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While I was conducting the Bradford Assay experiment to determine the concentration of my unknowns, I ended up by having an A595 for my unknown samples that is higher than the A595 of the standards. How can I resolve this issue? wishes… Micheal.

You could, depending on what you are doing, extrapolate your standard curve to include the absorbance of your samples. This, however, is not recommended since absorbance is linear only within a certain range. It's best to dilute your samples so that they fall within the range of the standard curve while also ensuring that your standard curve is, in fact, linear.

## Error bars in experimental biology

Error bars commonly appear in figures in publications, but experimental biologists are often unsure how they should be used and interpreted. In this article we illustrate some basic features of error bars and explain how they can help communicate data and assist correct interpretation. Error bars may show confidence intervals, standard errors, standard deviations, or other quantities. Different types of error bars give quite different information, and so figure legends must make clear what error bars represent. We suggest eight simple rules to assist with effective use and interpretation of error bars.

### What are error bars for?

Journals that publish science—knowledge gained through repeated observation or experiment—don't just present new conclusions, they also present evidence so readers can verify that the authors' reasoning is correct. Figures with error bars can, if used properly (1–6), give information describing the data (descriptive statistics), or information about what conclusions, or inferences, are justified (inferential statistics). These two basic categories of error bars are depicted in exactly the same way, but are actually fundamentally different. Our aim is to illustrate basic properties of figures with any of the common error bars, as summarized in Table I, and to explain how they should be used.

### What do error bars tell you?

#### Descriptive error bars.

Range and standard deviation (SD) are used for descriptive error bars because they show how the data are spread (Fig. 1). Range error bars encompass the lowest and highest values. SD is calculated by the formula

where *X* refers to the individual data points, *M* is the mean, and Σ (sigma) means add to find the sum, for all the *n* data points. SD is, roughly, the average or typical difference between the data points and their mean, *M*. About two thirds of the data points will lie within the region of mean ± 1 SD, and ∼95% of the data points will be within 2 SD of the mean.

It is highly desirable to use larger

n, to achieve narrower inferential error bars and more precise estimates of true population values.

Descriptive error bars can also be used to see whether a single result fits within the normal range. For example, if you wished to see if a red blood cell count was normal, you could see whether it was within 2 SD of the mean of the population as a whole. Less than 5% of all red blood cell counts are more than 2 SD from the mean, so if the count in question is more than 2 SD from the mean, you might consider it to be abnormal.

As you increase the size of your sample, or repeat the experiment more times, the mean of your results (*M*) will tend to get closer and closer to the true mean, or the mean of the whole population, μ. We can use *M* as our best estimate of the unknown μ. Similarly, as you repeat an experiment more and more times, the SD of your results will tend to more and more closely approximate the true standard deviation (σ) that you would get if the experiment was performed an infinite number of times, or on the whole population. However, the SD of the experimental results will approximate to σ, whether *n* is large or small. Like *M*, SD does not change systematically as *n* changes, and we can use SD as our best estimate of the unknown σ, whatever the value of *n*.

#### Inferential error bars.

In experimental biology it is more common to be interested in comparing samples from two groups, to see if they are different. For example, you might be comparing wild-type mice with mutant mice, or drug with placebo, or experimental results with controls. To make inferences from the data (i.e., to make a judgment whether the groups are significantly different, or whether the differences might just be due to random fluctuation or chance), a different type of error bar can be used. These are standard error (SE) bars and confidence intervals (CIs). The mean of the data, *M*, with SE or CI error bars, gives an indication of the region where you can expect the mean of the whole possible set of results, or the whole population, μ, to lie (Fig. 2). The interval defines the values that are most plausible for μ.

Because error bars can be descriptive or inferential, and could be any of the bars listed in Table I or even something else, they are meaningless, or misleading, if the figure legend does not state what kind they are. This leads to the first rule. **Rule 1:** when showing error bars, always describe in the figure legends what they are.

### Statistical significance tests and P values

If you carry out a statistical significance test, the result is a P value, where P is the probability that, if there really is no difference, you would get, by chance, a difference as large as the one you observed, or even larger. Other things (e.g., sample size, variation) being equal, a larger difference in results gives a lower P value, which makes you suspect there is a true difference. By convention, if P < 0.05 you say the result is statistically significant, and if P < 0.01 you say the result is highly significant and you can be more confident you have found a true effect. As always with statistical inference, you may be wrong! Perhaps there really is no effect, and you had the bad luck to get one of the 5% (if P < 0.05) or 1% (if P < 0.01) of sets of results that suggests a difference where there is none. Of course, even if results are statistically highly significant, it does not mean they are necessarily biologically important. It is also essential to note that if P > 0.05, and you therefore cannot conclude there is a statistically significant effect, you may not conclude that the effect is zero. There may be a real effect, but it is small, or you may not have repeated your experiment often enough to reveal it. It is a common and serious error to conclude “no effect exists” just because P is greater than 0.05. If you measured the heights of three male and three female Biddelonian basketball players, and did not see a significant difference, you could not conclude that sex has no relationship with height, as a larger sample size might reveal one. A big advantage of inferential error bars is that their length gives a graphic signal of how much uncertainty there is in the data: The true value of the mean μ we are estimating could plausibly be anywhere in the 95% CI. Wide inferential bars indicate large error short inferential bars indicate high precision.

### Replicates or independent samples—what is n?

Science typically copes with the wide variation that occurs in nature by measuring a number (*n*) of independently sampled individuals, independently conducted experiments, or independent observations.

**Rule 2:** the value of *n* (i.e., the sample size, or the number of independently performed experiments) must be stated in the figure legend.

It is essential that *n* (the number of independent results) is carefully distinguished from the number of replicates, which refers to repetition of measurement on one individual in a single condition, or multiple measurements of the same or identical samples. Consider trying to determine whether deletion of a gene in mice affects tail length. We could choose one mutant mouse and one wild type, and perform 20 replicate measurements of each of their tails. We could calculate the means, SDs, and SEs of the replicate measurements, but these would not permit us to answer the central question of whether gene deletion affects tail length, because *n* would equal 1 for each genotype, no matter how often each tail was measured. To address the question successfully we must distinguish the possible effect of gene deletion from natural animal-to-animal variation, and to do this we need to measure the tail lengths of a number of mice, including several mutants and several wild types, with *n* > 1 for each type.

