CHEM 370 | Week 3

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<h1 style = "text-align: left; font-weight: bold; margin-left: 175px;">Week 3A: Describing Uncertainty</h1>
<h5 style = "text-align: left; font-weight: bold; margin-left: 175px;">Harvey Chs 3, 4</h5>

???

- Goal of class, esp. lab, is to learn how to know when you're likely right -- or, more importantly -- when you're not right!
- Tylenol Example
  - If you get close to the label conc., are you right? Is the difference important?
  - If you are far from the label conc., are you wrong?

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# Accuracy vs. Precision

.image-credit[[Arbeck via Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Accuracy_and_Precision.svg), [CC BY 4.0](https://creativecommons.org/licenses/by/4.0) ]

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<h1 style = "text-align: left; font-weight: bold; margin-left: 175px;">Rethinking Accuracy and Precision</h1>

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$$ e = \overline{X} - \mu $$

$$ e = \text{measured} - \text{expected} $$

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$$ e\text{ (%)} = \frac{\overline{X} - \mu}{\mu} \times 100 $$

$$ e\text{ (%)} = \frac{\text{measured} - \text{expected}}{\text{expected}} \times 100 $$

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# Propagation of Uncertainty

Addition / subtraction

$$ R = A + B - C $$

$$ u\_R = \sqrt{ u\_A^2 + u\_B^2 + u\_C^2 } $$

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# Propagation of Uncertainty

Multiplication / division

$$ R = \frac{A \times B}{C} $$

$$ u\_R = \sqrt{ \left(\frac{u\_A}{A}\right)^2 + \left(\frac{u\_B}{B}\right)^2  + \left(\frac{u\_C}{C}\right)^2  } $$

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# Practice

You analyze a verified solution known to contain $0.55000 \pm 0.00002$ ppm Cu using a new method and determine the concentration of Cu according to:

$$ S\_{Cu} = k\_{Cu} C\_{Cu} + S_{mb} $$

with

$$ S\_{Cu} = 0.9648 \pm 0.0002$$

$$ k\_{Cu} = 1.571 \pm 0.001 \text{ ppm}^{-1}$$

$$ S_{mb} = 0.0501 \pm 0.0002 $$

What is the concentration of Cu and it's uncertainty?  What is the error on your new method?

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# Part I: Concentration & Uncertainty

## Concentration

1. Plug in values and solve for $C\_{Cu}$!

$$ C\_{Cu} = 0.5822 \text{ppm} $$

## Uncertainty

1. Uncertainties for $S\_{Cu}$ and $S\_{mb}$ are combined according to addition / subtraction rule (absolute uncertainty).

$$ u = \sqrt{ 0.0002^2 + 0.0002^2 } = 0.0002828 $$
    
1. Uncertainties for $[S\_{Cu} - S\_{mb}]$ and $k\_{Cu}$ are combined according to multiplication / division rule (relative uncertainty).

$$ u = \sqrt{ \left( \frac{0.0002828}{ 0.91147 } \right)^2 + \left( \frac{0.001}{ 1.571 } \right)^2 } = 0.00070765 $$
    
    Note that $ 0.9648 - 0.0501 $ is $0.91147$.

1. This is *relative error*!  Convert back to absolute error!

$$ 0.5822 \times 0.00070765 = 0.000411993 \text{ppm} $$

## Combine it all together

$$ C\_{Cu} = 0.5822 \pm 0.0004 \text{ppm} $$

# Part II: Error

Remember error = $\frac{\overline{X} - \mu}{\mu}$

$$e = \frac{0.5822 - 0.55000}{0.55000} \times 100 = 5.855 \%$$

Always set up this way so the +/- signs give info about direction high or low!

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# Confidence Intervals

> A **confidence interval** gives a range of values for an unknown parameter (e.g. a population mean).

If we select a random member from a population, what is its most likely value?

$$x\_i = \mu \pm z \sigma$$

.image-credit[David Harvey / [Analytical Chemistry 2.1](https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Book%3A_Analytical_Chemistry_2.1_%28Harvey%29) / [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/3.0/at/deed.en)]

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# Confidence Intervals

> A **confidence interval** gives a range of values for an unknown parameter (e.g. a population mean).

Or, if we select a random member of a population, what is the most likely mean?

$$\mu = x\_i \pm z \sigma$$

???

For a sample, this becomes:

$$\mu = \overline{X}\_i \pm t \frac{s}{\sqrt{n}} $$

???

gives a range of values for an unknown parameter (for example, a population mean)

This means that the confidence level represents the theoretical long-run frequency (i.e., the proportion) of confidence intervals that contain the true value of the unknown population parameter. In other words, 90% of confidence intervals computed at the 90% confidence level contain the parameter, 95% of confidence intervals computed at the 95% confidence level contain the parameter, 99% of confidence intervals computed at the 99% confidence level contain the parameter, etc.[2]

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# Comparing Means

> If confidence intervals do not overlap there is a signficant difference.

<img src="img/chapter-4/barplot_sigDiff.png" style = "height:350px; margin-left: auto; margin-right: auto; display: block;">
*The mean concentration of three samples. Error bars represent the 95% confidence interval. Dashed line is the known true concentration.*

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# Practice

An analyst determined the zinc content of a solution and obtained the data below.  What is the concentration of zinc in solution and the 95% CI?

Assuming the solution is known to contain 6.0 $\pm$ 0.2 ppm Zn, determine if there is a significant difference between the measured and expected concentrations.

| Replicate | [Zn] (ppm) |
| :-------: | :--------: |
| 1         | 5.4        |
| 2         | 2.9        |
| 3         | 4.7        |
| 4         | 2.7        |
| 5         | 5.1        |
| 6         | 4.8        |
| 7         | 7.2        |
| 8         | 6.6        |
| 9         | 3.8        |

???

- $\overline{X} = 4.8$
- $s = 1.5$
- $n = 9$ so D.O.F = $8$
- From *t*-table: $t = 2.306 (\alpha = 0.05)$

$$ CI = t \frac{s}{\sqrt{n}} = 2.306 \frac{1.5}{\sqrt{9}} = 1.2 \text{ppm} $$

$$\overline[Zn] = 4.8 \pm 1.2 \text{ppm} $$

Thus,

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# Limit of Detection

.image-credit[Fischer and Smith (2018) [AS&T](https://www.tandfonline.com/doi/full/10.1080/02786826.2017.1413231)]

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# Limit of Detection

???

- **Type 1 Error:** False positive
- **Type 2 Error:** False negative

# Limits of Detection

- Lower Limit of Detection

$$ LOD = S\_{mb} + 3s\_{mb} $$ or $$ LOD = \frac{3s\_{mb}}{k\_A} $$

- Lower Limit of Quantitation

$$ LOD = S\_{mb} + 10s\_{mb} $$ or $$ LOD = \frac{10s\_{mb}}{k\_A} $$