Observe the data and answer the questions that follow:
16, 15, 16, 16, 8, 15, 17
a)Which data value can be put in the data so that the mode remains the same?
b)At least how many and which value(s) must be put in to change the mode to 15?
c)What is the least number of data values that must be put in to change the mode to 17? Name them.

Hint: Here, we will be finding the data values that can change the mode. We will use the concept of measuring of central tendency to find the mode. Mode is defined as the value that appears the most frequently in the set of data values.

Complete step-by-step answer:
We are given the set of data values of 16, 15, 16, 16, 8, 15, 17.
First we will arrange the set of all data values in the ascending order. Therefore, we get
 8, 15, 15, 16, 16, 16, 17
So, the mode for the given data values is 16 which occurs the maximum times i.e., 3 times.
a)We will find the data values that can be put in the data so that the mode remains the same.
If 15 is put into the data value which occurred 2 times would change into 3 times, thus the mode changes into 15 and 16.
Thus, the data values of 8 or 16 or 17 can be put in the data such that the mode remains the same.

b)Now, we have to find the number of times that the value of 15 is added to change the mode into 15.
We know that the mode of the set is 16 which occurs three times. So, now to change the mode as 15 it has to be greater than 3 times which may be four. As 15 has occurred already 2 times, so it has to be added two times to change the mode as 15.

c)Now, we have to find the number of times that the value of 17 is added to change the mode into 17.
We know that the mode of the set is 16 which occurs three times. So, now to change the mode as 17 it has to be greater than 3 times which may be four. As 17 has occurred only one time, so it has to be added three times to change the mode as 17.

Therefore,
a)The data values of 8 or 16 or 17 can be put in the data such that the mode remains the same.
b)The data value 15 has to be added at least two times to change the mode to 15.
c)The data value 17 has to be added at least three times to change the mode to 17.


Note: We should know that the mode can be calculated for the raw data and for grouped and ungrouped frequency distribution. Mode is of four types, distribution with only one mode is unimodal, distribution with two modes is bimodal, distribution with three modes is trimodal and the distribution with more than three modes is multimodal. We should note that when all the numbers appeared for the same number of times, then we have no modes.