Question

Observations of raw data are 5, 28, 15, 10, 15, 8, and 24. Add four numbers so that the mean and median of the data remain the same, but mode increases by 1.

Answer 1

Hint: Mean is the average of all the numbers in the data set. To keep it the same the four numbers added must have the same average as of the raw data provided. Median is the middle value. To keep it unchanged we have to add numbers on both the sides of the median value. Mode is that value, which occurs most often. IN the given data the mode is 15, so the new mode will be 1 more than the previous which will be 16.

Complete step-by-step answer:
First of all let us arrange the given data in an ascending order to help find the median.
The data in ascending order will be 5, 8, 10, 15, 15, 24, and 28. Here, the number of values is 7 , so the median will be the middle value that is the \[{4^{th}}\] value. The \[{4^{th}}\] value here is 15, which will be the median.
Now, for mean, we will simply find the average of all the numbers as \[\dfrac{{5 + 8 + 10 + 15 + 15 + 24 + 28}}{7} = \dfrac{{105}}{7} = 15\].
Thus, the mean for the given data will be 15 too.
Mode of a data is that value which occurs most often. Here the mode will be 15 as it occurs twice which is the maximum by any other number.
Now, we have to add four numbers such that the mean and median remain the same while mode increases by 1. Thus the new mode will be 16. Also, as 16 has to be the mode it has to occur more than 2 times that is 3 or 4 times.
But as we have to keep the mean the same, the 4 numbers should sum up to \[4 \times 15 = 60\]. Thus 16 cannot occur 4 times as that will take the sum to 64. So 16 will occur three times and the remaining one number will be \[60 - (16 \times 3) = 60 - 48 = 12\]. Thus, we have the four numbers to be added as 12, 16, 16 and 16. Now, we have to keep the median also the same. For that when we count the new number of values it will \[7 + 4 = 11\] values, thus, 15 must occupy the \[{6^{th}}\] value. And as we arrange the new set of data in ascending order as 5, 8, 10, 12, 15, 15, 16, 16, 16, 24, and 28 we can easily observe that 15 is still occupying the middle position, thus proving that the numbers added are correct.

Note: These questions require a simple logical approach and the knowledge of where to begin. For the given question, the simple clue given was ‘mode increases by 1’, which helps us to know that 16 has to be added at least three times and after that we can eliminate the other options and get to all the numbers to be added.