3.5 – Statistics of error

Making sense of error in data.

This page explains how statistics of error help quantify differences between measured and true values, beginning with classical methods like standard error and confidence intervals that rely on large-sample assumptions. Related, statisticians talk about uncertainty — the quantification of doubt about a measurement. We introduce permutation-based approaches, including jackknife and bootstrap resampling, which avoid those assumptions and are especially useful with smaller or complex data sets. Observer variation (intra- and interobserver error) is also highlighted as an important source of measurement error.

Missing values are another common source of uncertainty in biological data sets. Data may be missing because of equipment failure, recording mistakes, sample loss, or biological factors that prevent a measurement from being collected. Even a small number of missing observations can influence summaries, graphs, and statistical tests, especially in smaller studies. In this chapter, we treat missing values mainly as a practical issue to identify and report during exploratory data analysis. Later, in the chapter on experimental design, we examine missing data in greater depth, including differences between random and non-random missingness, as well as approaches such as data imputation and the consequences these choices may have for interpretation.

We conclude with a discussion about confidence intervals as a measure of accuracy and emphasize proper use of significant figures to avoid overstating precision.

Statistics of error.

In statistics, error doesn’t mean mistake, an avoidable human misjudgement or oversight, eg, a typo. An error in statistics means there was a difference between the measured value and the actual value for an object. Efforts by many authors of notable classical statistics developed a large body of calculated statistics, eg, standard error the mean, which allows the user to quantify how large the error of measurement is given assumptions about the distribution of the errors. Thus, classical statistics requires user to make assumptions about the error distribution, the subject of our Chapter 6. A critical issue to understand is that these methods assume large sample sizes are available; they are called asymptotic statistics or large-sample statistics; the properties of the statistical estimates are evaluated as sample size approaches infinity.

Jackknife sampling and bootstrap sampling are permutation approaches to working with data when the Central Limit Theorem — as sample size increases, the distribution of sample means will tend to a normal distribution (see Chapter 6.7) — is unlikely to apply or, rather, we don’t wish to make that assumption (Chapter 19). The jackknife is a sampling method involving repeatedly sampling from the original data set, but each time leaving one value out. The estimator, for example, the sample mean, is calculated for each sample. The repeated estimates from the jackknife approach yield many estimates which, collected, are used to calculate the sample variance. Jackknife estimators tend to be less biased than those from classical asymptotic statistics.

Bootstrapping, and not jackknife resampling, may now be the preferred permutation approach (eg, Google Scholar search “bootstrap statistics” 36K hits; “jackknife statistics” 17K hits), but which method is best depends on qualities of the data set. Bootstrapping involves large numbers of permutations of the original data, which, in short, means we repeatedly take many samples of our data and recalculate our statistics on these sets of sampled data. We obtain statistical significance by comparing our result from the original data against how often results from our permutations on the resampled data sets exceed the originally observed results. By permutation methods, the goal is to avoid the assumptions made by large-sample statistical inference, ie, reaching conclusions about the population based on samples from the population. Since its introduction, “bootstrapping” has been shown to be superior in many — but, importantly, not all — cases for statistics of error compared to the standard, classical approach (Efron 1979, Hesterberg 2015).

There are many advocates for the permutation approaches, and, because we have computers now instead of the hand calculators our statistics ancestors used, permutation methods may be the approach you will take in your own work. However, the classical approach has its strengths — when the conditions, that is, when the assumptions of asymptotic statistics are met by the data, then the classical approaches tend to be less conservative than the permutation methods. By conservative, statisticians mean that a test performs at the level we expect it to. Thus, if the assumptions of classical statistics are met they return the correct answer more often than do the permutation tests.

Error and the observer.

Individual researchers make observations, therefore, we can talk about observer variation as a kind of error measurement. For repeated measures of the same object by an individual, we would expect the individual to return the same results. To the extent repeated measures differ, this is intraobserver error. In contrast, measures of the same object from different individuals is interobserver error. For a new instrument or measurement system, one would need to establish the reliability of the measure: confronted with the same object do researchers get the same measurement? Accounting for interobserver error applies in many fields, eg, histopathology of putative carcinoma slides (Franc et al 2003), liver biopsies for cirrhosis (Rousselet et al 2005), blood cell counts (Bacus 1973).

