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

Dataset SS consists of 11 distinct positive integers arranged in increasing order, with a median of 50, an interquartile range of 20, and a standard deviation of σ\sigma. A new dataset S′S' is created by adding 10 to each of the 5 integers in SS that are strictly greater than 50, while leaving the remaining 6 integers unchanged. Which of the following statements about dataset S′S' must be true? Select all such statements.

  1. The interquartile range of dataset S′S' is 30.Cevap
  2. The range of dataset S′S' is 10 greater than the range of dataset SS.Cevap
  3. The standard deviation of dataset S′S' is strictly greater than σ\sigma.Cevap
  4. D
    The median of dataset S′S' is 60.
  5. E
    The interquartile range of dataset S′S' remains 20.

Cevap

The statements asserting that the interquartile range of dataset S′S' is 30, that the range of dataset S′S' is 10 greater than the range of dataset SS, and that the standard deviation of dataset S′S' is strictly greater than σ\sigma are all correct.
In an ordered dataset of 11 elements, the median is the 6th element, Q1Q_1 is the 3rd element, and Q3Q_3 is the 9th element. When 10 is added only to the elements strictly above the median (elements 7 through 11): Q3Q_3 increases by 10 while Q1Q_1 is unchanged, making the new IQR equal to 20+10=3020 + 10 = 30. The maximum value increases by 10 while the minimum value remains unchanged, increasing the range by 10. Shifting values in the upper tail further right increases the overall distance of data points from the mean, causing the standard deviation to strictly increase.

Adım Adım Çözüm

1
Analyze the position of quartiles and median in a dataset of 11 ordered values.
For 11 ordered values x1<x2<⋯<x11x_1 < x_2 < \dots < x_{11}, the median is x6=50x_6 = 50, Q1=x3Q_1 = x_3, and Q3=x9Q_3 = x_9. The lower 6 elements (x1x_1 through x6x_6) are unchanged. The upper 5 elements (x7x_7 through x11x_{11}) each increase by 10.
Determining which specific data positions change allows us to evaluate median, IQR, and range.
2
Calculate the new interquartile range and range.
Q1,new=x3Q_{1,\text{new}} = x_3, Q3,new=x9+10Q_{3,\text{new}} = x_9 + 10. Thus IQRnew=(x9+10)−x3=IQRold+10=20+10=30\text{IQR}_{\text{new}} = (x_9 + 10) - x_3 = \text{IQR}_{\text{old}} + 10 = 20 + 10 = 30. The maximum element x11x_{11} increases by 10 while x1x_1 is unchanged, so Rangenew=(x11+10)−x1=Rangeold+10\text{Range}_{\text{new}} = (x_{11} + 10) - x_1 = \text{Range}_{\text{old}} + 10.
Interquartile range is Q3−Q1Q_3 - Q_1 and range is maximum−minimum\text{maximum} - \text{minimum}.
3
Evaluate the effect on the median and standard deviation.
The median remains x6=50x_6 = 50. Moving the upper values further away from the center increases the overall spread around the mean, which strictly increases the standard deviation beyond σ\sigma.
Standard deviation measures the average spread of values from the mean.

Anahtar Kavram

Effect of asymmetric data shifts on measures of central tendency and dispersion
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Konular

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

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