Stats Final
Equal Variance Violation
"Fanning out" of the residual plot indicates that errors have increasing variance as X increases.
Independence Violation
"Tracking" or "meandering" in the residual plot indicates that errors are correlated.
P-Value Interpretation
"given our sample size and the variation in the population, the chance of seeing a sample mean of nine or less under the null hypothesis is about one and 100." P-value is better than critical value because not only does it tell you whether it is significant, but how significant. Lower the stronger.
Confidence Intervals for the Population Mean (σ known) with Finite Population
*if the sample is taken from a finite population, a finite correction factor may be used to increase the accuracy of the solution. *Finite Correction Factor only required when n/N ≥ 5% See image.
Regression with Categorical Variables: Dummy Variables
- Involves categorical x-variable with 2 levels e.g., male-female; holiday/no holiday, weekend/weekday - Variable levels coded as 0 and 1 - Number of dummy variables is 1 less than number of levels of variable -May be combined with quantitative variables
VIF
- The most common rule: VIF > 10 - A simple rule without VIF: Multicollinearity exists if correlation (rij) between any two X variables is greater than R2 for the model (rij> R2).
What does a 95% confidence interval mean?
-"we are 95% confident that the true population mean falls within the range of CI" -If we drew 100 same sized random samples; 95 out of our 100 calculated intervals would include the population mean
Simple Regression Steps
1) Fit Regression Line 2) Check Model Assumptions 3) Quantify Model Fit (Se, R^2) 4) Test Hypotheses about the slope B1 5) Compute Confidence Intervals 6) Compute Prediction Intervals for Y given X
Steps for Multiple Regression
1) Fit regression model 2) check model assumptions 3) model summaries (se, R^2, Adj R^2) 4) F Test for overall model 5) t test for coefficients 6) confidence intervals for E(Y|X) 7) prediction intervals (Y|X)
When to use T statistic
1) The sample is selected randomly. 2) Sample can be any size 3) The population standard deviation (σ) is unknown. If the sample size is small (and the population is normally distributed), a t distribution should be used.
When to use Z statistic
1. Data are a random sample from the population 2. Population standard deviation (𝝈) is known. 3. Sample size (n) is at least 30, or underlying population is normal
Characteristics of a t distribution
1. t distributions are symmetric, unimodal, and a family of curves. 2. The t distributions are flatter in the middle and have more area in their tails than the standard normal distribution. 3. This means that we are more likely to see extreme values of the t statistic than the z statistic. 4.The smaller the sample size, the more extreme values of t. 5. As n becomes large, the t distribution approaches the z distribution.
Interval Estimate (aka Confidence Interval)
= a range of values that contains the unknown population parameter with a specified level of confidence. i.e. A random sample of 500 first-year MBAs is taken. Based on the sample data, I am 95% confident that µ = average age of all first-year MBAs lies between 28.1 and 28.5 years.
Point Estimate
= a single value that represents the best guess for the unknown population parameter based on the sample data. i.e. A random sample of 500 first-year MBAs is taken. Based on the sample data, the point estimate for µ = average age of all first-year MBAs is x̅= 28.3 years.
One-Sample z-Test for Population Proportion: Example using Critical Value Approach (two-tailed)
A manufacturer believes that 8% of its products contain at least one minor flaw. The company researcher tests this by selecting 200 items at random and finds that 24 items have flaws. Perform a hypothesis test using ⍺= .10. *Why ⍺=.10? Use when Type I error (false positive) is not too problematic.
Testing Hypotheses About a Population Mean Using the Z Statistic (𝝈 known): Example using Critical Value Approach (two-tailed)
A manufacturing process is considered "in control," when the population mean diameter of motor shafts produced on an assembly line is µ=815 mm. When the process is "out of control," the mean diameter of motor shafts is different from 815 mm. To determine whether the process is out of control, the quality control manager took a random sample of n=50 motor shafts from the assembly line and the mean diameter was x̅= 815.34 mm. Assume that the population standard deviation is σ=1 mm. Test whether the process is out of control. Use ⍺=.05
Multiple Regression: Adjusted R^2
As new predictor variables are added to a regression model, the value of R^2 never goes down and usually goes up. This is true even if useless predictors are added to the model. The Adjusted 𝐑𝟐 takes into consideration both the fit of the model (via SSE) and the number of predictors (k) in the model. The Adjusted R2 penalizes larger models by dividing SSE by its degrees of freedom (df = n-k-1). Thus it goes down as we add useless predictors.