Confidence in estimates.

A really useful concept in statistics is the idea that you can assign how confident you are to an estimate. This is another way to speak of the accuracy of an estimate. Clearly, we have more confidence in a sample estimate for a population parameter if many observations are made. Another factor in our ability to estimate is the magnitude of observation differences. In general, the larger the differences among values from repeated trials, the less confident we will be in out estimate, unless, again, we make our estimates from a large collection of observations. These two quantities, sample size and variability, along with our level of confidence, eg, 95%, are incorporated into a statistic called the confidence interval.

We will use this concept a lot throughout the course; for now, a simple but approximate confidence interval is to use the 2 x SEM rule (as long as sample size large): twice the standard error of the mean. Take your estimate of the mean, then add (upper limit) or subtract (lower limit) twice the value of the standard error of the mean (if you recall, that’s the standard deviation divided by the square-root of the sample size).

Example. Consider five magnetic darts thrown at a dart board (28 cm diameter, height of 1.68m from the floor) from a distance of 3.15 meters (Fig 1).

Figure 1. Magnetic dart board with 5 darts. Click image to view full sized image.

The distance in centimeters (cm) between where each of the five darts landed on the board compared to the bullseye is reported in (Table 1),

Table 1. Results of five darts thrown at a target (Figure 2)*

* distance in centimeters from the center. 

How precise was the dart thrower? We’ll use the coefficient of variation as a measure of precision. Second, how accurate were the throws? Use R to calculate

darts = c(7.5, 3.0, 1.0, 2.7, 7.4)
#use the coefficient of variation to describe precision of the throws
coefVar = 100*(sd(darts)/mean(darts)); coefVar
[1] 68.46141

Confidence Interval to describe accuracy.

For a game of darts, the true value would be a distance of zero — all bullseyes. We need to calculate the standard error of the mean (SEM), then, we calculate the confidence interval around the sample mean.

#Calculate the SEM
SEM <- sd(darts)/sqrt(length(darts)); SEM
[1] 1.322649
#now, get the lower and upper limit, subtract from the mean
confidence <- c(mean(darts)-2*SEM, mean(darts), mean(darts)+2*SEM); confidence
[1] 1.674702, 4.320000, 6.965298

The mean was 4.3 cm, therefore to get the lower limit of the interval subtract 2.65 (2*SEM = 2.645298) from the mean; for the upper limit add 2.65 to the mean. Thus, we report our approximate confidence interval as (1.7, 7.0), and we read this as saying we are about 95% confident the population value is between these two limits. Five is a very small sample number, so we shouldn’t be surprised to learn that our approximate confidence interval would be less than adequate. In statistical terms, we would use the t-distribution, and not the normal distribution, to make our confidence interval in cases like this.

We made a quick calculation of the confidence interval; we can get make this calculation by hand by incorporating the t distribution. We need to know the degrees of freedom, which in this case is 4 (n – 1, where n = 5). We look up critical value of t at 5% (to get our 95% confidence interval), t = 2.78. Subtract for lower limit t*SEM and add for upper limit t*SEM to the sample mean for the 95% confidence interval.

We can get help from R, by using the one-sample t-test with a test against the hypothesis that the true mean is equal to zero. Among a variety of output we will introduce later (see Chapter 8 and 10), the confidence interval is reported (yellow highlight),

#make an attach a data frame object
Ddarts <- data.frame(darts)
t.test(darts,mu=0)
One Sample t-test
data: darts
t = 3.2662, df = 4, p-value = 0.0309
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 0.6477381 7.9922619
sample estimates:
mean of x 
 4.32

t.test uses the function qt(), which provides the quantile function. To recreate the 95% CI without the additional baggage output from the t.test, we would simply write

#upper limit
mean(darts)+ qt(0.975,df=4)*sd(darts)/sqrt(5)

#lower limit
mean(darts)+ qt(0.025, df = 4, lower.tail=TRUE)*sd(darts)/sqrt(5)

where sd(darts)/sqrt(5) is the standard error of the mean.

