Mark B. answered • 11/19/17
Tutor
New to Wyzant
PhD Candidate in Psychology: Experienced Math, Statistics, Tutor
Hello Alex,
Okay, what I personally advise is that you make four blanks to represent the values. Remember you are seeking four numbers or values.
____ ____ _____ _____
Now, let me ask you a few questions:
First, you know the median is six, correct?
And, we know when the distribution of the values is even, as is the case here, we take the two middle values and divide them by two, correct? If you were working with an odd number of values, you would simply take the middle value after rank ordering your data. Do you understand so far? Great. Now, with that out of the way, we KNOW with utter certainty that the two middle values when added together and divided by two WILL be six.
Before proceeding further, let's remember some other characteristics and definitions to assist us, shall we?
We KNOW the range is the value of the maximum minus the value of the minimum and this has been identified as being 6, correct?
We KNOW the mean is the summation of the values divided by the number of values in the data set, correct? Great!
Therefore, where I personally would continue with this problem, is I would look at specific values for those two middle numbers which satisfy the above conditions I just mentioned regarding the mean and range, okay?
So, let's look at some values which when added together and divided by two provide a value of 6 for the median, fair enough?
6, and 6 work don't they? That produces a median of 6, right? Yes. But what about the two other values?
____ 6 6 ____
Which numbers can be on either side of these two middle values and still meet the conditions?
Let's try working on the RANGE first, shall we? Remember whichever two values we use when added to the two values of 6 and divided by 4 must equal 6, correct?
We DO know another thing about these two values too, don't we? The first value in the set MUST be either equal to or less than six, and the highest value must be in line with the being no more than six more than the lowest value to meet the criteria of the range, correct?
That means your FIRST value MUST BE A ONE, TWO OR THREE, FOUR OR FIVE BECAUSE the values are rank ordered. Then remember, once you use the value of 1, 2, 3, 4, or 5, you MUST consider whether you get the mean and range of six when solving for both.
So, try a 1. What happens when we do?
We have a set of values 1, 6, 6, meaning our last value MUST be 11 in order to meet the criteria of the mean being SIX correct? THIS WILL NOT WORK. Why? The range would then be 10.
Let's try a 2. What happens when we try that value? Well, the set of values; 2, 6, 6, MUST have a 10 in order for the mean to be SIX, correct? The problem is our range would be 8, correct? This won't work.
Let's try a 3, shall we? What happens when we try that value? Well, the set of values; 3, 6, 6 MUST have a 9 in order for the mean to be SIX, correct? Guess what? When we use the value of 9, with the value of 3, and added to the pre-existing two values of 6, we get a MEAN of 6. Better yet? Our range is the maximum value minus the minimum value which is 9 -3 which DOES equal 6.
THIS set of values, [3, 6, 6, 9] then satisfies ALL of the criteria. The median is six, the mean is six, and the range is 6.
Alex, I honestly do not know any equations other than what I have provided you - or a computer program - which would allow you to determine these missing values. The only way you can do so is through some logic and trial and error.
Please let me know if this all makes sense to you and whether you have any additional questions for me regarding the manner in which I resolved the problem. If there is an equation out there, great. But some good old fashioned thinking while understanding what the mean, median and range are, while solving gets you much farther down the road towards understanding statistics.
I hope you enjoy the rest of your Sunday and have a great short week at School.
~Mark B.
Okay, what I personally advise is that you make four blanks to represent the values. Remember you are seeking four numbers or values.
____ ____ _____ _____
Now, let me ask you a few questions:
First, you know the median is six, correct?
And, we know when the distribution of the values is even, as is the case here, we take the two middle values and divide them by two, correct? If you were working with an odd number of values, you would simply take the middle value after rank ordering your data. Do you understand so far? Great. Now, with that out of the way, we KNOW with utter certainty that the two middle values when added together and divided by two WILL be six.
Before proceeding further, let's remember some other characteristics and definitions to assist us, shall we?
We KNOW the range is the value of the maximum minus the value of the minimum and this has been identified as being 6, correct?
We KNOW the mean is the summation of the values divided by the number of values in the data set, correct? Great!
Therefore, where I personally would continue with this problem, is I would look at specific values for those two middle numbers which satisfy the above conditions I just mentioned regarding the mean and range, okay?
So, let's look at some values which when added together and divided by two provide a value of 6 for the median, fair enough?
6, and 6 work don't they? That produces a median of 6, right? Yes. But what about the two other values?
____ 6 6 ____
Which numbers can be on either side of these two middle values and still meet the conditions?
Let's try working on the RANGE first, shall we? Remember whichever two values we use when added to the two values of 6 and divided by 4 must equal 6, correct?
We DO know another thing about these two values too, don't we? The first value in the set MUST be either equal to or less than six, and the highest value must be in line with the being no more than six more than the lowest value to meet the criteria of the range, correct?
That means your FIRST value MUST BE A ONE, TWO OR THREE, FOUR OR FIVE BECAUSE the values are rank ordered. Then remember, once you use the value of 1, 2, 3, 4, or 5, you MUST consider whether you get the mean and range of six when solving for both.
So, try a 1. What happens when we do?
We have a set of values 1, 6, 6, meaning our last value MUST be 11 in order to meet the criteria of the mean being SIX correct? THIS WILL NOT WORK. Why? The range would then be 10.
Let's try a 2. What happens when we try that value? Well, the set of values; 2, 6, 6, MUST have a 10 in order for the mean to be SIX, correct? The problem is our range would be 8, correct? This won't work.
Let's try a 3, shall we? What happens when we try that value? Well, the set of values; 3, 6, 6 MUST have a 9 in order for the mean to be SIX, correct? Guess what? When we use the value of 9, with the value of 3, and added to the pre-existing two values of 6, we get a MEAN of 6. Better yet? Our range is the maximum value minus the minimum value which is 9 -3 which DOES equal 6.
THIS set of values, [3, 6, 6, 9] then satisfies ALL of the criteria. The median is six, the mean is six, and the range is 6.
Alex, I honestly do not know any equations other than what I have provided you - or a computer program - which would allow you to determine these missing values. The only way you can do so is through some logic and trial and error.
Please let me know if this all makes sense to you and whether you have any additional questions for me regarding the manner in which I resolved the problem. If there is an equation out there, great. But some good old fashioned thinking while understanding what the mean, median and range are, while solving gets you much farther down the road towards understanding statistics.
I hope you enjoy the rest of your Sunday and have a great short week at School.
~Mark B.
By the way, another four values WILL work. In fact, there are "several" data sets which will satisfy the problem you gave me. Can you figure one of those sets out using the same method I just used to determine those values?
Mark M.
11/19/17