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Difficulty: MediumMeasures of Central Tendency (Mean, Median, Mode)

A quality control manager records the durability ratings, on a scale from 11 to 5050, for 88 randomly selected component batches. The ratings are listed below:

34,18,42,27,18,39,45,2534, 18, 42, 27, 18, 39, 45, 25

Two additional component batches with identical ratings, xx and yy (where x=yx = y), are included in the dataset. If the median rating of the updated dataset of 1010 batches is equal to the median rating of the original dataset of 88 batches, what is the value of xx?

  1. A
    22.522.5
  2. B
    26.026.0
  3. 30.530.5Answer
  4. D
    31.531.5
  5. E
    36.536.5

Answer

The value of xx is 30.530.5.
To find the median of the original dataset, the numbers must first be ordered from smallest to largest: 18,18,25,27,34,39,42,4518, 18, 25, 27, 34, 39, 42, 45. Since there are 88 numbers, the median is the average of the 4th and 5th numbers: (27+34)/2=30.5(27 + 34)/2 = 30.5. When two equal values xx are added to form a 1010-element dataset, having x=30.5x = 30.5 places both new values directly between 2727 and 3434. The 5th and 6th terms of the updated set are both 30.530.5, making the new median (30.5+30.5)/2=30.5(30.5 + 30.5)/2 = 30.5, which matches the original median.

Step-by-Step Solution

1
Sort the original dataset of 8 durability ratings in ascending order.
The sorted dataset is 18,18,25,27,34,39,42,4518, 18, 25, 27, 34, 39, 42, 45.
Finding the median of a numerical dataset requires arranging the numbers in order from least to greatest.
2
Calculate the median of the original 8 ratings.
The 4th value is 2727 and the 5th value is 3434. The median is 27+342=30.5\frac{27 + 34}{2} = 30.5.
For a dataset with an even number of elements (n=8n = 8), the median is the average of the two middle elements (the 4th and 5th terms).
3
Analyze how adding two identical values xx and yy (x=yx = y) affects the median of the 10-element dataset.
Inserting two values equal to 30.530.5 places them as the 5th and 6th elements in the 10-element sorted array: 18,18,25,27,30.5,30.5,34,39,42,4518, 18, 25, 27, 30.5, 30.5, 34, 39, 42, 45. The new median is 30.5+30.52=30.5\frac{30.5 + 30.5}{2} = 30.5.
To keep the median unchanged at 30.530.5 when adding two identical values, the added values must fall between 2727 and 3434 and specifically equal 30.530.5 so that the middle two terms of the 10 values average to 30.530.5.

Key Concept

Median of a Dataset
Estimated Time:1m 30s
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Topics

Measures of Central Tendency (Mean, Median, Mode)

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