Similarly, a number of replicate cell cultures can be made by pipetting the same volume of cells from the same stock culture into adjacent wells of a tissue culture plate, and subsequently treating them identically. Although it would be possible to assay the plate and determine the means and errors of the replicate wells, the errors would reflect the accuracy of pipetting, not the reproduciblity of the differences between the experimental cells and the control cells. For replicates, *n* = 1, and it is therefore inappropriate to show error bars or statistics.

If an experiment involves triplicate cultures, and is repeated four independent times, then *n* = 4, not 3 or 12. The variation within each set of triplicates is related to the fidelity with which the replicates were created, and is irrelevant to the hypothesis being tested.

To identify the appropriate value for *n*, think of what entire population is being sampled, or what the entire set of experiments would be if all possible ones of that type were performed. Conclusions can be drawn only about that population, so make sure it is appropriate to the question the research is intended to answer.

In the example of replicate cultures from the one stock of cells, the population being sampled is the stock cell culture. For *n* to be greater than 1, the experiment would have to be performed using separate stock cultures, or separate cell clones of the same type. Again, consider the population you wish to make inferences about—it is unlikely to be just a single stock culture. Whenever you see a figure with very small error bars (such as Fig. 3), you should ask yourself whether the very small variation implied by the error bars is due to analysis of replicates rather than independent samples. If so, the bars are useless for making the inference you are considering.

Sometimes a figure shows only the data for a representative experiment, implying that several other similar experiments were also conducted. If a representative experiment is shown, then *n* = 1, and no error bars or P values should be shown. Instead, the means and errors of all the independent experiments should be given, where *n* is the number of experiments performed.

**Rule 3:** error bars and statistics should only be shown for independently repeated experiments, and never for replicates. If a “representative” experiment is shown, it should not have error bars or P values, because in such an experiment, *n* = 1 (Fig. 3 shows what not to do).

### What type of error bar should be used?

**Rule 4:** because experimental biologists are usually trying to compare experimental results with controls, it is usually appropriate to show inferential error bars, such as SE or CI, rather than SD. However, if *n* is very small (for example *n* = 3), rather than showing error bars and statistics, it is better to simply plot the individual data points.

### What is the difference between SE bars and CIs?

#### Standard error (SE).

Suppose three experiments gave measurements of 28.7, 38.7, and 52.6, which are the data points in the *n* = 3 case at the left in Fig. 1. The mean of the data is *M* = 40.0, and the SD = 12.0, which is the length of each arm of the SD bars. *M* (in this case 40.0) is the best estimate of the true mean μ that we would like to know. But how accurate an estimate is it? This can be shown by inferential error bars such as standard error (SE, sometimes referred to as the standard error of the mean, SEM) or a confidence interval (CI). SE is defined as SE = SD/√*n*. In Fig. 4, the large dots mark the means of the same three samples as in Fig. 1. For the *n* = 3 case, SE = 12.0/√3 = 6.93, and this is the length of each arm of the SE bars shown.

The SE varies inversely with the square root of *n*, so the more often an experiment is repeated, or the more samples are measured, the smaller the SE becomes (Fig. 4). This allows more and more accurate estimates of the true mean, μ, by the mean of the experimental results, *M*.

We illustrate and give rules for *n* = 3 not because we recommend using such a small *n*, but because researchers currently often use such small *n* values and it is necessary to be able to interpret their papers. It is highly desirable to use larger *n*, to achieve narrower inferential error bars and more precise estimates of true population values.

#### Confidence interval (CI).

Fig. 2 illustrates what happens if, hypothetically, 20 different labs performed the same experiments, with *n* = 10 in each case. The 95% CI error bars are approximately *M* ± 2xSE, and they vary in position because of course *M* varies from lab to lab, and they also vary in width because SE varies. Such error bars capture the true mean μ on ∼95% of occasions—in Fig. 2, the results from 18 out of the 20 labs happen to include μ. The trouble is in real life we don't know μ, and we never know if our error bar interval is in the 95% majority and includes μ, or by bad luck is one of the 5% of cases that just misses μ.

The error bars in Fig. 2 are only approximately *M* ± 2xSE. They are in fact 95% CIs, which are designed by statisticians so in the long run exactly 95% will capture μ. To achieve this, the interval needs to be *M* ± *t*_{(n–1)} ×SE, where *t*_{(n–1)} is a critical value from tables of the *t* statistic. This critical value varies with *n*. For *n* = 10 or more it is ∼2, but for small *n* it increases, and for *n* = 3 it is ∼4. Therefore *M* ± 2xSE intervals are quite good approximations to 95% CIs when *n* is 10 or more, but not for small *n*. CIs can be thought of as SE bars that have been adjusted by a factor (*t*) so they can be interpreted the same way, regardless of *n*.

This relation means you can easily swap in your mind's eye between SE bars and 95% CIs. If a figure shows SE bars you can mentally double them in width, to get approximate 95% CIs, as long as *n* is 10 or more. However, if *n* = 3, you need to multiply the SE bars by 4.

**Rule 5:** 95% CIs capture μ on 95% of occasions, so you can be 95% confident your interval includes μ. SE bars can be doubled in width to get the approximate 95% CI, provided *n* is 10 or more. If *n* = 3, SE bars must be multiplied by 4 to get the approximate 95% CI.

Determining CIs requires slightly more calculating by the authors of a paper, but for people reading it, CIs make things easier to understand, as they mean the same thing regardless of *n*. For this reason, in medicine, CIs have been recommended for more than 20 years, and are required by many journals (7).

Fig. 4 illustrates the relation between SD, SE, and 95% CI. The data points are shown as dots to emphasize the different values of *n* (from 3 to 30). The leftmost error bars show SD, the same in each case. The middle error bars show 95% CIs, and the bars on the right show SE bars—both these types of bars vary greatly with *n*, and are especially wide for small *n*. The ratio of CI/SE bar width is *t*_{(n–1)} the values are shown at the bottom of the figure. Note also that, whatever error bars are shown, it can be helpful to the reader to show the individual data points, especially for small *n*, as in Figs. 1 and 4, and rule 4.

