nusmavrik wrote:
A set of data consists of 4 integers. What is the standard deviation for this set of data?
(1) The average (arithmetic mean) and the median are both 2.
(2) The mode is 2 and the range is also 2.
Hi!
In order to calculate the standard deviation of a set, we need to know two things:
1) the number of terms; and
2) the exact spacing of the set.
Of course, if you know all the terms of the set, you have both things that you need.
From the stem, we know that there are 4 terms. So, in order to answer the question, we need the exact spacing of the set (or the 4 terms).
Now let's look at the statements:
(1) average and median = 2.
We can pick numbers to show that we can get more than 1 answer to the question. Our set could be:
{2, 2, 2, 2} or {1, 2, 2, 3},
each of which has a different standard deviation: insufficient.
(2) mode and range = 2.
Again, we can pick numbers to show that we can get more than 1 answer. Our set could be:
{1, 2, 2, 3} or {2, 2, 2, 4},
each of which has a different standard deviation: insufficient.
Taken together, we know that average=median=mode=range=2.
Since the mode and the median are both 2, the two middle terms in the set must be 2. Our set is now:
{x, 2, 2, y}
Since the average is also 2, the set must sum to 8, which means the first and fourth terms must sum to 4, i.e. x+y=4.
Since the range is 2, the first and fourth terms must be 2 apart, i.e. y-x = 2.
If x + y = 4 AND x - y = 2, we can solve for x and y. (Note: we neither need nor want to actually solve, just knowing that we can solve is enough.)
Accordingly, we know all 4 terms in the set and can answer ANY question about the set: together sufficient, choose (C).