Or, alternatively, download and take advantage of a small package called Rmisc (not to be confused with the RcmdrMisc package) and use the function CI

library(Rmisc)
CI(darts)
     upper     mean     lower 
7.9922619 4.3200000 0.6477381

The advantage of using the CI() command from the package Rmisc is pretty clear; I don’t have to specify the degrees of freedom or the standard error of the mean. By default, CI reports the 95% confidence interval. we can specify any interval simply by adding to the command. For example, 

CI(darts, ci=0.90)

reports upper and lower limits for the 90% confidence interval.

Try it. On your own, get the 90% confidence interval and the 99% confidence intervals for our dart example. Compare to the 95% confidence interval — what do you conclude about “confidence” in our estimate from 90% to 99%?

Confidence intervals from p-values.

And finally, it is not unusual for a report to provide summary statistics, differences between means (ie, effect size), and p-values (orange highlight), but not report confidence intervals. We discuss p-values and the concept of statistical significance beginning in Chapter 8. But for now, we show how to calculate approximate confidence intervals from the p-value. This may be of use for conducting meta-analysis (Chapter 20.15). See two helpful “how to” articles by Bland and Altman in 2011 BMJ (references listed in Chapter 3.6 – References). 

In brief, calculate the test statistic for a normal distribution (letZ), calculate a standard error, and then the lower and upper limits for the 95% confidence interval. We illustrate use of the calculator formulas with the dart example, the p-value was 0.0309; the estimate was 4.32.

R code

letZ = (-0.862 + sqrt(0.743 - 2.404*log(0.0309, base = exp(1)))) = 2.154903
letSE = 4.32/letZ = 2.00473
1.96*letSE = 3.929271 # Add for upper and subtract for lower

The lower limit is 0.39 (4.32 – 3.929271) and the upper limit is 8.25 (4.32 + 3.929271). Although in the ballpark, these calculated estimates are wider than the confidence interval estimated by the t.test command. Small sample size, n = 5, is to blame — the accuracy of these methods, based on asymptotic statistical estimation approaches, assumes the law of large numbers applies.

Significant figures.

And finally, we should respect significant figures, the number of digits which have meaning. Our data were measured to the nearest tenth of a centimeter, or two significant figures. Therefore, if we report the confidence interval as (0.6477381, 7.9922619), then we imply a false level of precision, unless we also report our random sampling error of measurement.

R has a number of ways to manage output. One option would be to set number of figures globally with the options() function — all values reported by R would hold for the entire session. For example, options(digits=3) would report all numbers to three significant figures. Instead, I prefer to use signif() function, which allows us to report just the values we wish and does not change reporting behavior for the entire session.

signif(CI(darts),2)
upper  mean lower 
 8.00  4.30  0.65

The R function signif() applies rounding rules. We apply rounding rules when required to report estimates to appropriate levels of precision. Rounding procedures are used to replace a number with a simplified approximation. Wikipedia provides a comprehensive list of rounding rules. Notable rules include 

• directed rounding to an integer, eg, rounding up or down
• rounding to nearest integer, eg, round half up (down) if the number ends with 5
• randomly rounding to an integer, eg, stochastic rounding.

With the exception of stochastic rounding, all rounding methods impose biases on the sets of numbers. For example, the round half up method applied for numbers above 5, round down for numbers below 5 will increase the variance of the sample. In R, use round() for most of your work. If you need one of the other approaches, for example, to round up, the command is ceiling(); to round down we use floor().

When to round?

No doubt your previous math classes have cautioned you about the problems of rounding error and their influence on calculation. So, as a reminder, if reporting calls for rounding, then always round after you’ve completed your calculations, never during the calculations themselves. 

A final note about significant figures and rounding. While the recommendations about reporting statistics are easy to come by (and often very proscriptive, eg, Table 1, Cole 2015), there are other concerns. Meta-analysis, which are done by collecting information from multiple studies, would benefit if more and not fewer numbers are reported, for the very same reason that we don’t round during calculations.

