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
On
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 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.
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.