Chapter 4 Variability and Uncertainty (Standard Deviations and Standard Errors)

Uncertainty is the stuff of science. A major goal of statistics is measuring uncertainty. What do we mean by uncertainty? Uncertainty is the error in estimating a parameter, such as the mean of a sample, or the difference in means between two experimental treatments, or the predicted response given a certain change in conditions. Uncertainty is measured with a variance or its square root, which is a standard deviation. The standard deviation of a statistic is also (and more commonly) called a standard error.

Uncertainty emerges because of variability. In any introductory statistics class, students are introduced to two measures of variability, the “standard deviation” and the “standard error.” These terms are absolutely fundamental to statistics – they are the start of everything else. Yet, many biology researchers confuse these terms and certainly, introductory students do too.

When a research biologist uses the term “standard deviation,” they are probably referring to the sample standard deviation which is a measure of the variability of a sample. When a research biologist uses the term “standard error,” they are probably referring to the standard error of a mean, but it could be the standard error of another statistics, such as a difference between means or a regression slope. An important point to remember and understand is that all standard errors are standard deviations. This will make more sense soon.

4.1 The sample standard deviation vs. the standard error of the mean

4.1.1 Sample standard deviation

The sample standard deviation is a measure of the variability of a sample. For example, were we to look at a histological section of skeletal muscle we would see that the diameter of the fibers (the muscle cells) is variable. We could use imaging software to measure the diameter of a sample of 100 cells and get a distribution like this

The mean of this sample is 69.4 and the standard deviation is 2.8. The standard deviation is the square root of the variance, and so computed by

\[\begin{equation} s_y = \sqrt{\frac{\sum_{i=1}^n{(y_i - \overline{y})^2}}{n-1}} \tag{4.1} \end{equation}\]

Memorize this equation. To understand the logic of this measure of variability, note that \(y_i - \overline{y}\) is the deviation of the \(i\)th value from the sample mean, so the numerator is the sum of squared deviations. The numerator is a sum over \(n\) items and the denominator is \(n-1\) so the variance is (almost!) an averaged squared deviation. More variable samples will have bigger deviations and, therefore, bigger average squared deviations. Since the standard deviation is the square root of the variance, a standard deviation is the square root of an average squared deviation. This makes it similar in value to the averaged deviation (or average of the absolute values of the deviations since the average deviation is, by definition of a mean, zero).

Notes on the variance and standard deviation

  1. Variances are additive but standard deviations are not. This means that the variance of the sum of two independent (uncorrelated) random variables is simply the sum of the variances of each of the variables. This is important for many statistical analyses.
  2. The units of variance are the square of the original units, which is awkward for interpretation. The units of a standard deviation is the same as that of the original variable, and so is much easier to interpet.
  3. For variables that are approximately normally distributed, we can map the standard deviation to the quantiles of the distribution. For example, 68% of the values are within one standard deviation of the mean, 95% of the values are within two standard deviations, and 99% of the values are within three standard deviations.

4.1.2 Standard error of the mean

A standard error of a statistic is a measure of the precision of the statistic. The standard error of the mean is a measure of the precision of the estimate of the mean. The smaller the standard error, the more precise the estimate. The standard error of the mean (SEM) is computed as

\[\begin{equation} SEM = \frac{s_y}{\sqrt{n}} \tag{4.2} \end{equation}\]

The SEM is often denoted \(s_{\bar{y}}\) to indicate that it is a standard deviation of the mean (\(\bar{y}\)). In what sense is a standard error a measure of variability? This is kinda weird. If we sample 100 cells in the slide of muscle tissue and compute the mean diameter, how can the mean have a standard deviation? There is only one value! To understand how the SEM is a standard deviation, imagine 1) resampling 100 cells and 2) recomputing a mean from the re-sampled data, then repeating this resampling and recomputation an infinite number of times and each time, you write down the newly computed mean. The true standard error of the mean is the standard deviation of this infinitely long column of means. This means that a standard error of the mean, computed from a single sample using equation (4.2) is itself a sample statistic.

