Statistics and Data Science: A Modeling Approach

6.3 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:

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

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

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

[1] 4.049691

There are actually a few different ways you can get the standard deviation for a variable. One is 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(tidyverse) require(mosaic) require(Lock5Data) require(supernova) Empty.model <- lm(Thumb ~ NULL, data = Fingers) # calculate the standard deviation of Thumb from Fingers with sd() # calculate the standard deviation with sqrt() and var() # calculate the standard deviation with favstats() sd(Fingers$Thumb) sqrt(var(Fingers$Thumb)) favstats(~Thumb, data = Fingers) ex() %>% { check_function(., "sd") %>% check_result() %>% check_equal() check_function(., "sqrt") %>% check_result() %>% check_equal() check_function(., "favstats") %>% check_result() %>% check_equal() }
Make sure to calculate SD all three separate ways, and Run before Submitting to make sure you are getting the correct values
DataCamp: ch6-6

[1] 8.726695
[1] 8.726695
 min Q1 median Q3 max     mean       sd   n missing
  39 55     60 65  90 60.10366 8.726695 157       0

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.

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.

Histogram of weight lost in MindsetMatters data frame

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(tidyverse) require(mosaic) require(Lock5Data) require(supernova) MindsetMatters$PoundsLost <- MindsetMatters$Wt - MindsetMatters$Wt2 # create an empty model of PoundsLost from MindsetMatters Empty.model <- # find SS, var, and sd # there are multiple correct solutions Empty.model <- lm(PoundsLost ~ NULL, data = MindsetMatters) sum(resid(Empty.model)^2) var(MindsetMatters$PoundsLost) sd(MindsetMatters$PoundsLost) ex() %>% { check_object(., "Empty.model") %>% check_equal() check_output(., 556.7) check_output(., 7.52) check_output(., 2.74) }
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.

Analysis of Variance Table

Response: PoundsLost
          Df Sum Sq Mean Sq F value Pr(>F)
Residuals 74 556.73  7.5233
[1] 7.523333
[1] 2.74287
SS:  556.7267
Variance:  7.523333
Standard Deviation:  2.74287

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 (or DGP), 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 for symbols are in parentheses).

Sample (or estimate) DGP (or population
Mean \(\bar{Y}\) (y bar) \(\mu\) (mu)
Variance \(s^2\) (s squared) \(\sigma^2\) (sigma squared)
Standard Deviation \(s\) (s) \(\sigma\) (sigma)

Variance is the mean squared error around the empty model. 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.

Analysis of Variance Table

Response: Thumb
           Df Sum Sq Mean Sq F value Pr(>F)
Residuals 156  11880  76.155
[1] 76.1552