Questions

1. Calculate the correct 90% and 99% confidence intervals for the dart data using the t-distribution
   • by hand
   • by one alternative method in R, demonstrated with examples in this page
2. How many significant figures should be used for the volumetric pipettor p1000? The p200? The p20 (data at end of this page)?
3. Another function, round(), can be used. Try

round(CI(darts),2)

1. • and report the results: vary the significant figures from 1 to 10 (signif() will take digits up to 22).
   • See any output differences between signif() and round()? Don’t forget to take advantage of R help pages (eg, enter ?round at the R prompt) and see Wikipedia.
2. Compare rounding by signif() and round() for the number 0.12345. Can you tell which rounding method the two functions use? 
3. Calculate the coefficient of variation (CV) for each of the three volumetric pipettors from the data at end of this page. Rank the CV from smallest to largest. Which pipettor had the smallest CV and would therefore be judged the most precise? 
4. (data provided at end of this page, click here).
5. Standards distinguish between within run precision and between run precision of a measurement instrument (see Chapter 16.5 – Instrument reliability and validity). The data in Table 1 were all recorded within 15 minutes by one operator. What kind pf precision was measured?
6. Calculate the standard error of the means for each of the three pipettors from the data
7. (data provided at end of this page, click here).
8. Calculate the approximate confidence interval using the 2SE rule and judge which of the three pipettors is the most accurate (narrowest confidence interval)
   • Repeat, but this time apply your preferred R method for obtaining confidence intervals.
   • Compare approximate and R method confidence intervals. How well did the approximate method work?
9. Return to the dart example, but now assume the p-value was p = 0.003 and again for p = 0.0003. Estimate the 95% confidence interval for each case; the difference remains 4.32. How does the change in p-value, from 0.03 to 0.0003 — a two orders of magnitude difference — affect the estimates for the confidence interval (hint: describe the width of the CI)?

Quiz Chapter 3.5

Statistics of error

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Data sets

Pipette calibration

Chapter 3 continues

• Introduction
• Data types
• Exploring data
• Measures of Central Tendency
• Measures of dispersion
• Estimating parameters
• Statistics of error
• References and suggested readings

3.3 – Measures of dispersion

More about summary statistics.

The statistics of data exploration involve calculating estimates of the middle, central tendency, and the variability, or dispersion, about the middle. Statistics about the middle were presented in the previous section, Chapter 3.1. Statistics about measures of dispersion, and how to calculate them in R, are presented in this page. Use of Z score to standardize or normalize scores is introduced. Statistical bias is also introduced. 

Describing the middle of the data gives your reader a sense of what was the typical observation for that variable. Next, your reader will want to know something about the variation about the middle value — what was the smallest value observed? What was the largest value observed? Were data widely scattered or clumped about the middle?

Measures of dispersion or variability are descriptive statistics used to answer these kinds of questions. Variability statistics are very important and we will use them throughout the course. A key descriptive statement of your data, how variable?

Examine the two figures below (Fig 1 and Fig 2): the two sample frequency distributions (put data into several groups, from low to high, and membership is counted for each group) have similar central tendencies (mean), but they have different degrees of variability (standard deviation).

Figure 1. A histogram which displays a sampling of data with a mean of 10 (arrow marks the spot) and standard deviation (sd) of 50 units.

Figure 2. A histogram which displays a sampling of data with the same mean of 10 (arrow marks the spot) displayed in Fig. 1, but with a smaller standard deviation (sd) of 5 units.

Clearly, knowing something about the middle of a data set is only part of the required information we need when we explore a data set; we need measures of dispersion as well. Provided certain assumptions about the frequency of observations hold, estimates of the middle (eg, mean, median) and the dispersion (eg, standard deviation) are adequate to describe the properties of any observed data set.

For measures of dispersion or variability, the most common statistics are: Range, Mean Deviation, Variance, Standard Deviation, and Coefficient of Variation.

The range.

The range is reported as one number, the difference between maximum and minimum values.

Arrange the data from low to high, subtract the minimum, X_min, from the maximum, X_max, value and that’s the range.

For example, the minimum height of a collection of fern trees might be 1 meter whereas the maximum height might be 2.2 meters. Therefore, the range is 1.2 meters (= 2.2 – 1).

