chance variation

Examples of chance variation in the following topics:

• The Sum of Draws

  • Your sum of draws is, therefore, subject to a force known as chance variation.
  • To better see the affects of chance variation, let us take 25 draws from the box.

• Comparing Two Sample Averages

  • Very different means can occur by chance if there is great variation among the individual samples.
  • In order to account for the variation, we take the difference of the sample means,

• Chance Error and Bias

  • Chance error and bias are two different forms of error associated with sampling.
  • The variations in the possible sample values of a statistic can theoretically be expressed as sampling errors, although in practice the exact sampling error is typically unknown.
  • In sampling, there are two main types of error: systematic errors (or biases) and random errors (or chance errors).
  • Of course, this is not possible, and the error that is associated with the unpredictable variation in the sample is called random, or chance, error.

• The F-Distribution and the F Ratio

  • The variance is also called variation due to treatment or explained variation.
  • The variance is also called the variation due to error or unexplained variation.
  • SSwithin = the sum of squares that represents the variation within samples that is due to chance.
  • Unexplained variation- sum of squares representing variation within samples due to chance: SSwithin = SStotal−SSbetween
  • Mean square (variance estimate) that is due to chance (unexplained): MSwithin = SSwithin /dfwithin

• Mean Squares and the F-Ratio

  • The variance is also called variation due to treatment or explained variation.
  • The variance is also called the variation due to error or unexplained variation.
  • $SS_{\text{within}}$ is the sum of squares that represents the variation within samples that is due to chance.
  • Unexplained variation: sum of squares representing variation within samples due to chance: $SS_{\text{within}} = SS_{\text{total}} = SS_{\text{between}}$
  • Mean square (variance estimate) that is due to chance (unexplained): $\displaystyle{ MS }_{ \text{within} }=\frac { { SS }_{ \text{within} } }{ { df }_{ \text{within} } }$

• Analysis of variance (ANOVA) and the F test

  • The method of analysis of variance in this context focuses on answering one question: is the variability in the sample means so large that it seems unlikely to be from chance alone?
  • This question is different from earlier testing procedures since we will simultaneously consider many groups, and evaluate whether their sample means differ more than we would expect from natural variation.
  • If the null hypothesis is true, any variation in the sample means is due to chance and should not be too large.
  • When the null hypothesis is true, any differences among the sample means are only due to chance, and the MSG and MSE should be about equal.

• Chance Models

  • A stochastic model is used to estimate probability distributions of potential outcomes by allowing for random variation in one or more inputs over time.
  • Therefore, accurately determining the standard error of the mean depends on the presence of chance.
  • A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time.
  • The random variation is usually based on fluctuations observed in historical data for a selected period using standard time-series techniques.
  • Distributions of potential outcomes are derived from a large number of simulations (stochastic projections) which reflect the random variation in the input(s).

• Coefficient of Determination

  • $r^2$, when expressed as a percent, represents the percent of variation in the dependent variable $y$ that can be explained by variation in the independent variable $x$ using the regression (best fit) line.
  • $1-r^2$ when expressed as a percent, represents the percent of variation in $y$ that is NOT explained by variation in $x$ using the regression line.
  • For example, if one is trying to predict the sales of a car model from the car's gas mileage, price, and engine power, one can include such irrelevant factors as the first letter of the model's name or the height of the lead engineer designing the car because the $r^2$ will never decrease as variables are added and will probably experience an increase due to chance alone.
  • Approximately 44% of the variation (0.4397 is approximately 0.44) in the final exam grades can be explained by the variation in the grades on the third exam.
  • Therefore approximately 56% of the variation ($1-0.44=0.56$) in the final exam grades can NOT be explained by the variation in the grades on the third exam.

• Inferential Statistics

  • In statistics, statistical inference is the process of drawing conclusions from data that is subject to random variation--for example, observational errors or sampling variation.
  • More substantially, the terms statistical inference, statistical induction, and inferential statistics are used to describe systems of procedures that can be used to draw conclusions from data sets arising from systems affected by random variation, such as observational errors, random sampling, or random experimentation.
  • These tests determine the probability that the results found were by chance, and therefore not representative of the entire population.

• Multiple Regression Models

  • A second use of multiple regression is to try to understand the functional relationships between the dependent and independent variables, to try to see what might be causing the variation in the dependent variable.
  • The main null hypothesis of a multiple regression is that there is no relationship between the $X$ variables and the $Y$ variables-- in other words, that the fit of the observed $Y$ values to those predicted by the multiple regression equation is no better than what you would expect by chance.
  • As you are doing a multiple regression, there is also a null hypothesis for each $X$ variable, meaning that adding that $X$ variable to the multiple regression does not improve the fit of the multiple regression equation any more than expected by chance.