Confidence Intervals for the Population Mean (𝝈 unknown)
If the population standard deviation is unknown and the sample size is small, the z distribution cannot be used. Must use t distribution.
Testing Hypotheses About a Population Mean Using the Z Statistic (𝝈 known): Example using Critical Value Approach (one-tailed)
In order to be approved by the FDA, a new COVID-19 drug must show a statistically significant decrease in recovery time compared to an existing drug. With the existing drug, the average recovery time from COVID-19 is 10 days. A new drug is introduced, and a clinical trial is completed with n = 100 patients. The mean recovery time is x̅= 9 days. Did the new drug show a significant decrease in mean recovery time from the old drug (µ0=10)? Use ⍺=.05. Assume that the population standard deviation σ = 4 days is known.
Testing Hypotheses About a Population Mean Using the Z Statistic (𝝈 known): Example using P- Value Approach (one-tailed)
In order to be approved by the FDA, a new COVID-19 drug must show a statistically significant decrease in recovery time compared to an existing drug. With the existing drug, the average recovery time from COVID-19 is 10 days. A new drug is introduced, and a clinical trial is completed with n = 100 patients. The mean recovery time is x̅= 9 days. Did the new drug show a significant decrease in mean recovery time from the old drug (µ0=10)? Use ⍺=.05. Assume that the population standard deviation σ = 4 days is known.
Determining Sample Size for Proportion
See image for equation. *Similar studies can be used to estimate p; if pis unknown, .5 is usually used.
The Regression Line
See image for formulas. Slope of line must be calculated before the intercept.
Confidence Intervals for the Population Mean (σ known): Finding the z critical value
See image.
Example of using degrees of freedom to calculate t critical value
See image.
Interpreting Multiple Regression
See image.
Multiple Regression: Se, R^2 and Adj. R^2 in Excel and Minitab
See image.
Se and R^2 in Excel and Minitab
See image.
Summary of Regular Confidence Intervals
See image.
T Test v. Z Test
See image.
Testing Hypotheses About a Population Mean Using the Z Statistic (𝝈 unknown): Example using P-Value Approach (one-tailed)
See image. For two-tailed test, multiply p-value by 2. Also, use T.TEST() in excel as alternative.
Confidence Intervals for the Population Mean (σ unknown): Finding the t critical value
See image. *t intervals are always larger than z. *The t confidence interval is robust to the assumption of normality.• So, the t-type confidence intervals are reliable even if the data are not perfectly bell-shaped (i.e., normally distributed).
Standard Error of the Estimate (Se):
Standard error tells us how accurate our predictions are. How much variability is there around the regression line? To determine whether the Se is big or small, compare it to the mean of the dependent variable. 5% is a good target. *Standard Error is the estimate for standard deviation. It's used to: - identify outliers - create prediction intervals - create confidence intervals Interpretation: "The standard deviation of the error term of the regression model is 0.316 or 316,600 bottles." SSE is the SS residual error on excel output
Eight Step Process for Testing Hyptheses
Step 1: Establish a null and alternative hypothesis. Step 2: Determine the appropriate statistical test. Step 3: Set value of alpha, Type 1 error rate. Step 4: Establish the decision rule. Step 5: Gather sample data. Step 6: Analyze the data. Step 7: Reach a statistical conclusion. Step 8: Make a business decision.
