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

A retail analyst recorded the number of online orders fulfilled per day over a period of 99 days. The data points, listed in non-decreasing order, are:

3,5,8,x,y,17,21,24,z3, 5, 8, x, y, 17, 21, 24, z

The median of the 99 daily order counts is 1414, the arithmetic mean is 1616, and the dataset has a unique mode of 88. If zz represents the highest number of orders fulfilled in a single day, what is the value of zz?

  1. A
    28
  2. B
    38
  3. 44Answer
  4. D
    52
  5. E
    54

Answer

44
For a sorted set of 9 values, the median is the 5th value, giving y=14y = 14. To make 8 the unique mode, 8 must appear at least twice, which forces x=8x = 8. With an arithmetic mean of 16 across 9 numbers, the total sum is 9×16=1449 \times 16 = 144. Subtracting the sum of the eight known numbers (3+5+8+8+14+17+21+24=1003+5+8+8+14+17+21+24 = 100) gives z=44z = 44.

Step-by-Step Solution

1
Determine the value of yy using the median definition.
y=14y = 14
For a dataset of n=9n = 9 numbers arranged in non-decreasing order, the median is the 9+12=5th\frac{9+1}{2} = 5\text{th} element. Thus, y=14y = 14.
2
Determine the value of xx using the unique mode condition.
x=8x = 8
The dataset is ordered as 3≤5≤8≤x≤14≤17≤21≤24≤z3 \le 5 \le 8 \le x \le 14 \le 17 \le 21 \le 24 \le z. For 88 to be a unique mode, it must appear more than once. Since all other given numbers are distinct, xx must equal 88 so that 88 occurs twice.
3
Calculate the required total sum of all 9 data points from the given arithmetic mean.
\text{Total Sum} = 144
\text{Mean} = \frac{\text{Total Sum}}{9} \implies \text{Total Sum} = 9 \times 16 = 144.
4
Sum the known values and solve for zz.
z=44z = 44
3+5+8+8+14+17+21+24+z=144  ⟹  100+z=144  ⟹  z=443 + 5 + 8 + 8 + 14 + 17 + 21 + 24 + z = 144 \implies 100 + z = 144 \implies z = 44.

Key Concept

Combining mean, median, and mode definitions to solve for unknown elements in a ordered dataset.
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Topics

Measures of Central Tendency (Mean, Median, Mode)

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