### Using inferential intervals to compare groups

When comparing two sets of results, e.g., from *n* knock-out mice and *n* wild-type mice, you can compare the SE bars or the 95% CIs on the two means (6). The smaller the overlap of bars, or the larger the gap between bars, the smaller the P value and the stronger the evidence for a true difference. As well as noting whether the figure shows SE bars or 95% CIs, it is vital to note *n*, because the rules giving approximate P are different for *n* = 3 and for *n* ≥ 10.

Fig. 5 illustrates the rules for SE bars. The panels on the right show what is needed when *n* ≥ 10: a gap equal to SE indicates P ≈ 0.05 and a gap of 2SE indicates P ≈ 0.01. To assess the gap, use the average SE for the two groups, meaning the average of one arm of the group C bars and one arm of the E bars. However, if *n* = 3 (the number beloved of joke tellers, Snark hunters (8), and experimental biologists), the P value has to be estimated differently. In this case, P ≈ 0.05 if double the SE bars just touch, meaning a gap of 2 SE.

**Rule 6:** when *n* = 3, and double the SE bars don't overlap, P < 0.05, and if double the SE bars just touch, P is close to 0.05 (Fig. 5, leftmost panel). If *n* is 10 or more, a gap of SE indicates P ≈ 0.05 and a gap of 2 SE indicates P ≈ 0.01 (Fig. 5, right panels).

Rule 5 states how SE bars relate to 95% CIs. Combining that relation with rule 6 for SE bars gives the rules for 95% CIs, which are illustrated in Fig. 6. When *n* ≥ 10 (right panels), overlap of half of one arm indicates P ≈ 0.05, and just touching means P ≈ 0.01. To assess overlap, use the average of one arm of the group C interval and one arm of the E interval. If *n* = 3 (left panels), P ≈ 0.05 when two arms entirely overlap so each mean is about lined up with the end of the other CI. If the overlap is 0.5, P ≈ 0.01.

**Rule 7:** with 95% CIs and *n* = 3, overlap of one full arm indicates P ≈ 0.05, and overlap of half an arm indicates P ≈ 0.01 (Fig. 6, left panels).

### Repeated measurements of the same group

The rules illustrated in Figs. 5 and 6 apply when the means are independent. If two measurements are correlated, as for example with tests at different times on the same group of animals, or kinetic measurements of the same cultures or reactions, the CIs (or SEs) do not give the information needed to assess the significance of the differences between means of the same group at different times because they are not sensitive to correlations within the group. Consider the example in Fig. 7, in which groups of independent experimental and control cell cultures are each measured at four times. Error bars can only be used to compare the experimental to control groups at any one time point. Whether the error bars are 95% CIs or SE bars, they can only be used to assess between group differences (e.g., E1 vs. C1, E3 vs. C3), and may not be used to assess within group differences, such as E1 vs. E2.

Assessing a within group difference, for example E1 vs. E2, requires an analysis that takes account of the within group correlation, for example a Wilcoxon or paired t analysis. A graphical approach would require finding the E1 vs. E2 difference for each culture (or animal) in the group, then graphing the single mean of those differences, with error bars that are the SE or 95% CI calculated from those differences. If that 95% CI does not include 0, there is a statistically significant difference (P < 0.05) between E1 and E2.

**Rule 8:** in the case of repeated measurements on the same group (e.g., of animals, individuals, cultures, or reactions), CIs or SE bars are irrelevant to comparisons within the same group (Fig. 7).

### Conclusion

Error bars can be valuable for understanding results in a journal article and deciding whether the authors' conclusions are justified by the data. However, there are pitfalls. When first seeing a figure with error bars, ask yourself, “What is *n*? Are they independent experiments, or just replicates?” and, “What kind of error bars are they?” If the figure legend gives you satisfactory answers to these questions, you can interpret the data, but remember that error bars and other statistics can only be a guide: you also need to use your biological understanding to appreciate the meaning of the numbers shown in any figure.

## Introduction

Deep learning techniques have recently achieved remarkable successes, especially in vision and language applications [1, 2]. In particular, state-of-the-art deep generative models can generate realistic images or sentences from low-dimensional latent variables [3]. The generated images and text data are often nearly indistinguishable from real data, and data generating performance is rapidly improving [4, 5]. The two most widely types of deep generative models are variational autoencoders (VAEs) and generative adversarial networks (GANs). VAEs use a Bayesian approach to estimate the posterior distribution of a probabilistic encoder network, based on a combination of reconstruction error and the prior probability of the encoded distribution [6]. In contrast, the GAN framework consists of a two-player game between a generator network and a discriminator network [7]. GANs and VAEs possess complementary strengths and weaknesses: GANs generate much better samples than VAEs [8], but VAE training is much more stable and learns more useful “disentangled” latent representations [9]. GANs outperform VAEs in generating sharp image samples [7], while VAEs tend to generate blurry images [10]. GAN training is generally less stable than VAE training, but some recent derivations of GAN like Wasserstein GAN [11–13] significantly improve the stability of GAN training, which is particularly helpful for non-image data.

Achieving a property called “disentanglement”, in which each dimension of the latent representation controls a semantically distinct factor of variation, is a key focus of recent research on deep generative models [14–20]. Disentanglement is important for controlling data generation and generalizing to unseen latent variable combinations. For example, disentangled representations of image data allow prediction of intermediate images [21] and mixing images’ styles [22]. For reasons that are not fully understood, VAEs generally learn representations that are more disentangled than other approaches [23–28]. The state-of-the-art methods for learning disentangled representations capitalize on this advantage by employing modified VAE architectures that further improve disentanglement, including *β*-VAE, FactorVAE, and *β*-TCVAE [9, 29–31]. In contrast, the latent space of the traditional GAN is highly entangled. Some modified GAN architectures, such as InfoGAN [32], encourage disentanglement using purely unsupervised techniques, but these approaches still do not match the disentanglement performance of VAEs [33–40].

