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. So 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 \( x_i \) here 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. 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. So, click on the next video and see how I approach this question which deals with standard deviation.

Mean Evaluation

So here it says the 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 ml. 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 calculate the difference from the mean. So each one of these measurements will be subtracted from the mean. That gives me 0.2 here, 0.3 there, 0.1, and so forth. The differences here would give me negative 0.1, negative 0.1, and negative 0.4. Then we square each of these values, giving us 0.042, 0.092, 0.012, 0.012, 0.012, 0.162. So now, we add all these values together. This summation here means I am adding up all of these totals together. When we add them all up together, it gives us 0.32.

Now we need to finally figure out our standard deviation. Remember, your standard deviation equals 0.325, where 5 is n - 1 since n = 6 which is the total number of measurements we had initially. Within that, it would give us 0.064. So your standard deviation will be approximately 0.253 as each of our volume measurements had 3 significant figures, hence we round to that. Again, remember, the smaller your standard deviation, the greater the precision or closeness of each of your measurements to one another. Again, this does not necessarily mean that they are accurate. We would need to compare these values to some true value to determine accuracy. All we can conclude at this point is that our standard deviation is quite small, indicating some degree of precision in the measurements.