**Chapter 4: Evaluating Analytical Data**

Range

: largest value – smallest value

Standard deviation (s)

:

Variance

is the value of standard deviation squared.

Percent relative standard deviation (%rsd)

:

Measurement of Spread

Mean

: average obtained by dividing the sum of the measurement by the number of measurements.

Median

: the middle value when data is in numerical order from lowest to highest value. For an even number of values, the median is the mean of the two central values.

Measurement of Central Tendency

Measurement errors

: all measurements include an error. Manufactures of equipment and glassware state the maximum measurement error (tolerance) of that piece of equipment.

Method errors

: introduced when assumptions about the relationship between the signal and the analyte are invalid.

Sampling errors

: occur when sampling strategy does not produce a representative sample.

Personal errors

: measurements are subject to human error. For example, going over the equivalence point in a titration.

Sources of Determinate Errors

Determinate Error

(systematic error): affects accuracy only, deviations in one direction (results too high or too low). Can be identified and eliminated.

Indeterminate Error

(random error): affects precision only, deviations in both directions. Cannot be eliminated. Can be reduced by the use of more precise equipment and techniques.

Types of Error

Two types:

Repeatability

: the precision for one set of conditions. E.g. same analyst, same day, same session, same equipment.

Reproducibility

: the precision under different set of conditions. E.g. different days, different analysts etc.

Precision

**Characterizing Results and Errors**

When measurements are multipled or divided, the relative uncertainty of the result is the sum of the squares of the relative uncertainties of the individual measurements.

For mixed operations, the rules are applied in steps in the same order as the calculation.

Uncertainty When Multiplying or Dividing

Often multiple measurements are combined to obtain a result.

When measurements are added or subtracted, the absolute uncertainty in the result is the square root of the sum of the squares of the individual absolute uncertainties.

Uncertainty When Adding or Subtracting

Determined by experimentally measuring precision or using the stated tolerance.

Absolute uncertainty (sR) is the standard deviation or tolerance value e.g. a class A 10 mL volumetric pipet has a stated tolerance of +/- 0.02 mL (it is expected to deliver between 9.98 mL and 10.02 mL if used correctly)

Relative uncertainly is the absolute error divided by the measurement value e.g. a 10 mL class A volumetric pipet has a relative error of 0.002 (0.02 mL/ 10 mL)

Uncertainty

**Propagation of Uncertainties**

A more useful way to represent the precision of a result.

For a sample we use the standard error of the mean (s.e.m.) to estimate the population standard deviation.

In addition the value of 2 times the standard deviation is only true for an infinite numbers of values. For a finite number we look up a value of t (from a student t table) instead.

The confidence interval for a sample becomes:

Confidence Intervals

Populations can be distributed around the mean value in a number of ways.

For analytical chemistry, we usually assume a normal (or Gaussian) distribution.

For a population 95% of the values are within +/- 2 standard deviations of the mean. We can be 95% confident that a value from taken randomly from the population is in the range:

Population Distributions

A population refers to all the objects in a system system under investigation.

A sample is a collection of objects taken from the population, chosen to represent the population.

e.g. a sample of one hundred pennies may be measured in order to represent all pennies in circulation (the population)

Population statistics are represented by Greek letters.

Populations and Samples

**The Distribution of Measurements and Results**

An outlier is a data point that is so different from the other values in the data set that it cannot be explained by indeterminate error.

The most common method to test for an outlier is the Dixon’s Q test.

An experimental value of Q is calculated using the equation:

If the experiment values is larger than the tabulated value of Q, the value can be rejected as an outlier.

The Q-test can only be applied once for a set of data.

Outliers

Paired data is generated when different samples are measured by two different techniques.

In this case we test if the difference between the two results is significantly different than zero.

An experimental value of t is calculated using the equation:

The experimental value is compared to the tabulated value.

Paired Data

An experimental value of t is calculated using the equation:

A pooled standard deviation is calculated using the equation:

If the experimental value is great than the tabulated value, the null hypothesis is rejected.

Comparing two sample means

Comparing sample and population means

To compare precision between two sets of data we use the F test.

An experimental value of F is calculated.

If the experimental value is larger than the tabulated value, the null hypothesis is rejected indicating significant difference.

Values of F are equal of greater than one.

Comparing Precision

A significance test can be one or two tailed depending on the question.

If question implies direction, it is a one-tailed test e.g. does a catalyst increase the rate of a reaction? Or does a medication improve a patients condition?

If question is independent of direction, it is two tailed e.g. Does a new method give different (higher or lower) results than the old one?

For a fixed confidence interval, a two-tailed test is more conservative (less likely to show significant difference) than a one-tailed test.

Values of t and F and different for one and two-tailed tests

One and Two Tailed

2.5%

5%

Two- Tailed

One-tailed

95% Confidence

**Significance Testing**

An experimental value of t is calculated using the equation:

If this value is greater than the tabulated value, the Null hypothesis is rejected indicating a significant difference in means.

Often used to test if a method is accurate by measuring a standard reference material.

The smallest concentration or absolute amount of substance that can be measured with statistical confidence

The signal at the detection limit is given by:

Z is usually chosen to be 3.

Detection Limit

Detection limit is usually stated as concentration or mass.

To convert the signal to a concentration we use a calibration curve.

Combining the equation for the signal detection limit with the equation of a straight line calibration curve we get (assuming reagent blank is the same as the intercept value)

Detection Limit

The precision of measurements at the detection limit is too poor to allow for quantitation.

To find the minimum concentration that can be used for quantitative measurements, we use the limit of quantitation, LOQ.

The equation for LOQ is the same as detection limit except that we use a value for z of 10.

Limit of Quantitation

There are variations to the way detection limits are calculated but in general a detection limit is calculated by:

Running a reagent blank several times (should state how many, usually about 10) to get the standard deviation

Running standards to construct a calibration curve to find the sensitivity

Dividing the signal at the detection limit by the calibration sensitivity, k

Calculating Detection Limit

**Detection Limit**

Using Excel