Disentanglement performance is usually quantitatively evaluated on standard image datasets with known ground truth factors of variation [41–44]. In addition, disentangled representations can be qualitatively assessed by performing traversals or linear arithmetic in the latent space and visually inspecting the resulting images [45–49].

Recently, molecular biology has seen the rapid growth of single-cell RNA-seq technologies that can measure the expression levels of all genes across thousands to millions of cells [50]. Like image data, for which deep generative models have proven so successful, single-cell RNA-seq datasets are large and high-dimensional. Thus, it seems likely that deep learning will be helpful for single-cell data. In particular, deep generative models hold great promise for distilling semantically distinct facets of cellular identity and predicting unseen cell states.

Several papers have already applied VAEs [51–61] and GANs [62] to single-cell data. A representative VAE method is scGen, which uses the same objective function as *β*-VAE [9]. The learned latent values in scGen are utilized for out-of-sample predictions by latent space arithmetic. The cscGAN paper adapts the Wasserstein GAN approach for single-cell data and shows that it can generate realistic gene expression profiles, proposing to use it for data augmentation.

Assessing disentanglement performance of models on single-cell data is more challenging than image data, because humans cannot intuitively understand the data by looking at it as with images. Previous approaches such as scGen have implicitly used the properties of disentangled representations [51], but disentanglement performance has not been rigorously assessed on single-cell data.

Here, we systematically assess the disentanglement and generation performance of deep generative models on single-cell RNA-seq data. We show that the complementary strengths and weaknesses of VAEs and GANs apply to single-cell data in a similar way as image data. We develop MichiGAN, a neural network that combines the strengths of VAEs and GANs to sample from disentangled representations without sacrificing data generation quality. We employ MichiGAN and other methods on simulated single-cell RNA-seq data [63, 64] and provide quantitative comparisons through several disentanglement metrics [29, 30]. We also learn disentangled representations of three real single-cell RNA-seq datasets [65–67] and show that the disentangled representations can control semantically distinct aspects of cellular identity and predict unseen combinations of cell states.

Our work builds upon that of Lotfollahi et al. [51], who showed that a simple VAE (which they called scGen) can predict single-cell perturbation responses. They also showed several specific biological contexts in which this type of approach is useful. First, they predicted the cell-type-specific gene expression changes induced by treating immune cells with lipopolysaccharide. Second, they predicted the cell-type-specific changes that occur when intestinal epithelial cells are infected by *Salmonella* or *Heligmosomoides polygyrus*. Finally, they showed that scGen can use mouse data to predict perturbation responses in human cells or across other species. For such tasks, one can gain significant biological insights from the generated scRNA-seq profiles.

Our method, MichiGAN, can make the same kinds of predictions and yield the same kinds of biological insights as scGen, but we show that MichiGAN has significant benefits compared to scGen (including disentanglement and data generation performance). In addition, we show that MichiGAN can predict single-cell response to drug treatment, a biological application that was not demonstrated in the scGen paper.

## What Is a Calibration Curve? (with picture)

A calibration curve is a method used in analytical chemistry to determine the concentration of an unknown sample solution. It is a graph generated by experimental means, with the concentration of solution plotted on the x-axis and the observable variable — for example, the solution’s absorbance — plotted on the y-axis. The curve is constructed by measuring the concentration and absorbance of several prepared solutions, called calibration standards. Once the curve has been plotted, the concentration of the unknown solution can be determined by placing it on the curve based on its absorbance or other observable variable.

Chemical solutions absorb different amounts of light based on their concentration. This fact is quantified in an equation known as Beer’s law, which shows a linear relation between a solution’s light absorbance and its concentration. Researchers can measure the absorbance of a solution using a laboratory instrument called a spectrophotometer. This process as a whole is called spectrophotometry.

Spectrophotometry can be useful in determining the concentration of an unknown solution. For example, if a researcher has a sample of river water and wants to know its lead content, he or she can determine it by using a spectrophotometer to plot a calibration curve. First, the researcher creates several standard solutions of lead, ranging from less to more concentrated. These samples are placed into the spectrophotometer, which records a different absorbance for each one.

The experimentally determined absorbance values are plotted on a graph against the known concentration of each calibration standard. A set of points is created, which in the case of absorbance should be roughly linear due to Beer’s law. A line is drawn to connect these data points, forming the calibration curve. In almost every case, the data points will not be mathematically exact, so the line should be drawn to intercept the maximum number of points — it is a line of best fit. Although the relationship of absorbance to concentration is linear, this is not always true for other experimentally determined variables, and occasionally curves must be employed to describe the relationship.

At this stage, the unknown solution can be analyzed. The sample is inserted into the spectrophotometer, and its absorbance is measured. Since this sample is being measured against several standards containing the same compound, its absorbance and concentration must fall somewhere along the calibration curve for that compound. This means that once the solution's absorbance is known, its concentration can be deduced mathematically or graphically.

A horizontal line can be drawn from the unknown solution’s y-value — its absorbance, which has just been measured. The point at which the line crosses the calibration curve will indicate the x-value — the concentration. A vertical line, drawn downwards from this point, gives the concentration of the unknown solution. The equation for the line of the calibration curve can also be used to mathematically determine the solution's concentration.

## Contents

In more general use, a calibration curve is a curve or table for a measuring instrument which measures some parameter indirectly, giving values for the desired quantity as a function of values of sensor output. For example, a calibration curve can be made for a particular pressure transducer to determine applied pressure from transducer output (a voltage). [3] Such a curve is typically used when an instrument uses a sensor whose calibration varies from one sample to another, or changes with time or use if sensor output is consistent the instrument would be marked directly in terms of the measured unit.

The data - the concentrations of the analyte and the instrument response for each standard - can be fit to a straight line, using linear regression analysis. This yields a model described by the equation *y = mx + y _{0}*, where

**y**is the instrument response,

**m**represents the sensitivity, and

**y**is a constant that describes the background. The analyte concentration (

_{0}**x**) of unknown samples may be calculated from this equation.

