Question

Can anyone explain #24 ???

Answer 1

Answers:

a) mean = 7.5

b) how many values? 35

===================================================

Explanation:

Let's go over an example that's similar to this.

Consider the set {1,2,3,10}

Add up the numbers and divide by the sample size (4) to get the mean

(1+2+3+10)/4 = 16/4 = 4

The mean is 4

It is purely coincidental that the mean and sample size are the same value for this example.

-----------------------

The standard deviation is a measure of spread. It tells us how closely clustered the values are, or if the values are more spread out. The higher the standard deviation, the more spread out the values will be.

How can we measure this "spread"? Well we basically use the number line distance formula. In other words, we subtract each x value from the mean of 4.

1-4 = -3

2-4 = -2

3-4 = -1

10-4 = 6

But negative distance makes no sense. We could apply absolute value to each result, but a much better practice is to square each result

(-3)^2 = 9

(-2)^2 = 4

(-1)^2 = 1

(6)^2 = 36

Then add up all of those values to get 9+4+1+36 = 50 which is the Sum of the Squared Error (SSE). The larger the SSE, the more spread out the data items will be. In other words, a larger SSE tells us that the values are further from the mean.

What does this have to do with the formula your teacher mentioned?

Well the fancy looking E symbol is the Greek letter sigma to represent "sums". It tells us to add up a bunch of terms.

Writing something like $\sum (\text{x}-4)^2$ tells us to add up a bunch of terms in the form $(\text{x}-4)^2$

As you can probably see, the $(\text{x}-4)^2$ refers to the idea of us subtracting the mean (4) from each x data value, then squaring the result.

For your specific math question, replace the "4" with "7.5" and it's the same basic idea. You are subtracting the mean 7.5 from each x value, then squaring the result. Afterward you add up all those deviations to get the SSE.

-----------------------

After we've calculated the SSE, we divide it over n-1 to get the sample variance.

$\text{sample variance} = \frac{\text{SSE}}{n-1}$

In the example I posted above, we have n = 4 so n-1 = 4-1 = 3 is in the denominator.

For your specific question, n = 35 is the sample size.

Then we apply the square root to the sample variance to get the sample standard deviation.

$\text{sample standard deviation} = \sqrt{\text{sample variance}}$

or we can write it like this

$\text{sample standard deviation} = \sqrt{\frac{\text{SSE}}{n-1}}$

where,

$\text{SSE} = \sum(\text{x}-\text{mean})^2$

Side note: if you wanted the population standard deviation (instead of sample standard deviation), then replace n-1 with n. Everything else is the same.