In this multi-part blog series, we will cover various topics relevant to the Federal Contractor community as it relates to statistical analyses. We begin our series with a topic that is often misunderstood by both practitioners and enforcement agencies. This past year we heard some crazy interpretations regarding the meaning of a statistically significant standard deviation test.
Our favorite was:. What would the numbers look like across ten requisitions? When a difference between two groups is statistically significant e. The greater the number of standard deviations, the less likely we are to believe the difference is due to chance. Some things to keep in mind:. Stay tuned for additional blogs. The range is measured in the same units as the variable of reference and, thus, has a direct interpretation as such.
This can be useful when comparing similar variables but of little use when comparing variables measured in different units. However, because the information the range provides is rather limited, it is seldom used in statistical analyses. For example, if you read that the age range of two groups of students is 3 in one group and 7 in another, then you know that the second group is more spread out there is a difference of seven years between the youngest and the oldest student than the first which only sports a difference of three years between the youngest and the oldest student.
The mid-range of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set, defined as:. The mid-range is the midpoint of the range; as such, it is a measure of central tendency. The mid-range is rarely used in practical statistical analysis, as it lacks efficiency as an estimator for most distributions of interest because it ignores all intermediate points.
The mid-range also lacks robustness, as outliers change it significantly. Indeed, it is one of the least efficient and least robust statistics. Variance is the sum of the probabilities that various outcomes will occur multiplied by the squared deviations from the average of the random variable.
When describing data, it is helpful and in some cases necessary to determine the spread of a distribution. In describing a complete population, the data represents all the elements of the population. When determining the spread of the population, we want to know a measure of the possible distances between the data and the population mean. These distances are known as deviations. The variance of a data set measures the average square of these deviations.
More specifically, the variance is the sum of the probabilities that various outcomes will occur multiplied by the squared deviations from the average of the random variable. When trying to determine the risk associated with a given set of options, the variance is a very useful tool. Calculating the variance begins with finding the mean.
Once the mean is known, the variance is calculated by finding the average squared deviation of each number in the sample from the mean. For the numbers 1, 2, 3, 4, and 5, the mean is 3. The calculation for finding the mean is as follows:. Once the mean is known, the variance can be calculated.
The variance for the above set of numbers is:. A clear distinction should be made between dealing with the population or with a sample from it. When dealing with the complete population the population variance is a constant, a parameter which helps to describe the population. When dealing with a sample from the population the sample variance is actually a random variable, whose value differs from sample to sample.
Population of Cheetahs : The population variance can be very helpful in analyzing data of various wildlife populations. Standard deviation is a measure of the average distance between the values of the data in the set and the mean. Since the variance is a squared quantity, it cannot be directly compared to the data values or the mean value of a data set. It is therefore more useful to have a quantity that is the square root of the variance.
The standard error is an estimate of how close to the population mean your sample mean is likely to be, whereas the standard deviation is the degree to which individuals within the sample differ from the sample mean. This quantity is known as the standard deviation. More precisely, it is a measure of the average distance between the values of the data in the set and the mean. A low standard deviation indicates that the data points tend to be very close to the mean; a high standard deviation indicates that the data points are spread out over a large range of values.
A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data. In statistics, the standard deviation is the most common measure of statistical dispersion.
However, in addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. To calculate the population standard deviation, first compute the difference of each data point from the mean, and square the result of each:. This quantity is the population standard deviation, and is equal to the square root of the variance.
The formula is valid only if the eight values we began with form the complete population. In cases where the standard deviation of an entire population cannot be found, it is estimated by examining a random sample taken from the population and computing a statistic of the sample. Unlike the estimation of the population mean, for which the sample mean is a simple estimator with many desirable properties unbiased, efficient, maximum likelihood , there is no single estimator for the standard deviation with all these properties.
Therefore, unbiased estimation of standard deviation is a very technically involved problem. However, other estimators are better in other respects:. The mean and the standard deviation of a set of data are usually reported together.
This is because the standard deviation from the mean is smaller than from any other point. Variability can also be measured by the coefficient of variation, which is the ratio of the standard deviation to the mean. Often, we want some information about the precision of the mean we obtained.
We can obtain this by determining the standard deviation of the sampled mean, which is the standard deviation divided by the square root of the total amount of numbers in a data set:. Standard Deviation Diagram : Dark blue is one standard deviation on either side of the mean. For the normal distribution, this accounts for The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the mean.
A large standard deviation, which is the square root of the variance, indicates that the data points are far from the mean, and a small standard deviation indicates that they are clustered closely around the mean.
Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. Standard deviation may serve as a measure of uncertainty.
