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Difficulty: MediumData Distributions and Measures

A marine biologist records the dive depths, in meters, of a seal on 6 consecutive dives. The recorded depths are 52, 68, 57, 65, 48, and xx. The median of these 6 dive depths is 58 meters. What is the value of xx?

Answer: 59

Answer

59
To find the median of a set of 6 numbers, the numbers must be listed in ascending order, and the median will be the average of the 3rd and 4th numbers. The five known numbers in ascending order are 48, 52, 57, 65, and 68. If the unknown value xx is less than or equal to 57, the 3rd and 4th numbers in the sorted list would be at most 57, which would make the median at most 57. If xx is greater than or equal to 65, the 3rd and 4th numbers in the sorted list would be 57 and 65, which would make the median 57+652=61\frac{57+65}{2} = 61. Since the median is given as 58, xx must be between 57 and 65. Thus, when the 6 numbers are sorted, the 3rd number is 57 and the 4th number is xx. Setting their average equal to 58 gives 57+x2=58\frac{57+x}{2} = 58. Multiplying both sides by 2 yields 57+x=11657 + x = 116. Subtracting 57 from both sides gives x=59x = 59.

Step-by-Step Solution

1
Order the five known depths from least to greatest.
48, 52, 57, 65, 68
To find or use the median of a dataset, the data points must first be arranged in ascending order.
2
Analyze the position of the median in a dataset of 6 values.
The median is the average of the 3rd and 4th values in the ordered list.
For an even number of data points, the median is the arithmetic mean of the two middle values.
3
Determine the constraints on xx to achieve a median of 58.
The value of xx must be between 57 and 65, making 57 the 3rd value and xx the 4th value.
If x≤57x \le 57, the 3rd and 4th values are both 57 or less, yielding a median of at most 57. If x≥65x \ge 65, the 3rd and 4th values are 57 and 65, yielding a median of 61. Thus, xx must be the 4th value.
4
Set up and solve the equation for the median.
57+x2=58  ⟹  57+x=116  ⟹  x=59\frac{57 + x}{2} = 58 \implies 57 + x = 116 \implies x = 59
Since the 3rd value is 57 and the 4th value is xx, their average must equal the given median of 58.

Key Concept

Calculating and interpreting the median of a dataset with an even number of values, including handling unknown variables.
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Topics

Data Distributions and Measures

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