQR; the distribution is symmetrical, or close to it, and there aren’t any outliers in the dataset.
IQR; the distribution is symmetrical, or close to it, and there aren’t any outliers in the dataset.

Range; the distribution is symmetrical, or close to it, and there aren’t any outliers in the dataset.
Range; the distribution is symmetrical, or close to it, and there aren’t any outliers in the dataset.

Range; the distribution is skewed, and there are outliers in the dataset.

1 answer

To determine which measure of variability is more appropriate for the given dataset, we need to analyze the distribution of the grades:

The grades provided are: 85, 88, 90, 90, 92, 92, 95, 96, 99.

Analyzing the Data:

  1. Distribution Shape:

    • If we visualize or consider the distribution, we notice that it is relatively symmetrical and clustered in the higher range (from 85 to 99), without any extreme values that would skew it in one direction or another.

  2. Outliers:

    • There do not appear to be any outliers in this dataset. All grades lie within a close range with no extreme high or low values compared to the rest.

  3. Range:

    • The range is calculated as the difference between the maximum and minimum values: \[ \text{Range} = 99 - 85 = 14 \]

  4. Interquartile Range (IQR):

    • To calculate the IQR, we find the first (Q1) and third quartiles (Q3):
      • The median (Q2) is 92.
      • Q1 (median of the first half: 85, 88, 90, 90) = (88 + 90)/2 = 89
      • Q3 (median of the second half: 92, 95, 96, 99) = (95 + 96)/2 = 95.5
    • So, IQR = Q3 - Q1 = 95.5 - 89 = 6.5.

Conclusion:

Given that the distribution is symmetrical (or close to it) and does not contain outliers, the IQR is a better measure of variability in this case because it is less affected by small fluctuations in the dataset compared to the range. The range can be misleading, especially if there are extreme values.

Correct Response:

IQR; the distribution is symmetrical, or close to it, and there aren’t any outliers in the dataset.