Many different variables can be used as the analytical signal. For instance, chromium (III) might be measured using a chemiluminescence method, in an instrument that contains a photomultiplier tube (PMT) as the detector. The detector converts the light produced by the sample into a voltage, which increases with intensity of light. The amount of light measured is the analytical signal.

Most analytical techniques use a calibration curve. There are a number of advantages to this approach. First, the calibration curve provides a reliable way to calculate the uncertainty of the concentration calculated from the calibration curve (using the statistics of the least squares line fit to the data). [4] [5]

Second, the calibration curve provides data on an empirical relationship. The mechanism for the instrument's response to the analyte may be predicted or understood according to some theoretical model, but most such models have limited value for real samples. (Instrumental response is usually highly dependent on the condition of the analyte, solvents used and impurities it may contain it could also be affected by external factors such as pressure and temperature.)

Many theoretical relationships, such as fluorescence, require the determination of an instrumental constant anyway, by analysis of one or more reference standards a calibration curve is a convenient extension of this approach. The calibration curve for a particular analyte in a particular (type of) sample provides the empirical relationship needed for those particular measurements.

The chief disadvantages are (1) that the standards require a supply of the analyte material, preferably of high purity and in known concentration, and (2) that the standards and the unknown are in the same matrix. Some analytes - e.g., particular proteins - are extremely difficult to obtain pure in sufficient quantity. Other analytes are often in complex matrices, e.g., heavy metals in pond water. In this case, the matrix may interfere with or attenuate the signal of the analyte. Therefore, a comparison between the standards (which contain no interfering compounds) and the unknown is not possible. The method of standard addition is a way to handle such a situation.

As expected, the concentration of the unknown will have some error which can be calculated from the formula below. [6] [7] [8] This formula assumes that a linear relationship is observed for all the standards. It is important to note that the error in the concentration will be minimal if the signal from the unknown lies in the middle of the signals of all the standards (the term y u n k − y ¯

## P-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the **test statistic will produce values at least as extreme as the t-score produced for your sample**. As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf_{t,d} we denote the **cumulative distribution function** of the t-Student distribution with d degrees of freedom:

p-value from **left-tailed** t-test:

p-value from **right-tailed** t-test:

p-value from **two-tailed** t-test:

or, equivalently: p-value = 2 - 2 * cdf_{t,d}(|t_{score}|)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

## ELISA analysis

**Qualitative ELISA**only determines whether the antigen is present or not in the sample. It requires a blank well containing no antigen or an unrelated control antigen.**Semi-quantitative ELISA**allows the relative comparison of the antigen levels between the samples.**Quantitative ELISA**allows calculating the amount of antigen present in the sample. It requires comparison of the values measured for the samples with a standard curve prepared from a serial dilution of a purified antigen in a known concentration. This is the most commonly reported ELISA data.

**ELISA standard curve**The standard or calibration curve is the element of the quantitative ELISA that will allow calculating the concentration of antigen in the sample.

The standard curve is derived from plotting known concentrations of a reference antigen against the readout obtained for each concentration (usually optical density at 450 nm).

Most ELISA plate readers will incorporate a software for curve fitting and data analysis. The concentration of the antigen in the sample is calculated by extrapolation of the linear portion of the standard curve.

**Figure 1:** Example of a quantitative ELISA standard curve from Human ICAM1 SimpleStep ELISA® Kit (ab174445).

**Curve fitting software** **allow** **using different models to plot your data.**

## How can I fit the A595 of my unknown samples to the data? - Biology

**Water Chemistry II: Spectrophotmetry & Standard Curves**

**Standard curves are graphs of light absorbance versus solution concentration which can be used to figure out the solute concentration in unknown samples. We generated a standard curve for a set of albumin samples.**

** **

** Interpreting a Standard Curve A spectrophotometer measures light quantity. It tells you how much light is passing through a solution (transmittance) or how much light is being absorbed by a solution (absorbance).**

If you graph absorbance versus concentration for a series of known solutions, the line, or **standard curve**, which fits to your points can be used to figure out the concentrations of an unknown solution. Absorbance, the dependent variable, is placed on the y-axis (the vertical axis). Concentration, the independent variable (because it was set by you when setting up the experiment), is graphed on the x-axis. When you measure the absorbance of an unknown sample, find that y-value on the standard curve. Then trace downward to see which concentration matches up to it. Mouse over the graph below to see an example of this.

Below is a standard curve generated from absorbance data similar to what we generated in class. Notice that as concentration increases, absorbance increases as well. While you can estimate concentration of an unknown from just looking at the graph, a more accurate way to determine concentration to actually use the equation of the line which fits to your data points. This equation is given in the y-intercept form: y=mx+b

where m is the slope of the line and b is the y-intercept (where the line touches the y-axis).

The equation y=mx+b can be translated here as "absorbance equals slope times concentration plus the y-intercept absorbance value." The slope and the y-intercept are provided to you when the computer fits a line to your standard curve data. The absorbance (or y) is what you measure from your unknown. So, all you have to do is pop those three numbers into the equation and solve for x (concentration).

An example: your unknown's absorbance (y) is 6.00

Based on the curve below, slope (m) = 8 and b=0

if y=mx+b then 6.00 = 8*x + 0

and 6/8=x

and 0.75=x

so the concentration of the unknown would be 75% the original stock, which was 100 ug/ml. 75% of 100 ug/ml = 75 ug/ml concentration.

The units on the graph below are absorbance (y) versus the dilution factor (x) of each solution used (0=water to 1=undiluted stock).

## Using the line equation to calculate unknown sample values

So you now have the equation for the line. To determine unknown values based on this, so in this example it would be using background-corrected absorbance values for unknown samples (‘**y’**) to work out the total protein concentration (‘**x**‘), you have to re-organise the equation to determine what ‘**x**‘ is. Let’s do this in the example:

Now you can just use the rearranged equation to input the ‘**y**‘ values in order to work out ‘**x**‘. For the example, let’s say we have a sample that we do not know the total protein concentration. We measure the absorbance of the sample at 562 nm and subtract the background value to account for the background signal. The sample has a background-corrected absorbance of ‘**0.497**‘. This would be our ‘**y**‘ value in the equation. Entering this would give:

So, the total protein concentration for the unknown sample is ‘**550.33 μg/mL**‘.