The multiple regression model
The partial regression coefficient, βi, represents the increase in y from a one-unit increase in xi, if all other x variables are held constant
Type 1 and Type 2 Errors
Type 1 = "false positive" - reject the null when null is true - ⍺= Probability of Type I error. Also called the level of significance. Type 2 = "False negative" - fail to reject null when null is false - β = Probability of Type II error. Power (= 1 - β): Probability of rejecting the null hypothesis when it is false. The Type I and II error probabilities are inversely related. As ⍺ increases, β decreases and vice versa
Pearson Correlation Coefficient
a statistical measure of the strength of a linear relationship between two metric variables bounded by -1 and 1. Value close to +1 indicates strong positive relationship. Value close to -1 denotes strong negative relationship. Steeper slope does not mean larger correlation
Confidence Intervals for the Population Mean (σ known): Example
A cellular telephone company would like to estimate the population monthly mean number of texts in the 18-to-24-year-old age category. The company will use the estimated number to forecast revenue next year. 1. From a sample of 85 bills, the sample mean is 1300 texts. 2. Using this sample mean, a confidence interval will be calculated within which we are relatively confident that the population mean is located. 3. Suppose that, from previous studies, the population standard deviation is known to be about 160. 95% Confidence
Testing Hypotheses About a Population Mean Using the Z Statistic (𝝈 unknown): Example using Critical Value Approach (one-tailed)
A firm that manages rental properties is considering a move to an expensive area in San Francisco. The earnings are proportional to the rents of the properties it manages. To cover its costs, the firm needs rents in this area to average more than $2000 per month. Are rents in San Francisco high enough to justify the expansion? To test this hypothesis, the firm obtained individual property rents for a sample of n=115 rental units in downtown San Francisco. Among these, the average rent is x̅ = $2,157 with sample standard deviation s = $581. Use a significance level of ⍺=.05.
Statistical Hyotheses
A formal structure derived from the scientific method. Composed of two parts: 1) Null Hypothesis- aka the old theory is true 2) Alternative Hypothesis- aka the new theory has changed. Something has changed.
Confidence Intervals for the Population Mean (𝝈 unknown): Example
A manager wants to estimate the average amount of comp time accumulated per week for engineers in the aerospace industry. Historically, engineers worked an average of 8 overtime hours per week, and the manager believes they are working more than that now. He randomly samples 18 engineers and measures the amount of extra time they work during a specific week. 1) The sample mean is 13.56 hours, and the sample standard deviation is 7.80 hours. 2) The manager would like a 90% confidence interval. 3) Since n < 30, and σ is unknown, use the t distribution.
Confidence Intervals for the Population Proportion: Example
A random sample of 50 people were asked to compare two smart voice assistants: Amazon Echo and Google Home. After trying both devices, each person was asked which device they preferred. 29 out of 50 preferred Google Home. Find a 95% confidence interval for the proportion of all people that prefer Google Home to Amazon Echo.
Determining Sample Size: Example
A researcher wants to estimate the average monthly expenditure on bread by a family in Chicago. She wants to be 90% confident of her results, and she wants the estimate to be within $1.00 of the actual figure (error) and the standard deviation of average monthly bread purchases is $4.00. What is the sample size estimation for this problem?
Research Hypotheses
A statement of what the researcher believes will be the outcome of an experiment or a study. i.e. Larger companies spend a higher percentage of their annual budget on advertising than do smaller companies.
Confidence Intervals for the Population Mean (σ known) with Finite Population: Example
A study is conducted in a company that employs 800 engineers (*this is where we see that pop is finite). A random sample of 50 of these engineers reveals that the average sample age is 34.30 years. Historically, the population standard deviation of the age of the company's engineers is around 8 years. Construct a 98% confidence interval to estimate the average age of all the engineers in this company.
Connection b/w Confidence Intervals and Hypothesis Tests
A two-sided hypothesis test for μ (H0: μ=μ0 vs Ha: μ≠μ0) at significance level ⍺=.05 is the same as checking if the 95% confidence interval for μ includes the value μ0.
Multiple Regression: R^2
Analogous to simple regression r^2 R#represents the proportion of variation of the dependent variable, y, explained by the independent variables in the regression model. Interpretation: 74.1% of the variability in house price is explained by the age and square footage of the house. 25.9% is explained by other factors.
Coefficient of Determination (R^2)
Answers the question: How well does my model fit the data? Tells us the proportion of variability of the dependent variable (y) explained by the independent variable (x). For simple regression R^2 = correlation (r) squared. Inversely, correlation = square root of R^2. We wont know sign (+/-) unless we look at slope.
Testing Hypotheses About a Population Mean Using the Z Statistic (𝝈 known): Finite Populations
As in previous cases, if the sample size is greater than or equal to 5% of the population size (n/N > .05), the correction factor should be used.
