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Difficulty: EasyMeasures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

A constant value of 77 is added to each number in a dataset to form a new dataset. Which of the following statistical measures of the new dataset are equal to the corresponding measures of the original dataset? Select all that apply.

  1. The rangeAnswer
  2. The standard deviationAnswer
  3. The interquartile rangeAnswer
  4. D
    The mean
  5. E
    The median

Answer

The range, the standard deviation, and the interquartile range of the new dataset are equal to those of the original dataset.
Adding a constant to every value in a dataset shifts the entire distribution uniformly without altering its overall shape or spread. Consequently, measures of dispersion—including the range, standard deviation, and interquartile range—remain completely unchanged.

Step-by-Step Solution

1
Evaluate the effect of adding a constant on measures of central tendency
If each data point xix_i is replaced by xi+7x_i + 7, the mean μ\mu becomes μ+7\mu + 7 and the median MM becomes M+7M + 7.
Measures of central tendency shift directly by the added constant, so they are not equal to the original measures.
2
Evaluate the effect of adding a constant on measures of dispersion
The new range is (xmax+7)−(xmin+7)=xmax−xmin(x_{\text{max}} + 7) - (x_{\text{min}} + 7) = x_{\text{max}} - x_{\text{min}}. The new interquartile range is (Q3+7)−(Q1+7)=Q3−Q1(Q_3 + 7) - (Q_1 + 7) = Q_3 - Q_1. Each deviation from the mean is (xi+7)−(μ+7)=xi−μ(x_i + 7) - (\mu + 7) = x_i - \mu, so the standard deviation is unchanged.
Measures of dispersion reflect the relative distances between data points or distance from the mean, which are unaffected by shifting the whole distribution.

Key Concept

Effect of adding a constant on measures of central tendency versus measures of dispersion
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Topics

Measures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

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