Q1.
Will including a new value to the data increase or decrease the median?
Answer
The data shown in the dot plot is 4, 7, 8, 12, 14, whose median is the middle value, 8.
Include 11 (a value greater than 8):
Sorted: 4, 7, 8, 11, 12, 14 → six values, so the median is the average of the two middle ones
Median = (8 + 11) ÷ 2 = 9.5 (it increased)
Include 5 instead (a value less than 8):
Sorted: 4, 5, 7, 8, 12, 14
Median = (5 + 7) ÷ 2 = 6 (it decreased)
Sorted: 4, 7, 8, 11, 12, 14 → six values, so the median is the average of the two middle ones
Median = (8 + 11) ÷ 2 = 9.5 (it increased)
Include 5 instead (a value less than 8):
Sorted: 4, 5, 7, 8, 12, 14
Median = (5 + 7) ÷ 2 = 6 (it decreased)
So a new value greater than the median pushes the median up, and one less than the median pushes it down — but only by a little, and it can never jump past the neighbouring value.
Why it happens: the median only cares about position. Adding a value above 8 puts one more number on the high side, so 8 is no longer in the middle — the middle slides one place towards the larger numbers. Notice the crucial difference from the mean: it does not matter whether you add 11 or 1100, the median moves to the same place, because the median never asks how far a value is, only which side it is on.
Tip: that is why the median is the safer summary when a data set has one wild value — a single crorepati in a village changes the mean income a lot and the median income hardly at all.