What is the median of the following string of values?
$16,95,3,37,97,13,27,42,64,1^4$
A. 32
B. 37
C. 27
D. 42
Answer (2)
Calculate 1 4 = 1 .
Sort the values in ascending order: 1 , 3 , 13 , 16 , 27 , 37 , 42 , 64 , 95 , 97 .
Since there are 10 values (an even number), the median is the average of the two middle values, which are 27 and 37.
Calculate the median: 2 27 + 37 = 32 . The median is 32 .
Explanation
Problem Analysis We are given the following values: 16 , 95 , 3 , 37 , 97 , 13 , 27 , 42 , 64 , 1 4 . Our goal is to find the median of these values.
Calculate 1^4 First, we need to calculate 1 4 . Since any number 1 raised to any power is 1, we have 1 4 = 1 .
Sort the values Next, we need to sort the values in ascending order. The sorted values are: 1 , 3 , 13 , 16 , 27 , 37 , 42 , 64 , 95 , 97 .
Determine the middle values Now, we need to determine the number of values, n . We have 10 values, so n = 10 . Since n is even, the median is the average of the two middle values. The two middle values are the 2 n and 2 n + 1 values. In this case, 2 10 = 5 and 2 10 + 1 = 6 . So, the two middle values are the 5th and 6th values in the sorted list.
Calculate the median The 5th value in the sorted list is 27, and the 6th value is 37. Therefore, the median is the average of 27 and 37, which is 2 27 + 37 = 2 64 = 32 .
Final Answer The median of the given values is 32.
Examples
The median is a useful measure in real life when you want to find the middle value of a dataset that may contain outliers. For example, if you have a list of salaries and want to know the 'typical' salary, the median is often more representative than the mean because it is not affected by extremely high or low salaries. Another example could be finding the median test score in a class to understand the central performance level, without being skewed by a few very high or very low scores. Understanding the median helps in making informed decisions based on central tendencies.
The median of the given values is calculated to be 32 by first sorting the values and then averaging the two middle values, 27 and 37. Since there are an even number of values, we take the average of these two. The answer choice corresponding to this result is therefore A. 32.
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