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Zorluk: OrtaMeasures of Central Tendency (Mean, Median, Mode)

A sports scientist records the recovery times, in minutes, for 5 subjects after an intense workout. Four of the times are 44,16,52,44, 16, 52, and 3232 minutes, and the fifth time is xx minutes. The 5 times are listed in no particular order. If 32<x<4432 < x < 44 and the arithmetic mean of all 5 recovery times is equal to the median of the 5 recovery times, what is the value of xx?

  1. A
    26
  2. B
    29
  3. C
    34
  4. 36Cevap
  5. E
    48

Cevap

The value of xx is 36.
Because 32<x<4432 < x < 44, the 5 numbers listed in ascending order are 16,32,x,44,5216, 32, x, 44, 52. The median of these 5 numbers is the middle value, xx. Setting the mean 16+32+x+44+525=144+x5\frac{16 + 32 + x + 44 + 52}{5} = \frac{144 + x}{5} equal to the median xx gives 144+x5=x\frac{144 + x}{5} = x, which simplifies to 4x=1444x = 144 and yields x=36x = 36.

Adım Adım Çözüm

1
Order the dataset in ascending numerical order.
The five data values in sorted order are 16,32,x,44,5216, 32, x, 44, 52 because it is given that 32<x<4432 < x < 44.
Finding the median of a dataset requires arranging all elements in ascending or descending order first.
2
Identify the median of the 5 values.
The median is the 3rd element in the sorted 5-element list, which is xx.
For an odd number of items (n=5n = 5), the median is the middle value at position 5+12=3\frac{5+1}{2} = 3.
3
Express the arithmetic mean in terms of xx.
\text{Mean} = \frac{16 + 32 + x + 44 + 52}{5} = \frac{144 + x}{5}$.
The arithmetic mean is defined as the sum of all values divided by the total count of values (n=5n = 5).
4
Set the mean equal to the median and solve for xx.
\frac{144 + x}{5} = x \implies 144 + x = 5x \implies 4x = 144 \implies x = 36$.
The problem states that the arithmetic mean equals the median.

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Measures of Central Tendency (Mean, Median, Mode)
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Measures of Central Tendency (Mean, Median, Mode)

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