In physical science, for example, the reported standard deviation of a group of repeated measurements gives the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance. If the mean of the measurements is too far away from the prediction with the distance measured in standard deviations , then the theory being tested probably needs to be revised.
This makes sense since they fall outside the range of values that could reasonably be expected to occur, if the prediction were correct and the standard deviation appropriately quantified. The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the average mean.
As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be farther from the average maximum temperature for the inland city than for the coastal one.
Another way of seeing it is to consider sports teams. In any set of categories, there will be teams that rate highly at some things and poorly at others. Chances are, the teams that lead in the standings will not show such disparity but will perform well in most categories.
The lower the standard deviation of their ratings in each category, the more balanced and consistent they will tend to be. Teams with a higher standard deviation, however, will be more unpredictable. Comparison of Standard Deviations : Example of two samples with the same mean and different standard deviations. The red sample has a mean of and a SD of 10; the blue sample has a mean of and a SD of Each sample has 1, values drawn at random from a Gaussian distribution with the specified parameters.
For advanced calculating and graphing, it is often very helpful for students and statisticians to have access to statistical calculators. Two of the most common calculators in use are the TI series and the R statistical software environment. The TI series of graphing calculators, shown in, is manufactured by Texas Instruments. Released in , it was one of the most popular graphing calculators for students.
TI : The TI series of graphing calculators is one of the most popular calculators for statistics students. R logo shown in is a free software programming language and a software environment for statistical computing and graphics. The R language is widely used among statisticians and data miners for developing statistical software and data analysis. R is an implementation of the S programming language, which was created by John Chambers while he was at Bell Labs.
R provides a wide variety of statistical and graphical techniques, including linear and nonlinear modeling, classical statistical tests, time-series analysis, classification, and clustering. Another strength of R is static graphics, which can produce publication-quality graphs, including mathematical symbols. Dynamic and interactive graphics are available through additional packages.
R is easily extensible through functions and extensions, and the R community is noted for its active contributions in terms of packages. Due to its S heritage, R has stronger object-oriented programming facilities than most statistical computing languages. The number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.
Consider this example: To compute the variance, first sum the square deviations from the mean. The mean is a parameter, a characteristic of the variable under examination as a whole, and a part of describing the overall distribution of values.
Knowing all the parameters, you can accurately describe the data. The more known fixed parameters you know, the fewer samples fit this model of the data. If you know only the mean, there will be many possible sets of data that are consistent with this model. However, if you know the mean and the standard deviation, fewer possible sets of data fit this model. In computing the variance, first calculate the mean, then you can vary any of the scores in the data except one.
This one score left unexamined can always be calculated accurately from the rest of the data and the mean itself. As an example, take the ages of a class of students and find the mean.
With a fixed mean, how many of the other scores there are N of them remember could still vary? The answer is N-1 independent pieces of information degrees of freedom that could vary while the mean is known. One piece of information cannot vary because its value is fully determined by the parameter in this case the mean and the other scores.
Each parameter that is fixed during our computations constitutes the loss of a degree of freedom. Imagine starting with a small number of data points and then fixing a relatively large number of parameters as we compute some statistic.
We see that as more degrees of freedom are lost, fewer and fewer different situations are accounted for by our model since fewer and fewer pieces of information could, in principle, be different from what is actually observed. If there is nothing that can vary once our parameter is fixed because we have so very few data points, maybe just one then there is nothing to investigate.
Degrees of freedom can be seen as linking sample size to explanatory power. In fitting statistical models to data, the random vectors of residuals are constrained to lie in a space of smaller dimension than the number of components in the vector. That smaller dimension is the number of degrees of freedom for error. In statistical terms, a random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.
The individual variables in a random vector are grouped together because there may be correlations among them. Often they represent different properties of an individual statistical unit e. A residual is an observable estimate of the unobservable statistical error. The sample mean could serve as a good estimator of the population mean.
The difference between the height of each man in the sample and the observable sample mean is a residual. Note that the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent.
Perhaps the simplest example is this. The sum of the residuals is necessarily 0. Specifically, the plotted hypothetical distribution is a t distribution with 3 degrees of freedom. The interquartile range IQR is a measure of statistical dispersion, or variability, based on dividing a data set into quartiles.
Quartiles divide an ordered data set into four equal parts. The values that divide these parts are known as the first quartile, second quartile and third quartile Q1, Q2, Q3.
The interquartile range is equal to the difference between the upper and lower quartiles:. As an example, consider the following numbers:. Divide the data into four quartiles by finding the median of all the numbers below the median of the full set, and then find the median of all the numbers above the median of the full set. Find the median of these numbers: take the first and last number in the subset and add their positions not values and divide by two.
This will give you the position of your median:. The median of the subset is the second position, which is two.
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