Question

42, 31, 48, 12, 21, 17, 27, 22, 35, 19, 17

Let's include one more data point in the data set. Suppose this new data value lies between the values 12 and 48, inclusively.

Find the smallest possible value of the median of the new data set

Find the largest possible value of the median of the new data set

Answer 1

The median is the middle most observation in the dataset i.e. median is that observation below which 50% of the observations lie.

Now, if we order the data points in an increasing order, then we have the dataset as: 12, 17, 17, 19, 21, 22, 27, 31, 35, 42, 48.

Since, we have odd number of observations (11), the median is the 6th ordered observation i.e. 22.

Now, if we add an observation, then the number of observations will be even.

Now, if the newly added observation is less than or equal to 21, then the median will be (21 + 22)/2 = 21.5.

If it is between 21 and 22 (say a), then median = (a + 22)/2 which is greater than 21.5 (as, 21 < a < 22).

If it is greater than or equal to 22 but less than 27 (say b), then median = (22 + b)/2.

If it is more than or equal to 27, then median = (22 + 27)/2 = 24.5 > (22 + b)/2 as, 22 < b < 27.

Hence, the smallest possible value of median in the new dataset is 21.5 and largest possible value of median in the new dataset is 24.5. (Ans).