One-Sample z-Test for Population Proportion
Binary outcomes: success/failure, defective/not defective - i.e iPhone/not iPhone, Coke/not Coke, Rain/No Rain.• For the central limit theorem to hold, np ≥ 5, and nq ≥ 5
Prediction Intervals for a Single Value
Calculated using Minitab A prediction interval estimates a single value of y for a given value of x. They contain about 95% of the points. They show us how much variation we have around the regression line. For x = 80 passengers $𝑦 = 1.57 + .0407𝑥 = 1.57 + .0407 80 = 4.826. A prediction interval is a range of values that is likely to contain the value of a single new observation given specified settings of the predictors. We are 95% confident that when x = 80, the next new observation of y will fall between $4414 and $5238.
Confidence Intervals for the Mean
Calculated using Minitab i.e. Airline Q: A point estimate of y is computed from the regression equation For x = 80 passengers $𝑦 = 1.57 + .0407𝑥 = 1.57 + .0407 80 = 4.826. - The point estimate of the predicted cost is 4.826, or $4,826. -A different sample might result in a different regression line and thus a different prediction.• Desirable to estimate a confidence interval for the prediction. This type of confidence interval is an estimate of the average value of y for a given x. "We are 95% confident that the mean value of y when x is 80 will be between $4,709 and $4942." Confidence intervals represent the mean of Y given Y in the population. They tell us how much uncertainty, we have when we estimate the true line. Confidence bands are bowed out because of the uncertainty in the slope.
Standardized Residuals (ri)
Calculated using the standard error. Standardized residuals can be used to detect outliers. Any observation with ri> 3 or ri< -3 is considered an outlier.
Simple Regression: Hypothesis Test for Slope and Overall Model, Example (Null = 0)
Can the cost of flying a commercial airliner be predicted using regression analysis? To do this, find t obs using the formula here. If we reject the null, then the dependent and independent variables are correlated. The most common test is whether the slope coefficient is different from zero. What does β1 = 0 imply about the relationship between X (Number of Passengers) and Y (Cost)? H0: β1 = 0 (X and Y have no linear relationship) Ha: B1 ≠ 0 (X and Y have a linear relationship)
Linearity Violation
Curved patterns in the residual plot indicates a nonlinear relationship between Y and X.
How to resolve multicolinearity
Eliminate one or more of the correlated x variables - take the one with highest VIF out. If VIF =, then take highest p-value out. Avoid inference on individual parameters Do not extrapolate. (i.e., Don't try to predict Y for values of X that are outside the range of the data.)
Interpretation of Regression Components
Example: 𝑦 = 34.85+ 0.5(x) and 𝑦 = 34.85+ 0.5(80)
Multiple Regression in Minitab
Example: Housing prices determined by sq ft and age
Multiple Regression: Standard Error of the Estimate
Example: Housing prices determined by sq ft and age Interpretation of SE: The estimated standard deviation of the errors is 11.96 or $11,960. Roughly: we can predict house prices to within ± $11,960. This can be used to check normality or find confidence intervals. If Se is too high, we may discard it.
F Test
F test for the overall significance of the model: Tests whether the regression model explains a significant amount of the variability in y. For simple regression (one x variable), the F test and the t test for the slope are testing the same thing. H0: β1= 0 (overall model has no explanatory power) Ha: β1≠ 0 (overall model does have explanatory power)
One-Sample z-Test for Population Proportion: Example using P-Value Approach (two-tailed)
For a two-tailed test, the p-value is twice the p-value for one-sided test.
Testing Hypotheses About a Population Mean Using the Z Statistic (𝝈 known): Example using P-Value Approach (two-tailed)
For a two-tailed test, the p-value is twice the p-value for the one-sided test.
Multiple Regression: F Test
For multiple regression, the F test tells us whether the x variables in the model provide significant explanatory power for y (accounting for the size of model). If we reject the null, we conclude that at least one of the x variables adds significant predictability for y. If we fail to reject the null, we conclude that the regression model has no significant predictability for y. A large Fobs value means the model explains a large amount of variability in y
Decision Making in Hypothesis Testing
Given the sample data, we will make one of two decisions: 1. Reject the null hypothesis (H0). 2. Fail to reject the null hypothesis (H0). If we reject the null hypothesis, we conclude that the alternative hypothesis is correct. If we fail to reject the null hypothesis, we do not accept H0. We conclude that we don't have enough evidence to reject H0.
Hypotheses for Parameter Estimation
H0, Ha are statements about population parameters: 1. μ (Population Mean) 2. p (population Proportion)
Path Diagrams
Here we want to know the total affect of age on price. To calculate the coefficient between age and sqft, we need to know the total effect of age on sqft. So, fun a simple regression with age as independent and sqft as dependent.