The range is useful, but may be misleading (as index of true variability in population) if there is one or more exceptional “outlier” (one individual that has an exceptionally large or small value). I often just report X_min and X_max. That’s the way R does it, too.

If you recall, we used the following as an example data set.

x = c(4,4,4,4,5,5,6,6,6,7,7,8,8,8,15)

The command for range in R is simply range().

range(x)
[1] 4 15

There is apparently no universally accepted symbol for “range,” so we report the statistic as range = 15 – 4 = 11.

You may run into an interquartile range, which is abbreviated as IQR. It is a trimmed range, which means, like the trimmed mean, it will be robust to outlier observations. To calculate this statistic, divide the data into fourths, termed quartiles. For our example variable, x, we can get the quartiles in R

quantile(x)
  0%  25%  50%  75% 100% 
 4.0  4.5  6.0  7.5 15.0 

Thus, we see that 25% of the observations are 4.5 (or less), the second quartile is the median, and 75% of the observations are less than 7.5. The IQR is the difference between the first quartile, Q1, and the third quartile, Q3.

The command for obtaining the IQR in R is simply IQR(). Yes. You have to capitalize “IQR” — R is case sensitive; that means IqR is not the same as IQR or iqr.

IQR(x)
[1] 3.5

and we report the statistic as IQR = 3.5. Thus, 75% of the observations are within 3.5 points of the median. The IQR is used in box plots (Chapter 4). Quartiles are special cases of the more general quantile; quantiles divide up the range of values into groups with equal probabilities. For quartiles, four groups at 25% intervals. Other common quantiles include deciles (ten equal groups, 10% intervals) and percentiles (100 groups, 1% intervals).

The mean deviation.

Subtract each observation from the sample mean; each  is called a deviate: some observations will be positive (greater than ) and some will be negative (less than ).

Take the absolute value of the deviation and then add up the absolute values of the deviations from the mean. At the end, divide by the sample size, n. Large values for this statistic implies that much of the data is spread out, far from the mean. Small values in turn imply that each observation is close to the mean.

In R, we can get the mean deviation with the mad() function. At the R prompt, then

mad(x)
[1] 2.9652

Population variance and population standard deviation.

The population variance is appropriate for describing variability about the middle for a census. Again, a census implies every member of the population was measured. Equation of the population variance is

The population standard deviation also describes the variability about the middle, but has the advantage of being in the same units as the quantity (ie, no “squared” term). Equation of the population standard deviation is

Sample variance and sample standard deviation.

The above statements about the population variance and standard deviation hold for the sample statistics. Equation of the sample variance

and the equation of the sample standard deviation

Of course, instead of taking the square-root of the sample variance, the sample standard deviation could be calculated directly.

*** QuickLaTeX cannot compile formula:
\begin{align*}s = \sqrt{\frac{\sum \left ( X_{i} - \bar{X} \right )^2}{n-1}\end{align*}

*** Error message:
File ended while scanning use of \align*.
Emergency stop.

Statistical bias is an important concept in its own right; bias is a problem because it refers to a situation in which an estimator consistently returns a value different from the population parameter for which it is intended to estimate. Thus, the sample mean is said to be an unbiased estimator of the population mean. Here, bias means that the formula will give you a good estimate of the value. This turns out not to be the case for the sample variance if you divide by n instead of n - 1. Now, for very large values of n, this is not much of an issue, but it shows up when n is small. In R you get what you ask for — if you ask for the sample standard deviation, the software will return the correct value; calculators, go to watch out for this — not all of them are good at communicating which standard deviation they are calculating, the population or the sample standard deviation.

Winsorized variances.

Like the trimmed mean and winsorized mean (Ch3.2), we may need to construct a robust estimate of variability less sensitive to extreme outliers.  Winsorized refers to procedures to replace extreme values in the sample with a smaller value. As noted in Chapter 3.2, we chose the level ahead of time, eg, 90%. Winsorized values then  The winsorized variance is just the sample variance of the winsorized values.  winvar() from the WRS2 package.

Making use of the sample standard deviation.

