# Easy GRE question: Statistics

I'm not sure how to set this statistics problem when they give me a group of arbitrary values. Can someone help?

A group of 20 values has a mean of 85 and a median of 80. A different group of 30 values has a mean of 75 and a median of 72.

a) What is the mean of the 50 values?

b) What is the median of the 50 values?

Thank you

Let $v_i$ denote the $i$th value.

For the 20 values, the mean is given by

$$85 = \mu = \frac{v_1+v_2+v_3+\cdots+v_{20}}{20}.$$

You don't know the individual amount for each value, but you can easily figure out what the sum of the values is:

$$20\times 85 = v_1+v_2+\cdots + v_{20}.$$

Now, do the same thing for the 30 values, $u_1$ to $u_{30}$.

Add those two numbers together, and divide by 50:

$$\mu_{\textrm{all values}} = \frac{1}{20+30}\cdot\left[\underbrace{v_1+v_2+\cdots+v_{20}}_{20\times 85}+\underbrace{u_1+u_2+\cdots+u_{30}}_{30\times 75}\right].$$

You can do something similar to find the median.

You can indeed compute the new mean easily as explained in the answer of Arkamis. However you cannot compute the new median: an assumption is missing for that.

ex (defining the median as the average of the two median values for even numbers of values): you can take for

• group 1: 8 times '50', 60,70,90,9 times '120'
• group 2: 13 times '60',66,70,74, 14 times '90'

group 1 has average 85, median 80, whereas group 2 has average 75, median 72

The new median when you merge groups is 72=(70+74)/2

Now, if you replace 70,90 by 75,85 in group 1, the average /median of group 1 remain the same, but when you merge groups you obtain median 74.5=(74+75)/2

The idea is that to compute the median you need some information about how the values are distributed inside your two first groups.

The formula to find mean is Mean= (total number of values) ÷ (quantity of values)

So, as we know the mean and the quantity values of the first group is, 85 and 20. (It is given) So we can substitute the values into the formula and get the (total number of values) Same goes to the other group that contains 30 values... The trick of the question is... They gave you the values, mean and median, of (group of 20 values) and (group of 30 values). And asked the mean and median of the combined group. (Group of 20 values and group of 30 values) So workouts are...

85=(sum of x values)÷20 20(85)= (sum of x values) X= 1700

75=(sum of x values)÷30 30(75)=(sum of x values) X=2250

So... 30+20=50

So... Mean of the group of 50values is..

Mean= (1700+2250)÷50 = 3950÷50 = 79

And.. For question b).. The median cannot be identified with the given information in the question.