Notes on standard errors

  1. The SEM is only one kind of standard error. A standard deviation can be computed for any statistic – these are all standard errors. For some statistics, such as the mean, the standard error can be computed directly using an equation, such as that for the SEM (equation (4.2). For other statistics, a computer intensive method such as the bootstrap is necessary to compute a standard error. We will return to the bootstrap at the end of this chapter.
  2. The units of a standard error are the units of the measured variable.
  3. A standard error is proportional to sample variability (the sample standard deviation, \(s_y\)) and inversely proportional to sample size (\(n\)). Sample variability is a function of both natural variation (there really is variation in diameter among fibers in the quadriceps muscle) and measurement error (imaging software with higher resolution can measure a diameter with less error). Since the SEM is a measure of the precision of estimating a mean, this means this precision will increase (or the SEM will decrease) if 1) an investigator uses methods that reduce measurement error and 2) an investigator computes the mean from a larger sample.
  4. This last point (the SEM decreases with sample size) seems obvious when looking at equation (4.2), since \(n\) is in the denominator. Of course \(n\) is also in the denominator of equation (4.1) for the sample standard deviation but the standard deviation does not decrease as sample size increases. First this wouldn’t make any sense – variability is variability. A sample of 10,000 cell diameters should be no more variable than a sample of 100 cell diameters (think about if you agree with this or not). Second, this should also be obvious from equation (4.1). The standard deviation is the square root of an average and averages don’t increase with the number of things summed since both the the numerator (a sum) and denominator increase with \(n\).

4.2 Using Google Sheets to generate fake data to explore uncertainty

In statistics we are interested in estimated parameters of a population using measures from a sample. The goal in this section is to use Google Sheets (or Microsoft Excel) to use fake data to discover the behavior of sampling and to gain some intuition about uncertainty using standard errors.

4.2.1 Steps

  1. Open Google Sheets
  2. In cell A1 type “mu”. mu is the greek letter \(\mu\) and is very common notation for the poplation value (the TRUE value!) of the mean of some hypothetical measure. In cell B1, insert some number as the value of \(\mu\). Any number! It can be negative or positive.
  3. In cell A2 type “sigma”. sigma is the greek letter \(\sigma\). \(\sigma^2\) is very common (universal!) notation for the population (TRUE) variance of some measure or parameter. Notice that the true (population) values of the mean and variance are greek letters. This is pretty standard in statistics. In cell B2, insert some positive number (standard deviations are the positive square roots of the variance).
  4. In cell A8 type the number 1
  5. In cell A9 insert the equation “=A8 + 1”. What is this equation doing? It is adding the number 1 to to the value in the cell above, so the resulting value should be 2.
  6. In Cell B8, insert the equation “=normsinv(rand())*$B$2 + $B$1“. The first part of the equation creates a random normal variable with mean 0 and standard deviation 1. multiplication and addition transform this to a random normal variable with mean \(\mu\) and standard deviation \(\sigma\) (the values you set in cells B1 and B2).
  7. copy cell B8 and paste into cell B9. Now Higlight cells A9:B9 and copy the equations down to row 107. You now have 100 random variables sampled from a infinite population with mean \(\mu\) and standard deviation \(\sigma\).
  8. In cell A4 write “mean 10”. In cell B4 insert the equation “=average(B8:B17)”. The resulting value is the sample mean of the first 10 random variables you created. Is the mean close to \(\mu\)?
  9. In cell A5 write “sd 10”. In cell B5 insert the equation “stdev(B8:B17)”. The result is the sample standard deviation of the first 10 random variables. Is this close to \(\sigma\)?
  10. In cell A6 write “mean 100”. In cell B6 insert the equation “=average(B8:B107)”. The resulting value is the sample mean of the all 100 random variables you created. Is this mean closer to \(\mu\) than mean 10?
  11. In cell A7 write “sd 100”. In cell B7 insert the equation “=stdev(B8:B107)”. The resulting value is the sample standard deviation of the all 100 random variables you created. Is this SD closer to \(\sigma\) than sd 10?

The sample standard deviation is a measure of the variability of the sample. The more spread out the sample (the further each value is from the mean), the bigger the sample standard deviation. The sample standard deviation is most often simply known as “The” standard deviation, which is a bit misleading since there are many kinds of standard deviations!