Given an estimate of the mean and an estimate of the standard deviation, one can quickly determine the kinds of observations made and how frequently they are expected to be found in a sample from a population (assuming a particular population distribution). For example, it is common in journal articles for authors to provide a table of summary statistics like the mean and standard deviation to describe characteristics of the study population (aka the reference population), or samples of subjects drawn from the study population (aka the sample population). The CDC provides reports of attributes of for a sample of adults (more than forty thousand) from the USA population (Fryar et al 2018). Table 1 shows a sample of results for height, weight, and waist circumference for men aged 20 – 39 years.  

Table 1. Summary statistics mean ( standard deviation) of height, weight, and waist circumference of 20-39 year old men USA.

We introduced the normal deviate as a way to normalize scores, and which we we will use extensively in our discussion of the normal distribution in Chapter 6.7, as a way to standardize a sample distribution, assuming a normal curve.

For data that are normally distributed, the standard deviation can be used to tell us a lot about the variability of the data:

​ 62.26% of the data will lie between of

​ 95.46% of the data will lie between of  

​ 99.0% of the data will lie between of

This is known as the empirical rule, where 68% of the observations will be within one standard deviation, 95% of observations will be within two standard deviations, and 99% of observations will be within three standard deviations. 

For example, men aged 20 years in the USA are on average μ = 5 feet 11 inches tall, with a standard deviation of σ = 3 inches. Consider a sample of 1000 men from this population. Assuming a normal distribution, we predict that 623 (62.26%) will be between 5 ft. 8 in. and 6 ft. 2 in. ().

Where did the 5 ft. 8 in. and the 6 ft. 2 in. come from /? We add or subtract multiples of standard deviations. Thus, 6 ft. 2 in. = 5 ft. 11 in. (replace σ = 3) and 5 ft. 8 in. = 5 ft. 11 in. (again, replace σ = 3″).

Can generalize to any distribution. Chebyshev’s inequality (or theorem) guarantees that no more than a particular fraction of observations can be a specified standard deviations distance away from the mean ( needs to be greater than 1). Thus, for standard deviations, we expect a minimum of 75% of values within () two standard deviations away from the mean, or for , then 89% of values will be within () three standard deviations from the mean.

Hopefully you are now getting a sense how this knowledge allows you to plan an experiment.

For example, for a sample of 1000 observations of height of men, how many do we expect to be greater than 6 ft. 7 in. tall? Apply the empirical rule and do a little math — 79 inches (6 ft. 7 in) minus 71 inches (the mean) is equal to 8. Divide 8 by 3 (our value of ) you’ll get 2.66667; so, 6 ft. 7 in. tall is about 2.67 X standard deviations greater than the mean. From Chebyshev’s inequality we have , or 14% of observations less than or greater than the mean ().

Our question asks only about expected number of observations greater than ; divide 14% in half — we therefore expect about 70 individuals out of 1000 men sampled to be 6 ft. 7 in. or taller. Note we made no assumption about the shape of the distribution. If we assume the sample of observations comes from the normal distribution, then we can apply the Z score: about 4 individuals out of 1000 expected to be from the mean (see R code in Note box). We extend the Z-score work in Chapter 6.7.

round(1000*(pnorm(c(79), mean=71, sd=3,lower.tail=FALSE)),0)

round(1000*(pnorm(c(2.67), mean=0, sd=1,lower.tail=FALSE)),0)

[1] 4

Another example, this time with a data set available in R, the diabetic retinopathy data set in the survival package.

data(diabetic, package="survival")

Ages of the 394 subjects ranged from 1 to 58. Mean was 20.8 + 14.81 years. How many out of 100 subjects do we expect greater than 50 years old? With Chebyshev’s inequality we have , or 25.7% of observations less than or greater than the mean, so about 13. If we assume normality, then Z score (Upper tail) is 28.5% and we expect 28 subjects older than 50. Checking the data set, only eight subjects were older than 50. Our estimate by Chebyshev’s inequality was closer to the truth. Why /? Take a look at the histogram (Ch4) of the ages (Fig 3).

Figure 3. Histogram of ages of subjects in the diabetic retinopathy data set in the survival package.

Doesn’t look like a normal distribution, does it/?. 