## Calibration Curves

Calibration curves are used to understand the instrumental response to an analyte and predict the concentration in an unknown sample. Generally, a set of standard samples are made at various concentrations with a range than includes the unknown of interest and the instrumental response at each concentration is recorded. For more accuracy and to understand the error, the response at each concentration can be repeated so an error bar is obtained. The data are then fit with a function so that unknown concentrations can be predicted. Typically the response is linear, however, a curve can be made with other functions as long as the function is known. The calibration curve can be used to calculate the limit of detection and limit of quantitation.

When making solutions for a calibration curve, each solution can be made separately. However, that can take a lot of starting material and be time consuming. Another method for making many different concentrations of a solution is to use serial dilutions. With serial dilutions, a concentrated sample is diluted down in a stepwise manner to make lower concentrations. The next sample is made from the previous dilution, and the dilution factor is often kept constant. The advantage is that only one initial solution is needed. The disadvantage is that any errors in solution making—pipetting, massing, etc.—get propagated as more solutions are made. Thus, care must be taken when making the initial solution.

### Principles

Calibration curves can be used to predict the concentration of an unknown sample. To be completely accurate, the standard samples should be run in the same matrix as the unknown sample. A sample matrix is the components of the sample other than the analyte of interest, including the solvent and all salts, proteins, metal ions, etc. that might be present in the sample. In practice, running calibration samples in the same matrix as the unknown is sometimes difficult, as the unknown sample may be from a complex biological or environmental sample. Thus, many calibration curves are made in a sample matrix that closely approximates the real sample, such as artificial cerebral spinal fluid or artificial urine, but may not be exact. The range of concentrations of the calibration curve should bracket that in the expected unknown sample. Ideally a few concentrations above and below the expected concentration sample are measured.

Many calibration curves are linear and can be fit with the basic equation y=mx+b, where m is the slope and b is the y-intercept. However, not all curves are linear and sometimes to get a line, one or both set of axes will be on a logarithmic scale. Linear regression is typically performed using a computer program and the most common method is to use a least squares fitting. With a linear regression analysis, an R 2 value, called the coefficient of determination, is given. For a simple single regression, R 2 is the square of the correlation coefficient (r) and provides information about how far away the y values are from the predicted line. A perfect line would have an R 2 value of 1, and most R 2 values for calibration curves are over 0.95. When the calibration curve is linear, the slope is a measure of sensitivity: how much the signal changes for a change in concentration. A steeper line with a larger slope indicates a more sensitive measurement. A calibration curve can also help define the linear range, the range of concentrations that the instrument gives a linear response. Outside this range, the response may taper off due to instrumental considerations, and the equation from the calibration cannot be used. This is known as the limit of linearity.

Limit of detection is the lowest amount that can be statistically determined from the noise. Generally this is defined as a signal that is 3 times the noise. The limit of detection can be calculated from the slope of the calibration curve and is generally defined as LOD=3*S.D./m, where S.D. is the standard deviation of the noise. The noise is measured by taking the standard deviation of multiple measurements. Alternatively, in one trace, noise can be estimated as the standard deviation of the baseline. The limit of quantitation is the amount that can be differentiated between samples and is usually defined as 10 times the noise.

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### Procedure

1. Making the Standards: Serial Dilutions

- Make a concentrated stock solution of the standard. Typically, the compound is accurately weighed out and then quantitatively transferred into a volumetric flask. Add some solvent, mix so the sample dissolves, then fill to the line with the proper solvent.
- Perform serial dilutions. Take another volumetric flask and pipette the amount of standard needed for the dilution, then fill to the line with solvent and mix. A ten-fold dilution is typically made, so for a 10-mL volumetric flask, add 1 mL of the previous dilution.
- Continue as needed for more dilutions, pipetting from the previous solution to dilute it to make the next sample. For a good calibration curve, at least 5 concentrations are needed.

2. Run the Samples for the Calibration Curve and the Unknown

- Run the samples with the UV-Vis spectrophotometer to determine the instrumental response needed for the calibration curve.
- Take the reading with the first sample. It is a good idea to run the samples in random order (
*i.e.*not highest to lowest or lowest to highest) in case there are any systematic errors. In order to get an estimate of noise, repeat the reading on any given sample 3x. - Run the additional standard samples, repeating the measurements for each sample to get an estimate of noise. Record the data to make a plot later.
- Run the unknown sample(s). Use as similar conditions to running the standards as possible. Thus, the sample matrix or buffer should be the same, the pH should be the same, and the concentration should be in the range of the standards run.

3. Making the Calibration Curve

- Record the data in a spreadsheet and use a computer program to plot the data vs concentration. If at least triplicate measurements were taken for each point, error bars can be plotted of the standard deviation of those measurements to estimate of the error of each point. For some curves, the data might need to be plotted with an axis as a log to get a line. The equation that governs the calibration curve is generally known ahead of time, so a log-plot is used when there is a log in the equation.
- Examine the calibration curve. Does it look linear? Does it have a portion that looks non-linear (
*i.e.*reached the limit of the instrumental response)? To check, fit all the data to a linear regression using the software. If the coefficient of determination (R 2 ) is not high, remove some of the points at the beginning or ending of the curve that do not appear to fit the line and perform the linear regression again. It is not acceptable to remove points in the middle just because they have a large error bar. From this analysis, decide what portion of the curve is linear. - The output of the linear should be an equation of the format y=mx+b, where m is the slope and b is the y-intercept. The units for the slope are the y-axis unit/concentration, in this example (
**Figure 1**) absorbance/μM. The units for the y-intercept are the y-axis units. A coefficient of determination (R 2 ) is obtained. The higher the R 2 the better the fit a perfect fit gives an R 2 of 1. The program may also be able to give an estimate of the error on the slope and the intercept.