Determining Sample Size for Proportion: Example
Hewitt Associates conducted a national survey to determine the extent to which employers are promoting health and fitness among their employees. One of the questions asked was, Does your company offer on-site exercise classes? Suppose it was estimated before the study that no more than 40% of the companies would answer yes. How large a sample would Hewitt Associates have to take in estimating the population proportion to ensure a 98% confidence in the results and to be within .03 of the true population proportion?
Substantive Hypotheses
If the null hypothesis is rejected and the alternative hypothesis is accepted, we would say that the result is statistically significant. Outcome is unlikely to occur by chance. A statistically significant outcome does not imply a material, substantive difference. It may not be a significant business outcome. A substantive result occurs when the outcome produces results that are important to the decision maker.
LINE assumptions continued
If we cant trust our models, we cant believe the predictions
T Tests for the Regression Coefficients
Individual significance tests can be computed for each regression coefficient using a t test. Remember, excel and minitab calculate these as H0 = 0 and H0 /= 0. If the null value is different, do by hand. If the t ratios for any predictor variables are not significant (fail to reject the null hypothesis), the researcher might decide to drop that variable(s) from the analysis as a nonsignificant predictor(s)
Simple Regression: Hypothesis Test for Slope and Overall Model, Example (Slope ≠ 0)
Instead, lets now say that the variable costs per flight must be less than $50/passenger. The observed slope is .0407 or $40.70. So, H0: β1 = 50 (flights not profitable) Ha: B1 < 50 (flights profitable)
Normality Violations
Large positive and small negative residuals indicate right skewness in residuals. The two outliers indicate a viola=on of the normality assumption
Minitab 4x1 Residual Plots
Left plots show no clear violations of normality Right plots show no violations of linearity, equal variance, or independence. Check outliers too!
LINE Assumptions
Linearity X and Y have a linear relationship Independence The errors are independent Normality The errors are normally distributed Equal Variance The errors have constant variances
Confidence Intervals for the Population Mean (σ known): Margin of Error
Margin of Error (E) determines the width of the confidence interval. It is the statistical uncertainty associated with our estimate. E depends on two factors: 1) Critical value (z sub ⍺/2): Number of standard errors used for the margin of error. Depends on the confidence level 1-⍺. 2) Standard error (σ/ sqrt(n)): = standard deviation of x̅; depends on the population standard deviation σ and sample size n. Width of E increases as CI increases. Width of E increases as σ increases. Width of E decreases as n increases.
Multicolinearity
Multicollinearity is when two or more of the X variables of a multiple regression model are highly correlated with each other. How do you know when multicollinearity exists? -Significant correlations between pairs of x variables are more than with y variable - Non-significant t-tests for most of the individual parameters, but overall model test is significant - Estimated parameters have wrong sign
Regression in Excel
Output description Slope (b1= .0407). For each person added to the flight, there is a $40.70 increase in the predicted cost of the flight. (Estimated variable cost.) Y-intercept (b0= 1.570 or $1,570): If no passengers are on the flight, the predicted cost of the flight is $1,570. (Estimated fixed cost.)
Normal Distribution and P values
P-value ≤ α: The data do not follow a normal distribution (Reject H0) P-value > α: Cannot conclude the data do not follow a normal distribution (Fail to reject H0)
Point vs Interval
Point is only good for sample, interval is better for populations.
Regression Analysis
Process of building a mathematical model or function to predict one variable using another variable or other variables. With no predictors, all variability in Y is random. Thus, predictors explain variability away, we add more to explain more variability. Simple regression model : most basic regression model. - Two variables; predict one variable using the other. - -- Variable to be predicted (Y) is called the dependent variable. - Variable used to predict (X) is called the independent variable. - Assumes straight-line relationship between X and Y.
Calculating Sb1 by hand
Se = standard error of model n = sample size s^2x = standard deviation of sample squared
One-Sample z-Test for Population Mean: Critical Values
See image for critical values.
Determining Sample Size
See image for equation.
Confidence Intervals for the Population Proportion
See image.
Using the p value to test hypotheses
Smaller p-values provide stronger evidence against H0. "If the p-value is low, the null must go" •
Critical Value vs. P-Value Approach
The critical value approach and p-value approach always give the same conclusion: reject H0 or don't reject H0.