Z-score or Chebyshev’s inequality, which to use /? Chebyshev’s inequality is more general — it can be used whether the variables is discrete or continuous and without assumption about the distribution. In contrast, the Z score assumes more is known about the variable: random, continuous, drawn from a normally distributed population. Thus, as long as these assumptions hold, the Z score approach will give a better answer. Makes intuitive sense — if we know more, our predictions should be better.

In summary from the above points, and perhaps more formally for our purposes, the standard deviation is a good statistic to describe variability of observations on subjects, it’s integral to the concept of precision of an estimate and is part of any equation for calculating confidence intervals (CI). For any estimated statistic, a good rule of thumb is to always include a confidence interval calculation. We introduced these intervals in our discussion of risk analysis (approximate CI of NNT), and we will return to confidence intervals more formally when we introduce t-tests.

When you hear people talk about “margin of error” in a survey, typically they are referring to the standard deviation — to be more precise, they are referring to a calculation that includes the standard deviation, the standard error and an accounting for confidence in the estimate (see also Chapter 3.5 – Statistics of error). 

Corrected Sums of Squares.

Now, take a closer look at the sample variance formula. We see a deviate

The variance is the average of the squared deviations from the mean. Because of the “squared” part, this statistic will always be positive and greater (or typically, equal to) zero. The variance has squared units (eg, if the units were grams, then the variance units are grams²). The sample standard deviation has the same units as the quantity (ie, no “squared” term). The numerator is called a sums of squares, and will be abbreviated as SS. Much like the deviate, will show up frequently as we move forward in statistics (eg, it’s key to understanding ANOVA).

Other standard deviations of means.

Just as we discussed for the arithmetic average, there will be corresponding standard deviations for the other kinds of means. With the geometric mean, one would calculate the standard deviation of the geometric mean, s_gm, as

where exp is the exponential function, ln refers to natural logarithm, and refers to the sample geometric mean.

For the sample harmonic mean, it turns out there isn’t a straight-forward formula, only an approximation (which I will spare you — it involves use of expectations and moments).

Base R, and therefore Rcmdr, doesn’t have built in functions for these, although you could download and install some R packages which do (eg, package NCStats, and the function is geosd() ).

If we run into data types appropriate for the geometric or harmonic means and standard deviations I will point these out; for now, I present these for completeness only.

Coefficient of variation (CV).

An unfortunate property of the standard deviation is that it is related (= “correlated”) to the mean — a fact recognized by Pearson (1896, p 276). Thus, if the mean of a sample of 100 individuals is 5 and variability is about 20%, then the standard deviation is about 1; compare to a sample of 100 individuals with mean = 25 and 20% variability, the standard deviation is about 5. For means ranging from 1 to 100, here’s a plot to show you what I am talking about.

Figure 4. Scatter plot of the standard deviation (StDev) by the mean. Data sets were simulated.

The data presented in the Figure 4 graph were simulated. Is this bias, a correlation between mean and standard deviation, something you’d see in “real” data? Here’s a plot of height in inches at withers for dog breeds (Fig 5). A line is drawn (ordinary linear regression, see Chapter 17) and as you can see, as the mean increases, the variability as indicated by the standard deviation also increases.

Figure 5. Plot of the standard deviation by the mean for heights of different breeds of dogs.

So, to compare variability estimates among groups when the means also vary, we need a new statistic, the coefficient of variation, abbreviated CV (Pearson 1896). The coefficient of variation is also known as the relative standard deviation, RSD. Many statistics textbooks not aimed at biologists do not provide this estimator (eg, check your statistics book).

This is simply the standard deviation of the sample divided by the sample mean, then multiplied by 100 to give a percent. The CV is useful when comparing the distributions of different types of data with different means.

Many times, the standard deviation of a distribution will be associated with the mean of the data.

Example. The standard deviation in height of elephants will be larger on the centimeter scale than the standard deviation of the height of mice. However, the amount of variability RELATIVE to the mean may be similar between elephants and mice. The CV might indicate that the relative variability of the two organisms is the same.