4. Results: Calibration Curve of Absorbance of Blue Dye #1

- The calibration curve for absorbance of blue dye #1 (at 631 nm) is shown below (
**Figure 1**). The response is linear from 0 to 10 µM. Above that concentration the signal begins to level off because the response is out of the linear range of the UV-Vis spectrophotometer. - Calculate the LOD. From the slope of the calibration curve, the LOD is 3*S.D. (noise)/m. For this calibration curve, the noise was obtained by taking a standard deviation of repeated measurements and was 0.021. The LOD would be 3*0.021/.109=0.58 µM.
- Calculate the LOQ. The LOQ is 10*S.D.(noise)/m. For this calibration curve, LOQ is 10*0.021/.109 =1.93 µM.
- Calculate the concentration of the unknown. Use the line equation to calculate the concentration of the unknown sample. The calibration curve is only valid if the unknown falls into the linear range of the standard samples. If the readings are too high, dilution might be necessary. In this example, the unknown sports drink was diluted 1:1. The absorbance was 0.243 and this corresponded to a concentration of 2.02 µM. Thus the final concentration of blue dye #1 in the in the sports drink was 4.04 µM.

**Figure 1. Calibration curves for UV-Vis absorbance of blue dye.** *Left:* The absorbance was measured of different concentrations of blue dye #1. The responses level off after 10 µM, when the absorbance is over 1. The error bars are from repeated measurements of the same sample and are standard deviations. *Right:* The linear portion of the calibration curve is fit with a line, y=0.109*x + 0.0286. The unknown data is shown in black. Please click here to view a larger version of this figure.

Calibration curves are used to understand the instrumental response to an analyte, and to predict the concentration of analyte in a sample.

A calibration curve is created by first preparing a set of standard solutions with known concentrations of the analyte. The instrument response is measured for each, and plotted vs. concentration of the standard solution. The linear portion of this plot can then be used to predict the concentration of a sample of the analyte, by correlating its response to concentration.

This video will introduce calibration curves and their use, by demonstrating the preparation of a set of standards, followed by the analysis of a sample with unknown concentration.

A set of standard solutions is used to prepare the calibration curve. These solutions consist of a range of concentrations that encompass the approximate concentration of the analyte.

Standard solutions are often prepared with a serial dilution. A serial dilution is performed by first preparing a stock solution of the analyte. The stock solution is then diluted by a known amount, often one order of magnitude. The new solution is then diluted in the same manner, and so on. This results in a set of solutions with concentrations ranging over several orders of magnitude.

The calibration curve is a plot of instrumental signal vs. concentration. The plot of the standards should be linear, and can be fit with the equation y=mx+b. The non-linear portions of the plot should be discarded, as these concentration ranges are out of the limit of linearity.

The equation of the best-fit line can then be used to determine the concentration of the sample, by using the instrument signal to correlate to concentration. Samples with measurements that lie outside of the linear range of the plot must be diluted, in order to be in the linear range.

The limit of detection of the instrument, or the lowest measurement that can be statistically determined over the noise, can be calculated from the calibration curve as well. A blank sample is measured multiple times. The limit of detection is generally defined as the average blank signal plus 3 times its standard deviation.

Finally, the limit of quantification can also be calculated. The limit of quantification is the lowest amount of analyte that can be accurately quantified. This is calculated as 10 standard deviations above the blank signal.

Now that you've learned the basics of a calibration curve, let's see how to prepare and use one in the laboratory.

First, prepare a concentrated stock solution of the standard. Accurately weigh the standard, and transfer it into a volumetric flask. Add a small amount of solvent, and mix so that the sample dissolves. Then, fill to the line with solvent. It is important to use the same solvent as the sample.

To prepare the standards, pipette the required amount in the volumetric flask. Then fill the flask to the line with solvent, and mix.

Continue making the standards by pipetting from the stock solution and diluting. For a good calibration curve, at least 5 concentrations are needed.

Now, run samples with the analytical instrument, in this case a UV-Vis spectrophotometer, in order to determine the instrumental response needed for the calibration curve.

Take the measurement of the first standard. Run the standards in random order, in case there are any systematic errors. Measure each standard 3x to get an estimate of noise.

Measure the rest of the standards, repeating the measurements for each. Record all data.

Finally, run the sample. Use the same sample matrix and measurement conditions as were used for the standards. Make sure that the sample is within the range of the standards and the limit of the instrument.

To construct the calibration curve, use a computer program to plot the data as signal vs. concentration. Use the standard deviation of the repeated measurements for each data point to make error bars.

Remove portions of the curve that are non-linear, then perform a linear regression and determine the best-fit line. The output should be an equation in the form y = m x + b. An R 2 -value near 1 denotes a good fit.

This is the calibration curve for blue dye #1, measured at 631 nm. The response is linear between 0 and 15 mM.

Calculate the concentration of the sample using the equation of the best-fit line. The absorbance for the sample was 0.141, and corresponded to a concentration of 6.02 mM.

Now that you've seen how a calibration curve can be used with a UV-Vis spectrophotometer, let's take a look at some other useful applications.

Calibration curves are often used with electrochemistry applications, as the electrode signal must be calibrated to the concentration of ions in the solution. In this example, data were collected for an ion-selective electrode for fluoride.

The concentration data must be plotted on the log scale to obtain a line. This calibration curve can be used to measure the concentration of fluoride in a solution, such as toothpaste or drinking water.

High-performance liquid chromatography, or HPLC, is a separation and analysis technique that is used heavily in analytical chemistry. HPLC separates components of a mixture based on the time required for the molecules to travel the length of the chromatography column. This time varies depending on a range of chemical properties of the molecules.

The elution of the molecules is measured using a detector, resulting in a chromatogram. The peak area can be correlated to concentration using a simple calibration curve of a range of standard solutions, like in this example of popular soda ingredients.

In some cases, where the solution matrix interferes with the measurement of the solute, a classical calibration curve can be inaccurate. In those cases, a modified calibration curve is prepared. For this, a range of standard solution volumes is added to the sample. The signal to concentration plot is created, where the x intercept is equal to the original concentration of the sample solution. For more detail on this technique, please watch the JoVE science education video, "The method of standard addition".

You've just watched JoVE's introduction to the calibration curve. You should now understand where the calibration curve is used, how to create it, and how to use it to calculate concentrations of samples.

As always, thanks for watching!

