Three values are missing from the data set blank, 29, 59, blank, 76, blank. Given that the median is 66, the mean is 55, and the range is 67, and the data is listed in ascending order, find the missing values.
We could try and find the missing values using trial and error. However, it is more sensible to use an algebraic method in this question. We will let the three missing numbers be 𝑎, 𝑏, and 𝑐. We are told that the median is 66. We also know that the median is the middle value when the numbers are listed in ascending order. As there are six values, the middle two values will be 59 and 𝑏. The median will be the midpoint of these. In order to find the midpoint of any two values, we find the sum of them and then divide by two, in this case, 59 plus 𝑏 divided by two. This is equal to the median value of 66.
Multiplying both sides of this equation by two gives us 59 plus 𝑏 is equal to 132. Subtracting 59 from both sides of this equation gives us a value of 𝑏 equal to 73. The second missing value in our data set is 73. We could check this using a number line as 59 and 73 are equal distance from 66. Adding seven to 59 and subtracting seven from 73 gives us 66. We’re also told that the mean is equal to 55. We can calculate the mean by summing all the values and then dividing by how many values there are. As there are six values in total, 𝑎 plus 29 plus 59 plus 73 plus 76 plus 𝑐 divided by six is equal to 55.
Multiplying both sides of this equation by six and simplifying the numerator on the left-hand side gives us 𝑎 plus 237 plus 𝑐 is equal to 330. Subtracting 237 from both sides of this equation gives us 𝑎 plus 𝑐 is equal to 93. We will call this equation one. We were also told in the question that the range of the data set is 67 and the range is the highest number minus the smallest number. 𝑐 minus 𝑎 is equal to 67. If we call this equation two, we now have a pair of simultaneous equations.
We will now clear some room and solve these to calculate the value of 𝑎 and 𝑐. As 𝑎 plus 𝑐 is the same as 𝑐 plus 𝑎, we have 𝑐 plus 𝑎 is equal to 93 and 𝑐 minus 𝑎 is equal to 67. We can then add equation one and equation two to eliminate 𝑎. 93 plus 67 is 160. So two 𝑐 is equal to 160. Dividing both sides of this by two gives us 𝑐 is equal to 80. We can now substitute this value into equation one to calculate 𝑎. As 80 plus 𝑎 is equal to 93, 𝑎 is equal to 13. The three missing values from the data set are 13, 73, and 80.