Remember that your computed mean and standard deviations are estimates computed from a sample. They are estimates of the true values \(\mu\) and \(\sigma\). Explore the behavior of the sample mean and standard deviation by re-calculating the spreadsheet. In Excel, a spreadsheet is re-calculated by simultaneously pressing the command and equal key. In Google, command-R recalculates but is painfully slow. Instead, if using Google Sheets, just type the number 1 into a blank cell, and the sheet recalculates quickly. Do it again. And again.

Each time you re-calculate, a new set of random numbers are generated and the new means and standard deviations are computed. Compare mean 10 and mean 100 each re-calculation. Notice that these estimates are variable. They change with each re-calculation. How variable is mean 10 compared to mean 100? The variability of the estimate of the mean is a measure of uncertainty in the estimate. Are we more uncertain with mean 10 or with mean 100? This variability is measured by a standard deviation. This standard deviation of the mean is also called the standard error of the mean. Many researchers are loose with terms and use “The” standard error to mean the standard error of the mean, even though there are many kinds of standard errors. In general, “standard error”" is abbreviated as “SE.” Sometimes “standard error of the mean” is specifically abbreviated to “SEM.”

The standard error of the mean is a measure of the precision in estimating the mean. The smaller the value the more precise the estimate. The standard error of the mean is a standard deviation of the mean. This is kinda weird. If we sample a population one time and compute a mean, how can the mean have a standard deviation? There is only one value! And we compute this value using the sample standard deviation: \(SEM = \frac{SD}{\sqrt{N}}\). To understand how the SEM is a standard deviation, Imagine recalculating the spread sheet an infinite number of times and each time, you write down the newly computed mean. The standard error of the mean is the standard deviation of this infinitely long column of means.

4.3 Using R to generate fake data to explore uncertainty

note that I use “standard deviation” to refer to the sample standard deviation and “standard error” to refer to the standard error of the mean (again, we can compute standard errors as a standard deviation of any kind of estimate)

4.3.1 part I

In the exercise above, you used Google Sheets to generate \(p\) columns of fake data. Each column had \(n\) elements, so the matrix of fake data was \(n \times m\) (it is standard in most fields to specify a matrix as rows by columns). This is much easier to do in R and how much grows exponentially as the size of the matrix grows.

To start, we just generate a \(n \times p\) matrix of normal random numbers.

# R script to gain some intuition about standard deviation (sd) and standard error (se)
# you will probably need to install ggplot2 using library(ggplot2) 
n <- 6 # sample size
p <- 100 # number of columns of fake data to generate
fake_data <- matrix(rnorm(n*p, mean=0, sd=1), nrow=n, ncol=p) # create a matrix

the 3rd line is the cool thing about R. In one line I’m creating a dataset with \(n\) rows and \(p\) columns. Each column is a sample of the standard normal distribution which by definition has mean zero and standard deviation of 1. But, and this is important, any sample from this distribution will not have exactly mean zero and standard deviation of 1, because it’s a sample, the mean and standard deviation will have some small errror from the truth. The line has two parts to it: first I’m using the function “rnorm” (for random normal) to create a vector of n*m random, normal deviates (draws from the random normal distribution) and then I’m organizing these into a matrix (using the function “matrix”)

To compute the vector of means, standard deviations, and standard errors for each column of fake_data, use the apply() function.

means <- apply(fake_data,2,mean) # the apply function is super useful
sds <- apply(fake_data,2,sd)
sems <- sds/sqrt(n)

apply() is a workhorse in many R scripts. Learn it. Know it. Live it.

The SEM is the standard deviation of the mean, so let’s see if the standard deviation of the means is close to the true standard error. We sampled from a normal distribution with SD=1 so the true standard is

1/sqrt(n)
## [1] 0.4082483

and the standard deviation of the \(p\) means is

sd(means)
## [1] 0.3731974

Questions

  1. how close is sd(means) to the true SE?
  2. change p above to 1000. Now how close is sd(means) to the true SE?
  3. change p above to 10,000. Now how close is sd(means) to the true SE?

