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Problem-Solving and Data Analysis / One-variable data: Distributions and measures of center and spread Difficulty: Medium

If a is the mean and b is the median of nine consecutive integers, what is the value of $the absolute value of, a, minus b, end absolute value$ ?

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Explanation

The correct answer is 0. Any nine consecutive integers can be written as $k$ , $k plus 1$ , $k plus 2$ , $k plus 3$ , $k plus 4$ , $k plus 5$ , $k plus 6$ , $k plus 7$ , $k plus 8$ . The mean of the integers is their sum divided by 9: $the fraction with numerator open parenthesis, k plus, k plus 1, plus, k plus 2, plus, dot dot dot, plus, k plus 8, close parenthesis, and denominator 9, equals, the fraction with numerator, open parenthesis, 9 k plus 36, close parenthesis, and denominator 9$ , which simplifies to $k plus 4$ . So $a, equals, k plus 4$ . Since there is an odd number of integers (nine), the median is the integer in the middle when all the integers are ordered from least to greatest: $k plus 4$ . So $b equals, k plus 4$ . Therefore, $the absolute value of a, minus b, end absolute value, equals, the absolute value of, open parenthesis, k plus 4, close parenthesis, minus, open parenthesis, k plus 4, close parenthesis, end absolute value$ , which is 0.