Mean Evaluation: Study with Video Lessons, Practice Problems & Examples

The standard deviation (s) quantifies the precision of data relative to the mean, with a smaller value indicating greater precision. The formula for standard deviation is $\sqrt{\frac{\sum x_i-\bar{x}{\left(}^{\right)}}{n-1}}$. Variance is the square of the standard deviation. While precision reflects measurement consistency, accuracy indicates closeness to a true value. Understanding these concepts is crucial for effective statistical analysis.

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.

Here’s what students ask on this topic:

What is the formula for standard deviation and how is it used in statistical analysis?

The formula for standard deviation (s) is:

$\sqrt{\frac{\sum {x-\overline{x}}^2}{n-1}}$

In this formula, $x$_i represents an individual measurement, $\overline{x}$ is the mean of the data set, and n is the number of measurements. Standard deviation quantifies the precision of data relative to the mean, with a smaller value indicating greater precision. It is used in statistical analysis to assess the variability or dispersion of a data set, helping to understand how spread out the measurements are from the mean.

What is the difference between precision and accuracy in statistical analysis?

Precision and accuracy are two important concepts in statistical analysis. Precision refers to the consistency of measurements, indicating how close the measurements are to each other. It is quantified by the standard deviation, with a smaller standard deviation indicating higher precision. Accuracy, on the other hand, refers to how close the measurements are to the true or actual value. A data set can be precise but not accurate if the measurements are close to each other but far from the true value. Conversely, a data set can be accurate but not precise if the measurements are close to the true value but spread out. Both precision and accuracy are crucial for reliable data analysis.

How is variance related to standard deviation?

Variance and standard deviation are closely related statistical measures. Variance is the square of the standard deviation. Mathematically, if the standard deviation is represented by s, then the variance (σ²) is given by:

$s^2$

Variance measures the average squared deviation of each data point from the mean, providing a sense of the data's spread. While standard deviation is in the same units as the data, variance is in squared units. Both metrics are used to assess the variability or dispersion of a data set, with variance being a key component in various statistical tests and analyses.

What is the coefficient of variation and how is it calculated?

The coefficient of variation (CV), also known as the relative standard deviation, is a measure of relative variability. It is calculated by dividing the standard deviation (s) by the mean (x̄) and multiplying by 100 to express it as a percentage:

$\frac{s}{\overline{x}}\times 100$

The CV is useful for comparing the degree of variation between different data sets, especially when the means are significantly different. A lower CV indicates less relative variability, while a higher CV indicates greater relative variability. It is particularly helpful in fields like finance and quality control where comparing variability across different scales is important.

What is the significance of the degree of freedom (n-1) in the standard deviation formula?

The degree of freedom (n-1) in the standard deviation formula accounts for the fact that we are estimating the population standard deviation from a sample. By using n-1 instead of n, we correct for the bias in the estimation of the population variance and standard deviation. This adjustment ensures that the sample variance is an unbiased estimator of the population variance. The degree of freedom represents the number of independent values that can vary in the data set after the mean has been calculated. It is crucial for accurate statistical analysis, especially when dealing with small sample sizes.