Calculate Range, Average Deviation, and Standard Deviation

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Joychap

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Can somebody please help me with this problem because I am really lost about how to do it?

Calculate the range, average deviation, and standard deviation for the following five distributions:

A. 1, 2, 3, 4, 5, 6, 7, 8, 9

B. -4, -3, -2, -1, 0, 1, 2, 3, 4

C. 10, 20, 30, 40, 50, 60, 70, 80, 90

D. 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

E. 100, 200, 300, 400, 500, 600, 700, 800, 900
 
Do you know the definitions of "range, average deviation, and standard deviation "?
 
Response to Range, Average Deviation, and Standard Deviation

Yes, I know what the definitions are but that doesn't explain to me exactly how to go about and solve the math problems. The definition of range is a measure of variation; the difference between the lowest and the highest scores in a distribution. The definition of standard deviation is a measure of variation; the average difference between the scores in the distribution and the mean or central point of the distribution, or, more precisely, the square root of the average squared deviation from the mean. The definition of average deviation is an alternative measure of variation that, like the standard deviation, indicates the average difference between the scores in a distribution and the mean of the distribution. This doesn't help me really to understand how to go about doing the problems and so I am quite lost on where to start and totally confused.

jsterkel said:
Do you know the definitions of "range, average deviation, and standard deviation "?
Click to expand...
 
Joychap said:
Calculate the range, average deviation, and standard deviation for the following five distributions:

A. 1, 2, 3, 4, 5, 6, 7, 8, 9
Click to expand...

Let's just focus on this one for now.

What is the range ("the difference between the lowest and the highest scores in a distribution")

What is the mean?
 
Calculating Range, Standard Deviation, Average Deviation

Please tell me if I am correct in this. In order to get the mean, I do as follows:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

45/9 = 5

Is 5 the mean?

As for range I am not sure because it says difference between lowest and highest scores in a distribution

1, 2, 3, 4, 5, 6, 7, 8, 9

I know the difference means to subtract.

The difference between the lowest and highest score would be 1 - 9 = -8.
The difference between the highest and lowest score would be 9 - 1 = 8.

Can you tell me which one of these are correct? Thanks for your patience with me.

jsterkel said:
Let's just focus on this one for now.

What is the range ("the difference between the lowest and the highest scores in a distribution")

What is the mean?
Click to expand...
 
Yes, you are correct, the mean is 5 and the range is 8.

Now, lets look at average deviation.

We take each entry, and find out how far it is from the mean.

So, 9 - 5 = 4.

Can you find the others? 8-5? 7-5? etc.

The tricky part is that this "distance" is always positive (called the absolute value). So we can look at this as 5-1 = 4. The more correct way to look at this is "the absolute value of 1-5" which is the absolute value of -4 which is 4.

We add up all of the differences, and then find the mean of them, just like you found the mean the first time.

Your answer will be the average deviation.
 
9-5 = 4, 8-5 = 3, 7-5 = 2, 6-5 = 1, 5-5 = 0
Okay you say the distance should be positive when trying to figure out the average deviation, so with the rest I came up with the following:
4 -5 = -1 (absolute value of 4-5 = -1, so absolute value of -1 =1).
3- 5 = -2 (absolute value of 3-5 = -2, so absolute value of -2 =2).
2- 5 = -3 (absolute value of 2-5 = -3, so absolute value of -3 = 3).
1- 5 = -4 (absolute value of 1-5 = -4, so absolute value of -4 = 4).

You said to add the differences, so here is what I did as follows:
4 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 4 = 20
20 / 9 = 2.222222222

Is this the average deviation? Can you tell me if I did this correctly?

jsterkel said:
Yes, you are correct, the mean is 5 and the range is 8.

Now, lets look at average deviation.

We take each entry, and find out how far it is from the mean.

So, 9 - 5 = 4.

Can you find the others? 8-5? 7-5? etc.

The tricky part is that this "distance" is always positive (called the absolute value). So we can look at this as 5-1 = 4. The more correct way to look at this is "the absolute value of 1-5" which is the absolute value of -4 which is 4.

We add up all of the differences, and then find the mean of them, just like you found the mean the first time.

Your answer will be the average deviation.
Click to expand...
 
Exactly! The average deviation is 2.22222222....

Now, let's look at standard deviation. We are almost there.

There is one tricky part here. Are we calculating standard deviation of a population, or a sample?

For instance, if it was test scores for a statistics class, and we sampled 10 tests of the 100 students, we would use a slightly different formula.

I'm going to ASSUME that this list of numbers is the entire population.

We will use the values you found for each difference in the last problem. We will square each number, add them together, divide by the number of data points, and take the square root.

Here is the list: 4, 3, 2, 1, 0, 1, 2, 3, 4

Square each value, add them together, divide by 9 (because there are 9 values in the list), and take the square root of the sum. The result is the standard deviation.
 
The list of numbers are: 4, 3, 2, 1, 0, -1, -2, -3, -4
The numbers squared are: 4 squared, 3 squared, 2 squared, 1 squared, 0 squared, -1 squared, -2, squared, -3 squared, and -4 squared
Answers to numbers squared are: 16, 9, 4, 1, 0, 1, 4, 9, 16
Add the following numbers together are: 16 + 9 + 4 + 1 + 0 + 1 + 4 + 9 + 16 = 60
Total divided by 9 (because of 9 numbers in list): 60 / 9 = 6.666666667
Square Root of 6.666666667: 2.58199

Did I do this correct? Is this answer supposed to be the Standard Deviation? Please let me know if any corrections need to be made.

jsterkel said:
Exactly! The average deviation is 2.22222222....

Now, let's look at standard deviation. We are almost there.

There is one tricky part here. Are we calculating standard deviation of a population, or a sample?

For instance, if it was test scores for a statistics class, and we sampled 10 tests of the 100 students, we would use a slightly different formula.

I'm going to ASSUME that this list of numbers is the entire population.

We will use the values you found for each difference in the last problem. We will square each number, add them together, divide by the number of data points, and take the square root.

Here is the list: 4, 3, 2, 1, 0, 1, 2, 3, 4

Square each value, add them together, divide by 9 (because there are 9 values in the list), and take the square root of the sum. The result is the standard deviation.
Click to expand...
 
Calculate Range, Standard Deviation, and Average Deviation

Thanks for all of your help. I hope that your able to help me with problems that I need help on if I have any more questions.

jsterkel said:
Correct, the standard deviation is 2.58199!
Click to expand...
 
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