A set of data consists of 4 integers. What is the standard

Question

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.

babusona · Accepted Answer

A set of data consists of 4 integers. What is the standard deviation of this data set?

(1) The arithmetic mean and the median are both 2
(2) The mode is 2, and the range is also 2

skovinsky · Answer

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).

praveenism · Answer

Its C.
For any question relating to SD, we need to figure out Range of Data Sets.
The option A gives that middle 2 terms are only 2 & 2. The second gives me information about the first and last data points. Possible set for this question
1,2,2,3.

Bunuel · Answer

A set of data consists of 4 integers. What is the standard deviation of this data set?

(1) The arithmetic mean and the median are both 2 --> if the set is {2, 2, 2, 2} then the SD=0 but if the set is {0, 2, 2, 4} then the SD>0. Not sufficient.

Notice that from this statement we know that since the mean of 4 integers is 2 then the sum of those integers is 2*4=8

(2) The mode is 2, and the range is also 2--> the standard deviation of {0, 2, 2, 2} is different from the standard deviation of {1, 2, 2, 3}. Not sufficient.

(1)+(2) We have that mean=median=mode=range=2. Now, median=mode=2 means that the two middle terms must be 2, so our set is {a, 2, 2, b}. Next, since range=2 --> b-a=2 and since mean=2 --> a+2+2+b=8 --> a+b=4. We can solve for a and b, so we'll have all terms in the set, hence we can calculate the standard deviation. Sufficient.

Answer: C.

babusona · Answer

Thanks Bunuel...That helps....

mvictor · Answer

1. 
say we have 0; 2; 2; 4 - mean is 2, and median is 2 - we have one SD
or 2; 2; 2; 2 - mean is 2, median is 2, and SD is 0.
1 alone not sufficient.

2. mode is 2, so most likely, we have at least two of 2's.
we can have 0; 2; 2; 2 - mode is 2, and range is 2 and one SD.
or we can have 1; 2; 2; 3 - mode is 2, and range is 2, but SD is different 
2 alone is not sufficient.

1+2
only option is 1; 2; 2; 3 - otherwise, we would not have range/mean/median equal to 2.

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A set of data consists of 4 integers. What is the standard

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