Thread: Construct a data set with mean/pop. standard dev given

1. Construct a data set with mean/pop. standard dev given

So I've been given a problem in my Bus. Stats. class and I'm not sure how to approach it after looking at it for several hours.

Construct a data set with the following characteristics:
a. one data point is 70
b. the mean is exactly 61
c. the population standard deviation is exactly 3

He gave a hint saying: use the defining formula. You may use as many data points as you wish.

I really don't know where to start and have looked through my notes, through a help book, and all over online with no luck.

2. G'day muttonchops

This is a neat little question

Here's a start

consider one data point is 70. To add one more point to the set and get the mean to be 61 the other point must be 52

i.e. mean = $\displaystyle \frac{70+52}{2} = 61$

Now we should ask what is the standard deviation for this set? Well its $\displaystyle \approx 12.73$

So that's is no good! Now adding another data point to the set might change the mean unless the data point is always 61

i.e. mean = $\displaystyle \frac{70+52+61}{3} = 61$

Now we should ask what is the standard deviation for this set? Well its $\displaystyle = 9$

Adding a point in the neighbourhood of the mean reduces the spread.

Now keep adding 61 as as additional data point, the standard deviation will reduce everytime until it is 3!

Does this make sense?

3. [tex]How did you get a standard deviation of 9 for the set of 52,61,70?
I did({52+61+70)/{3} = 61

Then I did (52-61)^2 + (61-61)^2 + (70-61)^2 = 162

162/3 = 54 sqrt 54 = 7.34 for standard deviation

4. $\displaystyle s = \sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}$

$\displaystyle s = \sqrt{\frac{162}{3-1}}=9$

5. I did what you said, though, and came up with 18 numbers
52, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 70

That's 16 61's, 18 numbers total. 61 is the mean. the standard deviation is 3, and 70 is a point.

That sound right?

6. Originally Posted by pickslides
$\displaystyle s = \sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}$

$\displaystyle s = \sqrt{\frac{162}{3-1}}=9$

isnt that the sample standard deviation formula?

I have in my notes the Population Standard Deviation formula as

$\displaystyle \sqrt{\frac{\sum (x-\bar{x})^2}{N}}$

7. I had 19 numbers, 70,52, 61 (17 of them!)

Gave me $\displaystyle \mu = 61, \sigma=3$

8. yea i think the difference is due to the different formulas we are using. the problem asked me to use the population standard deviation. the one you are quoting is the sample standard deviation. its a difference of N or n-1 for the denominator

9. Originally Posted by muttonchops
yea i think the difference is due to the different formulas we are using. the problem asked me to use the population standard deviation. the one you are quoting is the sample standard deviation. its a difference of N or n-1 for the denominator

I agree, but your is your standard deviation exactly 3? or is it 3.09?

I would check with your lecturer/teacher. I think they have advised on the wrong formula.

10. thanks again for the help. I spent all day trying to figure it out and you explained it very well.

11. Originally Posted by muttonchops
So I've been given a problem in my Bus. Stats. class and I'm not sure how to approach it after looking at it for several hours.

Construct a data set with the following characteristics:
a. one data point is 70
b. the mean is exactly 61
c. the population standard deviation is exactly 3

He gave a hint saying: use the defining formula. You may use as many data points as you wish.

I really don't know where to start and have looked through my notes, through a help book, and all over online with no luck.
This should be the easiest problem you start with. Do you know the definitions of "mean" and standard deviation. Find those first. You know that your mean must be exactly, 61, and that your standard deviation must be exactly 3. Using the definition of mean with one data point equal to 70, should make figuring out other values easy.

The only real "problem" here is making sure that the data points you select for the mean also end up giving you a standard deviation of 3.