4.3.2 part II - means

This is a visualization of the spread, or variability, of the sampled means

qplot(means)
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Compute the mean of the means

mean(means)
## [1] -0.039961

Questions

  1. Remember that the true mean is zero. How close, in general, are the sampled means to the true mean. How variable are the means? How is this quantified?
  2. change n to 100, then replot. Are the means, in general, closer to the true mean? How variable are the means now?
  3. Is the mean estimated with \(n=100\) closer to the truth, in general, then the mean estimated with \(n=6\)?
  4. Redo with \(n=10000\)

4.3.3 part III - how do SD and SE change as sample size (n) increases?

mean(sds)
## [1] 1.017144

Questions

  1. what is the mean of the standard deviations when n=6 (set p=1000)
  2. what is the mean of the standard deviations when n=100 (set p=1000)
  3. when n = 1000? (set p=1000)
  4. when n = 10000? (set p=1000)
  5. how does the mean of the standard deviations change as n increases (does it get smaller? or stay about the same size)
  6. repeat the above with SEM
mean(sems)
## [1] 0.4152472

Congratulations, you have just done a Monte Carlo simulation!

4.3.4 Part IV – Generating fake data with “for loops”

n <- 6 # sample size
n_iter <- 10^5 # number of iterations of loop (equivalent to p)
means <- numeric(n_iter)
sds <- numeric(n_iter)
sems <- numeric(n_iter)
for(i in 1:n_iter){
  y <- rnorm(n) # mean=0 and sd=1 are default so not necessary to specify
  means[i] <- mean(y)
  sds[i] <- sd(y)
  sems[i] <- sd(y)/sqrt(n)
}
sd(means)
## [1] 0.4090702
mean(sems)
## [1] 0.3883867

Questions

  1. What do sd(means) and mean(sems) converge to as n_iter is increased from 100 to 1000 to 10,000?
  2. Do they converge to the same number?
  3. Should they?
  4. What is the correct number?

Question number 4 is asking what is E(SEM), the “expected standard error of the mean”. There is a very easy formula to compute this. What is it?

4.4 Bootstrapped standard errors

The bootstrap is the best invention since duct tape. Really.

A standard error is the standard deviation of an infinite number of hypothetically re-sampled means. A bootstrap standard error of a statistic is the standard deviation of the statistic from a finite number of resamples of the data! Wait what?

Let’s download some data to explore this concept. The data are archived at Dryad Repository.

  1. URL: https://datadryad.org//resource/doi:10.5061/dryad.31cc4
  2. file: RSBL-2013-0432 vole data.xlsx
  3. sheet: COLD VOLES LIFESPAN

The data are the measured lifespans of the short-tailed field vole (Microtus agrestis) under three different experimental treatments: vitamin E supplementation, vitamin C supplementation, and control (no vitamin supplementation). Vitamins C and E are antioxidants, which are thought to be protective of basic cell function since they bind to the cell-damaging reactive oxygen species that result from cell metabolism.

I’ve read in the file using read_excel and converted to a data.table named vole. I used setnames to rename the columns to lifespan, control, vitamin_E, and vitamin_C. The data are in a wide format – that is instead of a single “treatment” column, there are three columns (“control”, “vitamin C”, “vitamin E”) with value = 1, if that row (or lifespan) was assigned the treatment of the column label and zero otherwise. In general, we want data.tables to be in long format. Wide formats can be useful for some computations but not really for these data.

Compute the standard error of the mean of the lifespan for the control group using equation (4.2). One simple way to do this for the control group is to extract the subset of the data satisfying the condition control = 1 (the value in the column “control” equals 1). In R, these conditional querries use ==.

y <- na.omit(vole[control==1, lifespan]) # subset of data satisfying condition and omitting missing data, if these exist
n <- length(y) # the sample size
se <- sd(y)/sqrt(n-1) # standard error of the mean

Okay, the SEM using equation (4.2) is 31.9. Let’s compare this with a bootstrap estimate of the SEM.

n_iter <- 2000 # number of bootstrap iterations
means <- numeric(n_iter) # we will save the means each iteration to this
inc <- 1:n # the first sample is the actual sample
for(iter in 1:n_iter){ # the for loop
  means[iter] <- mean(y[inc])
  inc <- sample(1:n, replace=TRUE) # re-sample for the next iteration
}
se_boot <- sd(means)

#compare
se
## [1] 31.89536
se_boot
## [1] 30.93843

dayum!