Question

1a.) If a constant c is added to each xi in a sample, yielding yi = xi + c, how do the sample mean and median of the yis relate to the mean and median of the xis? Verify your conjectures. Verify using a made-up example.

1b.) If each xi is multiplied by a constant c, yielding yi=cxi, answer the question of part (a). Again, verify your conjectures. Verify using a made-up example.

Answer 1

1a)

Suppose the constant is added to each observation of the sample then the mean and median will also increase by the same constant.

For example: Suppose a data set has values 8, 12, 14, 15 and 21

The mean and median of this data set are

Median is the middlemost observation of the arranged data.

The data is arranged in increasing order as 8, 12, 14, 15 and 21

So the median is the third observation that is 14

So mean = 14 and Median = 14

Now suppose constant 5 is added to this then the data set becomes 13, 17, 19, 20, 26

The mean and median are

That is new mean = old mean + constant = 14 + 5 = 19

Median is the third observation that is 19

That is new median = old median + constant = 14 + 5 = 19

Therefore the mean and median are also increase by the same constant.

1b)

Suppose each observation of the multiplied by a constant then the mean and median are also multiplied by the same constant.

Example: Suppose the data values 2, 4, 8, 10 and 14

The mean and median for this data set are

And the median is the middlemost observation of the arranged data set.

The third observation is middlemost among the 5.

So median = 8

Now multiply the data set by constant 10 then the observations are 20, 40, 80, 100, 140

The mean and median of the new data set are,

New mean = old mean * constant = 7.6 * 10 = 76

And the median is the middlemost observation that is third observation which is 80

So new median = old median * constant = 8 * 10 = 80

Therefore the mean and median are also multiplied by same constant.