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

Data set A consists of $10$ positive integers less than $60$ . The list shown gives $9$ of the integers from data set A.

$43$ , $45$ , $44$ , $43$ , $38$ , $39$ , $40$ , $46$ , $40$

The mean of these $9$ integers is $42$ . If the mean of data set A is an integer that is greater than $42$ , what is the value of the largest integer from data set A?

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Explanation

The correct answer is $52$ . The mean of a data set is calculated by dividing the sum of the data values by the number of values. It’s given that data set A consists of $10$ values, $9$ of which are shown. Let $x$ represent the $10 th$ data value in data set A, which isn’t shown. The mean of data set A can be found using the expression $\frac{43 + 45 + 44 + 43 + 38 + 39 + 40 + 46 + 40 + x}{10}$ , or $\frac{378 + x}{10}$ . It’s given that the mean of the $9$ values shown is $42$ and that the mean of all $10$ numbers is greater than $42$ . Consequently, the $10 th$ data value, $x$ , is larger than $42$ . It’s also given that the data values in data set A are positive integers less than $60$ . Thus, $42 less than x less than 60$ . Finally, it’s given that the mean of data set A is an integer. This means that the sum of the $10$ data values, $378 plus x$ , is divisible by $10$ . Thus, $378 plus x$ must have a ones digit of $0$ . It follows that $x$ must have a ones digit of $2$ . Since $42 less than x less than 60$ and $x$ has a ones digit of $2$ , the only possible value of $x$ is $52$ . Since $52$ is larger than any of the integers shown, the largest integer from data set A is $52$ .