The standard deviation will also be influenced by the scale of measurement. If you measure on the mm scale versus the meter scale the magnitude of the SD will change. However, the CV will be the same!

By dividing the standard deviation by the mean you remove the measurement scale dependence of the standard deviation and generally, you also remove the relationship of the standard deviation with the mean. Therefore, CV is useful when comparing the variability of the data from distributions with different means.

One disadvantage is that the CV is only useful for ratio scale data (ie, those with a true zero).

The coefficient of variation is also one of the statistics useful for describing the precision of a measurement. See Chapter 3.4 Estimating parameters.

Standard error of the mean (SEM).

All estimates should be accompanied by a statistic that describes the accuracy of the measure. Combined with the confidence interval (Chapter 3.4), one such statistic is called the standard error of the mean, or SEM for short. For the error associated with calculation of our sample mean, it is defined as the sample standard deviation divided by the square root of n, the sample size.

sem →

The concept of error for estimates is a crucial concept. All estimates are made with error and for many statistics, a calculation is available to estimate the error (see Chapter 3.4).

Although related to each other, the concepts of sample standard deviation and sample standard error have distinct interpretations in statistics. The standard deviation quantifies the amount of variation of observations from the mean, while the standard error quantifies the difference between the sample mean and the population mean. The standard error will always be smaller than the standard deviation and is best left for reporting accuracy of a measure and statistical inference rather than description.

Questions.

1. For a sample data set like myY <- c(1,1,3,6), you should now be able to calculate, by hand, the
   • range
   • mean
   • median
   • mode
   • standard deviation
   • coefficient of variation
2. For our example data set, myX <- c(4,4,4,4,5,6,6,6,7,7,8,8,8,8,8) calculate
   • IQR
   • sample standard deviation, s
   • coefficient of variation
3. Use the sample() command in R to draw samples of size 4, 8, and 12 from your example data set stored in myX. Repeat the calculations from question 4. For example x4 <- sample(x,4) will randomly select four observations from x, and will store it in the object x4, like so (your numbers probably will differ!) x4 <- sample(x,4); x4 [1] 8 6 8 6
4. Repeat the exercise in question 4 again using different samples of 4, 8, and 12. For example, when I repeat sample(x,4) a second time I get sample(x,4) [1] 8 4 8 6
5. For Table 1, determine how many multiples of the standard deviation for observations greater than 95-percentile (eg, determine the observation value for a person who is in the 95-percentile for Height in the different decades, etc.
6. Calculate the sample range, IQR, sample standard deviation, and coefficient of variation for the following data sets
   • Height in inches of mothers,
   mom <- c(67, 66.5, 64, 58.5, 68, 66.5) #(data from GaltonFamilies in R package HistData)
   and fathers,
   dad <- c(78.5, 75.5, 75, 75, 74, 74) #(data from GaltonFamilies in R package HistData)
   • Carbon dioxide (CO₂) readings from Mauna Loa for the month of December for
   years <-c (1960, 1965, 1970, 1975, 1980, 1985, 1990, 1995, 2000, 2005, 2010, 2015, 2020)
   years <-c (1960, 1965, 1970, 1975, 1980, 1985, 1990, 1995, 2000, 2005, 2010, 2015, 2020) #obviously, do not calculate statistics on years; you can use to make a plot
co2 <- c(316.19, 319.42, 325.13, 330.62, 338.29, 346.12, 354.41, 360.82, 396.83, 380.31, 389.99, 402.06, 414.26) #(data from Dr. Pieter Tans, NOAA/GML (gml.noaa.gov/ccgg/trends/) and Dr. Ralph Keeling, Scripps Institution of Oceanography (scrippsco2.ucsd.edu/))
   • Body mass of Rhinella marina (formerly Bufo marinus) (see Fig. 1),
   bufo <- c(71.3, 71.4, 74.1, 85.4, 85.4, 86.6, 97.4, 99.6, 107, 115.7, 135.7, 156.2) 

Quiz – Chapter 3.3

Measures of dispersion

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Chapter 3 contents

• • Exploring data
  • Data types
  • Measures of Central Tendency
  • Measures of dispersion
  • Estimating parameters
  • Statistics of error
  • References and suggested readings