00:01
Let's have some exercises involving standard deviations, variances, and normal distributions.
00:09
So there we go.
00:09
Let's go ahead and go through this.
00:10
So we have some data here, and we want to find the variance and standard deviation of it, of the sample.
00:20
That is.
00:21
So s squared is the variance, and that is given as the sum of all the x values, subtracted from the mean squared.
00:31
And take that number and divide it by n minus 1.
00:35
And the standard deviation s is given as the square root of the variance.
00:39
So once we find the variance, take the square root of it, and that will give us our standard deviation.
00:44
So the only other thing we need is x bar, the mean, and that is, of course, the mean we learn when we're in primary school.
00:52
In secondary school, here we go.
00:54
So it's x as equal to the sum of all the x values divided by the number of x values.
01:01
So let your spreadsheet do your work for you.
01:03
So first thing we do is find the mean.
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That's what we have here.
01:08
So add up the values.
01:09
We get 36.
01:11
Divide that by 9.
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We get our mean, which is 4.
01:14
That's our mean.
01:15
Our mean is 4 here.
01:19
So in this case, it's 4.
01:21
And then we take the, this is the x minus x bar squared.
01:29
So 2 minus the mean.
01:32
So 2 minus 4 squared is 4.
01:34
3 minus 4 is negative 1, squared is 1.
01:37
And do it all the way down the line.
01:38
Add those up we get 24 so in our work here this is going to be the variance the sum of those square differences is 24 and we're going to divide that by eight right because nine there's nine scores one two three four five six seven eight nine um so we divided by eight and that means our variance is three and then to get the standard deviation we take the square root of three so it's equal to the square root of 3, which is like approximately 1 .732.
02:22
All right.
02:23
All right, so let's get rid of this.
02:26
Because the next one involves, let's clean this up.
02:35
And these next ones are kind of fun because they kind of force us to think about these distributions as a whole.
02:40
Okay, so here we have the distribution where there are total of n scores, capital n, and the distribution has a mean of 50.
02:52
And a standard deviation, population standard deviation of 10.
02:56
So please note the difference between the sample standard deviation we're divided by n minus 1 versus the population standard deviation.
03:04
And this is just where we're dividing the sum of squares by n.
03:08
And if we have another five individuals who have a score of 50 or added to this distribution, what will happen to the mean? what about the standard deviation will increase or decrease? all right, so let's go ahead and go through this.
03:25
If we add these five individuals, our capital n will become 15.
03:30
Our mean is going to stay the same.
03:32
It doesn't add anything.
03:33
It's just, it's going to stay the same.
03:35
It's not going to change anything whatsoever.
03:38
It's going to stay 50, right? and that's because the mean is given as the sum of, right, the sum of the x's all over, this x bar is over the sum or the number.
03:55
So it would be over 10.
03:57
And we currently have the mean as 50, right? so then the sum of the values would be equal to 500.
04:14
And then what we're going to do is we're going to add five times 50.
04:22
So that's going to be, what's that, 250? so then this would become 250.
04:33
That we're going to add here.
04:34
So we'd add 250 to this.
04:37
So the new sum would be 750.
04:43
But then the new number to divide by would be 15.
04:47
And if we divide 750 by 15, we get 50.
04:56
So our mean stays the same.
05:00
The standard deviation, this changes.
05:02
This is going to become smaller.
05:07
This is going to decrease.
05:12
And the reason why is if we look at this, well, look at the formula, the standard deviation of the population, it's a slightly different formula than the one up here, but it's still the sum of the squared deviations about the mean.
05:31
So you take x bar.
05:33
I guess it would be mu, so it considers the population.
05:36
Then we're dividing by n this time.
05:38
That's the main difference.
05:39
We're divided by m .s .1.
05:40
It's n.
05:42
And so let's see what's happening here.
05:44
The difference is here, the x values minus the mean...