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Zorluk: OrtaMeasures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

A logistics company recorded the daily delivery processing times (in minutes) for a warehouse over a given period, forming Dataset XX. Dataset XX has a range of 4040 minutes, an interquartile range (IQR\text{IQR}) of 1515 minutes, and a standard deviation of 8.58.5 minutes. A new dataset, Dataset YY, is created by transforming each processing time xx in Dataset XX according to the formula y=1.5x−10y = 1.5x - 10. Which of the following statements regarding the measures of dispersion for Dataset YY must be true? Select all such statements.

  1. The range of Dataset YY is 6060 minutes.Cevap
  2. The standard deviation of Dataset YY is 12.7512.75 minutes.Cevap
  3. C
    The interquartile range of Dataset YY is 12.512.5 minutes.
  4. D
    The standard deviation of Dataset YY is 2.752.75 minutes.
  5. E
    The interquartile range of Dataset YY is equal to the interquartile range of Dataset XX.

Cevap

The statements asserting that the range of Dataset Y is 60 minutes and that the standard deviation of Dataset Y is 12.75 minutes are both correct.
Under a transformation of the form y=ax+by = ax + b (where a>0a > 0), any measure of dispersion DD transforms according to DY=a⋅DXD_Y = a \cdot D_X. The constant bb does not affect spread. Therefore, the range becomes 1.5×40=601.5 \times 40 = 60 minutes and the standard deviation becomes 1.5×8.5=12.751.5 \times 8.5 = 12.75 minutes.

Adım Adım Çözüm

1
Recall the effect of a linear transformation y=ax+by = ax + b on measures of dispersion.
Measures of dispersion (range, IQR, standard deviation) are scaled by ∣a∣|a| and are completely unaffected by the constant addition/subtraction bb.
Adding or subtracting a constant shifts all data points by the exact same amount without altering the relative distances between data points.
2
Calculate the range for Dataset Y.
RangeY=1.5×RangeX=1.5×40=60\text{Range}_Y = 1.5 \times \text{Range}_X = 1.5 \times 40 = 60 minutes.
The multiplicative factor is a=1.5a = 1.5.
3
Calculate the standard deviation for Dataset Y.
SY=1.5×SX=1.5×8.5=12.75S_Y = 1.5 \times S_X = 1.5 \times 8.5 = 12.75 minutes.
The standard deviation scales proportionally by 1.51.5.
4
Calculate the interquartile range (IQR) for Dataset Y.
IQRY=1.5×IQRX=1.5×15=22.5\text{IQR}_Y = 1.5 \times \text{IQR}_X = 1.5 \times 15 = 22.5 minutes.
The IQR also scales proportionally by 1.51.5.

Anahtar Kavram

Linear Transformations of Dispersion Measures
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Konular

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

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