Confidence Intervals for the Population Proportion: Margin of Error
The margin of error (E) determines the width of the interval. Phat = .05 gives largest margin of error (E)
Pearson Correlation Coefficient
The most common statistical measure of the strength of linear relationships among variables. r = 0 no correlation r = -1 perfect negative correlation r = 1 perfect positive correlation
F obs Minitab
The observed F value is Fobs= 28.63. The p-value is the area to the right of Fobs= 28.63 under an F distribution with numerator df = 2 and denominator df = 20. Because the p-value < 0.0001, the F value is significant at 𝛼= .001. The null hypothesis is rejected, and there is at least one significant predictor of house price in this analysis.
Residuals
The residuals (or estimated errors) are the differences between the actual and predicted y values Least squares regression line minimizes sum of squared residuals
Interval Width
The width (or margin of error) of the interval: -increases when confidence level (1-alpha) increases, because z(alpha/2) increases. -increases when sample size (n) decreases, since n is in the denominator of the margin of error. -increases when standard deviation (sigma or s) increases. -decreases when z is used instead of t, since critical value of z is always less than t. -increases when N increases, since the Finite Correction Factor gets closer to 1.
Multiple Regression Example Interpretations
The y-intercept is b0= 57.4. - Since y is in thousands, a house with 0 square feet and 0 age would have a value of $57,400. The coefficient of 𝑥1(sqft) is b1= .0177. - If the age of the house is held constant, the addition of 1 square foot of space in the house results in a predicted increase of $17.70 in the home price. The coefficient of x2(age) is b2= −.666. - If the square feet in the house is held constant, a one-year increase in the age of the house results in a predicted decrease of $666 in the home price. Point Prediction of Price: For a 12-year-old house with 2,500 square feet, predicted price is 𝑦hat = 57.4 + .0177 2500 − .666 12 = 93.658, or $93,658.
When to use z test for hypothesis testing about a proportion
The z-statistic is used when the following conditions are met: 1. Random sample from the population. 2. Both np0>5 and nq0 > 5.
Hypothesis Test Example
To get approval by the US Food and Drug Administration (FDA), a new drug must show statistically significant improvement over the existing drug. Under the existing COVID-19 drug treatment, the mean time to recovery is 10 days with a standard deviation of σ=4 days. A new drug is introduced. A clinical trial with n=100 patients is completed, and the mean time to recovery is x̅=8.3 days. Should the FDA approve the new drug? In other words, is the sample mean significantly less than μ0=10?
One-tailed vs. two-tailed tests
Two tailed= non directional, covers all hypotheses One tailed= one directional, covers 1 hypothesis
Multiple Regression LINE Assumptions
Use 4X1 Plots and Residual vs X plot. The latter will confirm violations of linearity and equal variance (curvature, fanning out). Check outliers too!
ANOVA
We use the ANOVA table to understand how much variation is explained by the model. If 90% is explained by the model, the other 10% is explained by other factors.
Confidence Intervals for the Population Mean (σ known): Confidence Interval to estimate Mean
When σ is known, we use z statistic. See image.
Variability in slope
Which factors contribute to variability in the slope b1? 1) Sample size (n) 2) Variability of the x variable: 3) Variability of the errors: Se^2Sx^2
Interpretation of R^2
i.e. R^2 = .90. About 90% of the variation ("uncertainty") in cost can be explained by the number of passengers. So, the number of customers (X) is an excellent predictor of cost (Y).
The P-Value Method
p-value = Probability of observing a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis (H0) is correct. Step 1: Establish a null and alternative hypothesis. Step 2: Determine the appropriate statistical test. Step 3: Set value of alpha, Type 1 error rate. Step 4: Establish the decision rule. SKIP Step 5: Gather sample data. Step 6: Analyze the data. Compute p-value Step 7: Reach a statistical conclusion. Reject if p-value < ⍺ Step 8: Make a business decision.
Hypothesis Tests
statistical procedures in which we decide between two competing hypotheses about the population, based on sample data. Three types of hypotheses: 1) Research Hypotheses 2) Statistical Hypotheses 3) Substantive Hypotheses
Degrees of freedom
the number of independent observations for a source of variation minus the number of independent parameters estimated in computing the variation.
