CourseKata - 6.3 Standard Deviation

Introduction to Statistics: A Modeling Approach

Standard Deviation

The standard deviation (written as \(s\)) is simply the square root of the variance. We generally prefer thinking about error in terms of standard deviation because it yields a number that makes sense using the original scale of measurement. So, for example, if you were modeling weight in pounds, variance would express the error in squared pounds (not something we are used to thinking about), whereas standard deviation would express the error in pounds.

Here are two formulas that represent the standard deviation:

\[s = \sqrt{s^_{2}}\]

\[\sqrt{\frac{\sum_{i=1}^n (Y_i-\bar{Y})^2}{n-1}}\]

L_Ch6_Standard_1

To calculate standard deviation in R, we use sd(). Here is how to calculate the standard deviation of our Thumb data from TinyFingers.

sd(TinyFingers$Thumb)

source: Ch6_Var3

There are actually a few different ways you can get the standard deviation for a variable. The function sd() obviously. But you can also square root the variance with a combination of the functions sqrt() and var(). Yet another, and possibly more useful, way is to use good old favstats(). Try all three of these methods to calculate the standard deviation of Thumb from the larger Fingers data frame.


                require(mosaic)
                require(ggformula)
                require(supernova)


                # calculate the standard deviation of Thumb from Fingers
                    sd( )

                # calculate the standard deviation with sqrt() and var()
                    sqrt(var( ))

                # calculate the standard deviation with favstats()
                    favstats( )


                
                sd(Fingers$Thumb)
                sqrt(var(Fingers$Thumb))
                favstats(~ Thumb, data = Fingers)


                test_function_result("sd")
                test_correct(test_function_result("sqrt"),
             {
                test_function("var")
                test_error()
              })
                test_function_result("favstats")
                test_error()

Make sure to calculate sd three separate ways

DataCamp: ch6-6

source: Ch6_ DF - StanDevThumb

L_Ch6_Standard_2

Sum of Squares, Variance, and Standard Deviation

We have discussed three ways of quantifying error around our model. All start with residuals, but they aggregate those residuals in different ways to summarize total error.

All of them are minimized at the mean, and so all are useful when the mean is the model for a quantitative variable.

L_Ch6_Standard_3

Thinking About Quantifying Error in MindsetMatters

Below is a histogram of amount of weight lost (PoundsLost) by each of the 75 housekeepers in the MindsetMatters data frame.

source: Ch6_ PoundsLost HIstogram

Use R to create an empty model of PoundsLost. Call it Empty.model. Then find the SS, Variance, and Standard Deviation around this model.


                require(mosaic)
                require(ggformula)
                MindsetMatters <- read.csv(file="https://raw.githubusercontent.com/UCLATALL/intro-stats-modeling/master/datasets/mindset-matters.csv", header=TRUE, sep=",")
                MindsetMatters$PoundsLost <- MindsetMatters$Wt - MindsetMatters$Wt2


                # create an empty model of PoundsLost from MindsetMatters
                    Empty.model <- 
  
                # find SS, var, and sd
                # there are multiple correct solutions


                # create an empty model of PoundsLost from MindsetMatters
                    Empty.model <- lm(PoundsLost ~ NULL, data = MindsetMatters)
  
                # find SS, var, and sd
                    anova(Empty.model)
                    var(MindsetMatters$PoundsLost)
                    sd(MindsetMatters$PoundsLost)


                test_object("Empty.model")
                test_error()
                success_msg("Nice job!")

Remember to use NULL for the empty model

DataCamp: ch6-7

There are multiple ways to compute these in R but here is one set of possible outputs.

source: Ch6 EM - PoundsLost1

L_Ch6_Thinking_1

source: Ch6_ EM PoundsLost2

L_Ch6_Thinking_2

Notation for Mean, Variance, and Standard Deviation

Finally, we also use different symbols to represent the variance and standard deviation of a sample, on one hand, and the population, on the other. Sample statistics are also called estimates because they are our best estimates of the DGP parameters. We have summarized these symbols in the table below (pronunciations are in parentheses).

source: Ch6_SamplePopTable

L_Ch6_Thinking_3

Variance is the mean squared error. It is an average of the squared deviations from the mean. In tables, it may be shortened to be Mean Square or MSE. You now know that just means variance. Remember in the output (see below) from the anova() function the column headed “Mean Sq”? That is, in fact, the variance.