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### Applications and Summary

Calibration curves are used in many fields of analytical chemistry, biochemistry, and pharmaceutical chemistry. It is common to use them with spectroscopy, chromatography, and electrochemistry measurements. A calibration curve can be used to understand the concentration of an environmental pollutant in a soil sample. It could be used determine the concentration of a neurotransmitter in a sample of brain fluid, vitamin in pharmaceutical samples, or caffeine in food. Thus, calibration curves are useful in environmental, biological, pharmaceutical, and food science applications. The most important part of making a calibration curve is to make accurate standard samples that are in a matrix that closely approximates the sample mixture.

An example of an electrochemistry calibration curve is shown below (**Figure 2**). The data were collected with an ion-selective electrode for fluoride. Electrochemical data follow the Nernst equation E=E 0 + 2.03*R*T/(nF) * log C. Thus, the concentration data (x-axis) must be plotted on a log scale to obtain a line. This calibration curve could be used to measure the concentration of fluoride in toothpaste or drinking water.

**Figure 2. Calibration curve for an ion-selective electrode. **The response of a fluoride selective electrode (in mV) to different concentrations of fluoride is plotted. The expected equation for the electrode response is y (in mV)=-59.2*log x+b at 25 °C. The actual equation is y=-57.4*log x +56.38. The R 2 value is 0.998. Please click here to view a larger version of this figure.

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### Transcript

Calibration curves are used to understand the instrumental response to an analyte, and to predict the concentration of analyte in a sample.

A calibration curve is created by first preparing a set of standard solutions with known concentrations of the analyte. The instrument response is measured for each, and plotted vs. concentration of the standard solution. The linear portion of this plot can then be used to predict the concentration of a sample of the analyte, by correlating its response to concentration.

This video will introduce calibration curves and their use, by demonstrating the preparation of a set of standards, followed by the analysis of a sample with unknown concentration.

A set of standard solutions is used to prepare the calibration curve. These solutions consist of a range of concentrations that encompass the approximate concentration of the analyte.

Standard solutions are often prepared with a serial dilution. A serial dilution is performed by first preparing a stock solution of the analyte. The stock solution is then diluted by a known amount, often one order of magnitude. The new solution is then diluted in the same manner, and so on. This results in a set of solutions with concentrations ranging over several orders of magnitude.

The calibration curve is a plot of instrumental signal vs. concentration. The plot of the standards should be linear, and can be fit with the equation y=mx+b. The non-linear portions of the plot should be discarded, as these concentration ranges are out of the limit of linearity.

The equation of the best-fit line can then be used to determine the concentration of the sample, by using the instrument signal to correlate to concentration. Samples with measurements that lie outside of the linear range of the plot must be diluted, in order to be in the linear range.

The limit of detection of the instrument, or the lowest measurement that can be statistically determined over the noise, can be calculated from the calibration curve as well. A blank sample is measured multiple times. The limit of detection is generally defined as the average blank signal plus 3 times its standard deviation.

Finally, the limit of quantification can also be calculated. The limit of quantification is the lowest amount of analyte that can be accurately quantified. This is calculated as 10 standard deviations above the blank signal.

Now that you've learned the basics of a calibration curve, let's see how to prepare and use one in the laboratory.

First, prepare a concentrated stock solution of the standard. Accurately weigh the standard, and transfer it into a volumetric flask. Add a small amount of solvent, and mix so that the sample dissolves. Then, fill to the line with solvent. It is important to use the same solvent as the sample.

To prepare the standards, pipette the required amount in the volumetric flask. Then fill the flask to the line with solvent, and mix.

Continue making the standards by pipetting from the stock solution and diluting. For a good calibration curve, at least 5 concentrations are needed.

Now, run samples with the analytical instrument, in this case a UV-Vis spectrophotometer, in order to determine the instrumental response needed for the calibration curve.

Take the measurement of the first standard. Run the standards in random order, in case there are any systematic errors. Measure each standard 3–5x to get an estimate of noise.

Measure the rest of the standards, repeating the measurements for each. Record all data.

Finally, run the sample. Use the same sample matrix and measurement conditions as were used for the standards. Make sure that the sample is within the range of the standards and the limit of the instrument.

To construct the calibration curve, use a computer program to plot the data as signal vs. concentration. Use the standard deviation of the repeated measurements for each data point to make error bars.

Remove portions of the curve that are non-linear, then perform a linear regression and determine the best-fit line. The output should be an equation in the form y = m x + b. An R2-value near 1 denotes a good fit.

This is the calibration curve for blue dye #1, measured at 631 nm. The response is linear between 0 and 15 mM.

Calculate the concentration of the sample using the equation of the best-fit line. The absorbance for the sample was 0.141, and corresponded to a concentration of 6.02 mM.

Now that you've seen how a calibration curve can be used with a UV-Vis spectrophotometer, let's take a look at some other useful applications.

Calibration curves are often used with electrochemistry applications, as the electrode signal must be calibrated to the concentration of ions in the solution. In this example, data were collected for an ion-selective electrode for fluoride.

The concentration data must be plotted on the log scale to obtain a line. This calibration curve can be used to measure the concentration of fluoride in a solution, such as toothpaste or drinking water.

High-performance liquid chromatography, or HPLC, is a separation and analysis technique that is used heavily in analytical chemistry. HPLC separates components of a mixture based on the time required for the molecules to travel the length of the chromatography column. This time varies depending on a range of chemical properties of the molecules.

The elution of the molecules is measured using a detector, resulting in a chromatogram. The peak area can be correlated to concentration using a simple calibration curve of a range of standard solutions, like in this example of popular soda ingredients.

In some cases, where the solution matrix interferes with the measurement of the solute, a classical calibration curve can be inaccurate. In those cases, a modified calibration curve is prepared. For this, a range of standard solution volumes is added to the sample. The signal to concentration plot is created, where the x intercept is equal to the original concentration of the sample solution. For more detail on this technique, please watch the JoVE science education video, "The method of standard addition".

You've just watched JoVE's introduction to the calibration curve. You should now understand where the calibration curve is used, how to create it, and how to use it to calculate concentrations of samples.