Statistical symbols & probability symbols (μ,σ,)
We use many kinds of variances, standard deviations, and standard errors in distributions of a mean, a proportion, a difference between two means, and a. Standard deviation is a square root of variance. Explanation: Variance. σ2=(1n)⋅ Σni=1(xi−¯x). Standard deviation. σ=√(1n)⋅Σni=1(xi−¯x). The standard deviation is expressed in the same units as the mean is, whereas the variance is expressed in squared units, but for looking at a.
That looks good and is the Mean Deviationbut what about this case: It also gives a value of 4, Even though the differences are more spread out!
So let us try squaring each difference and taking the square root at the end: The Standard Deviation is bigger when the differences are more spread out In fact this method is a similar idea to distance between pointsjust applied in a different way. And it is easier to use algebra on squares and square roots than absolute values, which makes the standard deviation easy to use in other areas of mathematics. Covariance and Correlation Covariance and correlation describe how two variables are related.
Variables are positively related if they move in the same direction.
Standard Deviation and Variance
Variables are inversely related if they move in opposite directions. Both covariance and correlation indicate whether variables are positively or inversely related. Correlation also tells you the degree to which the variables tend to move together.
You are probably already familiar with statements about covariance and correlation that appear in the news almost daily. For example, you might hear that as economic growth increases, stock market returns tend to increase as well. These variables are said to be positively related because they move in the same direction. You may also hear that as world oil production increases, gasoline prices fall.
- Standard Deviation and Variance
- Standard Deviation and Variance (1 of 2)
- Z-Stats / Basic Statistics
These variables are said to be negatively, or inversely, related because they move in opposite directions. The relationship between two variables can be illustrated in a graph. In the examples below, the graph on the left illustrates how the positive relationship between economic growth and market returns might appear.Standard Deviation Formula, Statistics, Variance, Sample and Population Mean
One common application of the variance is its use in the F-test to compare the variance of two methods and determine whether there is a statistically significant difference in the imprecision between the methods. In many applications, however, the SD is often preferred because it is expressed in the same concentration units as the data.
Using the SD, it is possible to predict the range of control values that should be observed if the method remains stable. As discussed in an earlier lesson, laboratorians often use the SD to impose "gates" on the expected normal distribution of control values. Normal or Gaussian distribution Traditionally, after the discussion of the mean, standard deviation, degrees of freedom, and variance, the next step was to describe the normal distribution a frequency polygon in terms of the standard deviation "gates.
This distribution shows the shape of a normal curve. Some authors call this polygon an error curve to illustrate that small errors from the mean occur more frequently than large ones.
Other authors refer to this curve as a probability distribution. Coefficient of variation Another way to describe the variation of a test is calculate the coefficient of variation, or CV. The CV expresses the variation as a percentage of the mean, and is calculated as follows: For example, the data from a replication experiment may show an SD of 4 units at a concentration of units and an SD of 8 units at a concentration of units. The CVs are 4. However, not all tests will demonstrate imprecision that is constant in terms of CV.
For some tests, the SD may be constant over the analytical range.
The CV also provides a general "feeling" about the performance of a method. However, you should look carefully at the mean value before judging a CV. At very low concentrations, the CV may be high and at high concentrations the CV may be low. For example, a bilirubin test with an SD of 0. Alternate formulae The lessons on Basic QC Practices cover these same terms see QC - The data calculationsbut use a different form of the equation for calculating cumulative or lot-to-date means and SDs.
Guidelines in the literature recommend that cumulative means and SDs be used in calculating control limits , therefore it is important to be able to perform these calculations. The idea is to add the xi and n terms from groups of data in order to calculate the mean of the combined groups.
The cumulative or lot-to-date standard deviation can be expressed as follows: This equation looks quite different from the prior equation in this lesson, but in reality, it is equivalent.
The cumulative standard deviation formula is derived from an SD formula called the Raw Score Formula. Instead of first calculating the mean or Xbar, the Raw Score Formula calculates Xbar inside the square root sign.
Oftentimes in reading about statistics, an unfamiliar formula may be presented.