Question

Lucas recorded his lunch expenditure each day for one week in the table below.
Day: Sunday ($4.85), Monday($5.10), Tuesday($5.50), Wednesday ($4.75), Thursday ($4.50), Friday($5.00) Saturday($6.00)

a. calculate the standard deviation using the formula
round to the nearest thousandth.

b. If the sample size increases but the sum of the squared differences stays the same, what will be the effect on the standard deviation?

Answer 1

@phi

Answer 2

@Mertsj

Answer 3

@Luis_Rivera

Answer 4

@godzgyrl2001

Answer 5

@bahrom7893

Answer 6

@katlin95 @geerky42 @blurbendy @e.mccormick

Answer 7

@Peter14

Answer 8

Do you understand the formula?

Answer 9

Not at all. I am completely lost.

Answer 10

With explanaitory notes:
http://0.tqn.com/d/statistics/1/0/M/-/-/-/standarddev.GIF

Answer 11

Does that help?

Answer 12

I get the first one but not the other two.

Answer 13

Sigma, \(\Sigma\), means sum.

Answer 14

OK. Let's jsut walk through it together.
So you need to start by finding the mean. Know how to do that?

Answer 15

Yeah, it's just the average right?

Answer 16

Yes. That is the arithmetic mean.

Answer 17

I got 5.1

Answer 18

In this formula that is \(\overline{x}\). I got 5.1 too. So we will remember that for now.

Answer 19

Next, we need the deviation of each value from the mean, that means for each item, or \(x_i\), we subtract 5.1 and write those new values down. The devations need to be squared, then added. That is the whole: \(\sum_1^n(x_i-\overline{x}})^2\) part.

Answer 20

okay, what next?

Answer 21

Okay that just confused me

Answer 22

Man that was a laggy bit....

OK, so. lets just start with the deveation part.

You have:
Sunday ($4.85),
Monday($5.10),
Tuesday($5.50),
Wednesday ($4.75),
Thursday ($4.50),
Friday($5.00)
Saturday($6.00)

For each one, what is that value \(-5.1\)

Answer 23

-.25
0
.4
-.35
-0.6
-.1
.9?

Answer 24

Devation means, "How far away from the mean." That is why we are subtracting the mean.

Yes. Those are the deveations. Now, square each one of them.

Answer 25

Then add up those values. The squared ones.
0.0625+
0+
0.16+
0.1225+
0.36+
0.01+
0.81=
1.525

I think you know basic math, so there is that one for you.

Answer 26

That entire process was:
Find the mean.
Find each deviation from the mean.
Square each deveation.
Sum the squares.

In math, that is done in shorthand like this:\[\sum_1^n(x_i-\overline{x})^2\]All that says is do what we just did.

OK?

That is probably the most complex part of this all. There is still some left, but that was the big part.

Answer 27

The site has been bumping people off for a while, and it looks like you got zapped.

OK, we started with this formula:\[S_x=\sqrt{\frac{\sum_1^n(x_i-\overline{x})^2}{n-1}}\]
Do far we found that \(\sum_1^n(x_i-\overline{x})^2=1.525\). So if we put that into the formula it becomes:\[S_x=\sqrt{\frac{1.525}{n-1}}\]
n is the number of entries, so 7 in this case.

That should be enough info for you to finish it. You got the hard part before getting zapped.

Answer 28

my brain is processing sorry lol.

Answer 29

You made it back! Wheee.

Answer 30

Give that all a read over, then see if you get it or if you have questions.

Answer 31

thank you so much!!!!!!!!

Answer 32

I take it that means it made sense. Like I said, the sigma notation sum is the conceptually hard part. The math of a sigma sum is not hard. We did it. But until you know the concept, how can you do the math?

Answer 33

Another thing to notice is that the absolute value of each of the devations is not too far from the answer. The answer is what is called the Standard Deveation. When you realize that the Standard Deveation is made of the individual deveations, the name makes mroe sense.

Answer 34

So what about the second part?

Answer 35

Well, you see the [\S_x=\sqrt{\frac{1.525}{n-1}}\]Can you solve that now?

Answer 36

oops. It neded to start that with slash brace. hehe. \[S_x=\sqrt{\frac{1.525}{n-1}}\]

Answer 37

I got part a but I'm not sure how to do part b.

Answer 38

Well, n is the number of days, so 7. Put that in for n. After that, it is just math.

Answer 39

I don't understand...

Answer 40

Just to clarify, the answer to part a is 0.46675 right?

Answer 41

OH! That part b... ooops.... I had not scrolled back all the way up....
I got 0.5041494487 for a.

Answer 42

Hmmmmmmm. how?

Answer 43

Oh wait! the formula I have on the bottom has n not n-1

Answer 44

\[S_x=\sqrt{\frac{1.525}{n-1}}\implies S_x=\sqrt{\frac{1.525}{7-1}} \implies S_x=\sqrt{\frac{1.525}{6}} \implies \\
\implies S_x=\sqrt{0.2541666667} \implies S_x=0.5041494487\]

Answer 45

OH. OK. The difference is for globals vs. locals. They aare both SD formulas. the one you linked in the picture is the n-1 version, which would be a local SD.

Answer 46

So I'm actually looking for the global SD?

Answer 47

It does not matter what it is called, as long as you use the right formula. If that formula you linked is what they designated, you MUST use n-1.

Answer 48

my worksheet has n on bottom

Answer 49

I got 0.5 for the answer.

Answer 50

OK, with n, I got 0.4667516929, which in thousanths is .467

Answer 51

Now, the sample size is n. So if n is larger... we did n-1, which would be smaller. So n being larger is 8....

Answer 52

okay now what about the 2nd part?

Answer 53

Try it with like n=8 or n=10. See what happens.

Answer 54

but how does the sample size get bigger and the sum of the squared differences stay the same
?

Answer 55

it is a theorhetical question.

Answer 56

I stink at this.):

Answer 57

would the SD stay the same?

Answer 58

It is a "what if" they pulled out of their anatomy in an effort to get the students to see something... even if the student is smart enough to say, "But that would change thing, so it is a bad question!" they still want it ansewered.

No, it would chamnge. Just change n and see.

Answer 59

I don't know how to add more numbers but keep the sum the same.

Answer 60

No, no. Just change it like this:
\[S_x=\sqrt{\frac{1.525}{10}}\]

Answer 61

The problem is you were too smart for the question! It is a stupid question and it does not make sense. But that is what they want you to do.

Answer 62

Oh! haha thank you! so it should go down right?

Answer 63

cause I got 0.391

Answer 64

yah. SD is gets smaller when that happens. Also, SD gets more accurate when the number gets larger. That second concept is something they are probably leading up to, but at this point they want you to see that it gets smaller.

Answer 65

Thank you so much! I will probably tag you in more tomorrow!

Answer 66

If the site works... this has been.... ugly tonight.

Answer 67

Definitely. One more thing, it asks for a real world application?

Answer 68

Wheee... got logged out again.

Well, find a range of money needed for a lunch budget... hmmm.... Stats is more along the lines of finding how one person or a small group in comparison to normal.

Answer 69

Alright, thanks for all your help!

Answer 70

np. Have fun!