Mean Evaluation - Online Tutor, Practice Problems & Exam Prep

Mean Evaluation

So here we're going to say that our standard deviation measures how close data results are in relation to the mean or average value. Basically, the smaller your standard deviation is, the more precise your measurements will be in relation to the mean or average value. So here, the formula for standard deviation is \( s \) which stands for standard deviation equals the square root. And here we have the summation of our measurements minus our average squared, divided by \( n - 1 \). In terms of this equation, we're going to say here that \( x_i \) represents an individual measurement that we're undertaking in terms of our data set.

We're going to say that our average or mean value is represented by \( \overline{x} \). Variance is just our standard deviation squared. Later on, when we get more into statistical analysis, we'll see that the \( F \)-test has a close relationship to the variance of our calculations. Next, we have \( n \) which represents our numbers of measurements, and \( n - 1 \) represents our degree of freedom. Finally, we have our relative standard deviation also called our coefficient of variation.

That is just our standard deviation divided by our mean or average value times 100. So, at some point, we're going to run to using one of these variables in terms of the standard deviation equation. Just remember, the smaller your standard deviation is, the more precise all your measurements are within your data set. Now, your measurements can be precise but that doesn't necessarily mean they will be accurate. Remember, accuracy is how close you are to a true value.

Your measurements themselves may be close to one another but still far off from the actual true value, so the accuracy may not be good. Now that you've known the basics of standard deviation, we'll take a look at the example left below. Click on the next video and see how I approach this question which deals with standard deviation.

Mean Evaluation

Here it says, "Data below gives the volumes obtained by a chemist from the use of a pipette. Determine the standard deviation. We have our measurements of 24.9 ml, 25 ml, 24.8 ml, 24.6 ml twice and then 24.3 milliliters." All right. So, what we're going to do first is we're going to fill out this chart.

We have each of our measurements which represents our volumes here. So that's 24.9, 25, 24.8, 24.6, 24.6, and then 24.3. Here we would get the mean or average. So remember, the average would just be each one of these numbers added up together, then divided by the total number of measurements. Okay, so you would do 24.9 plus 25.0 plus 24.8 as well as the others and divide it by the total number of measurements, which is 6.

That will give you at the end 24.7. So that represents our mean or average value. Now here we're going to do the difference from the mean. So each one of these measurements will subtract from the mean. So that gives me 0.2 here.

Gives me 0.3, 0.1, -0.1, -0.1, and -0.4. Then we do the difference from the mean squared. So basically, we're squaring each one of these.

Okay, and squaring each of them, this one gives us 0.04, 0.09, 0.01, 0.01, 0.01, 0.16. So now, we would add them all up together. This summation value here means basically I am adding up all of these totals together.

Okay, so that's all it means. I'm just adding up all of them together. And when I add them all up together, it gives me 0.32. So now we need to finally figure out our standard deviation.

So remember, your standard deviation equals the square root of the summation of each measurement minus the average, and then you square that divided by n minus 1. We found out that this top portion is 0.32. N is the number of measurements, which is 6 volumes that we had initially, minus 1. So then here inside that'll give me 0.064. So your standard deviation will be approximately 0.253.

Since each one of my volumes here has 3 significant figures, we'll just go with 3 significant figures at the end. So my standard deviation here is 0.253. Again, remember, the smaller your standard deviation is, the more precise or the closeness each of your measurements have to one another. Again, this does not necessarily mean that they are accurate. We'd have to compare these values here to some true value.

From there, then we'd be able to determine if it's accurate or not. All we can tell at this point is that our standard deviation is pretty small. The numbers are very close to one another, so there is some precision involved.