Statistical Analysis of Laboratory Data

Who Should Attend

Bench chemists, assayists, bioassayists, technicians, scientists, engineers, laboratory managers, R&D managers, manufacturing and production managers, research supervisors, project managers, vice presidents, and others who need to learn, understand, and apply statistical methods of data analysis.

The course is aimed at both beginning and experienced workers. The course assumes no previous knowledge of statistics.

Key Topics You Will Learn About

  • How to understand the strengths and weaknesses of data
  • How to recognize and reduce different types of statistical errors
  • Ways to carry out significance tests
  • How to correctly use outlier tests and when not to use them
  • Ways of defining the limits of detection, determination, and quantification
  • How to know what statistical test to use when
  • How to understand the influence of sample size on statistical significance and power
  • Why pooling variances gives stability to analytical results
  • How to set in-house specifications
  • How to apply statistical process control charts to measurement processes

How You Will Benefit From This Course

  • You will gain confidence in the use of basic statistical methods
  • You will enhance your ability to extract meaningful information from your data sets
  • You will improve your decision-making abilities
  • You will learn new ways to look at data
  • You will reduce the number of measurements required for certain applications
  • You will understand statistical terminology and be able to communicate more easily with statisticians
  • You will consult with seasoned experts about your data analysis problems

Day 1 Morning


  • Statistics as a science
  • Statistics as an art
  • The importance of measurement uncertainty
  • Confidence and risk as complements
  • The difference between statistical and practical significance


  • Measurement as a production process
  • Reference materials
  • The importance of a stable measurement process

Accuracy And Precision

  • Accuracy as a concept
  • Error, a measure of inaccuracy
  • Precision as a concept
  • Range, a measure of imprecision

The Mean

  • Central tendency
  • Median, mode, and mean
  • The mean as a least-squares estimate
  • Constant (or determinate) errors
  • The difference between the "true mean" and the truth
  • Validation standards for accuracy

The Standard Deviation

  • Degrees of freedom
  • Random (or indeterminate) errors
  • Overall accuracy requires minimum bias and minimum variability
  • The variance, s2
  • The standard deviation, s
  • Gaussian (normal) distributions
  • z scores
  • Probability densities
  • Homoscedasticity, heteroscedasticity, and the %RSD


  • Sources of variation
  • The pooled standard deviation
  • The "method standard deviation" for a measurement method
  • An alternative protocol for reporting measurement results

Day 1 Afternoon

z Decisions

  • Common z values
  • One-sided decisions based on the z distribution
  • Two-sided decisions based on the z distribution

Confidence Intervals

  • Two-sided (or two-tailed) confidence intervals
  • One-sided (or one-tailed) confidence intervals
  • The real meaning of confidence intervals

Statistical Samples

  • Samples from a conceptually infinite population of data (CIP DATA)
  • The effect of increasing n on the variation of means
  • The effect of increasing n on the variation of standard deviations
  • The effect of variation on confidence intervals for μ and σ


  • μ for the sample means is the same as μ for the raw data
  • The theoretical "inverse square root of n" effect on the variation of the sample means
  • How to calculate the sample size n
  • The cost of replicate measurements, and alternatives
  • Don't try to do with statistics what what you can do with improved measurements

Standard Deviations

  • The chi-square distribution
  • The usefulness of the "chi-square over nu" distribution
  • Confidence intervals for σ
  • The relative uncertainty in σ2 and σ as a function of n

Student's t

  • Student's t distributions
  • How to calculate confidence intervals for μ based on s, n and t
  • Sidedness

Day 2 Morning

Statistical Testing

  • A "different from" test using a two-sided confidence interval
  • A "between" test using a two-sided confidence interval
  • An "outside" test using a two-sided confidence interval
  • A "greater than" test using a one-sided confidence interval
  • A "less than" test using a one-sided confidence interval
  • Cautions about when to choose the sidedness of the test
  • Modified specifications

p Values

  • How to use confidence intervals to determine the exact confidence and exact risk
  • p values
  • p values vs. α values for statistical testing

Algebra And Logic

  • The algebra associated with confidence intervals
  • The logic of statistical testing

Hypothesis Testing

  • The alternative hypothesis Ha for α-testing
  • The null hypothesis H0 as the logical opposite of Ha
  • "Don't know" is a valid outcome of statistical testing
  • α-Testing and Type I errors
  • β-Testing and Type II errors
  • Contrast between Type I (α) and Type II (β) errors

Overview Of Formal Statistical Tests

  • One-sample t test and its applications
  • Two-sample t test and its applications
  • Paired t test and its applications
  • F test and its applications
  • One-way analysis of variance (ANOVA) and its applications

One-Sample t Test

  • How to carry out a one-sample t test
  • Decisions based on confidence intervals
  • Decisions based on algebra and logic
  • Sometimes "t tests" can be "z tests"

Day 2 Afternoon

Two-Sample t Test

  • How to carry out a two-sample two-sided "different from" t test
  • A two-sample one-sided "greater than" t test
  • A two-sample two-sided "greater than a certain amount" t test
  • Confidence interval for a sum
  • Standard deviation of a difference
  • Confidence interval for a difference

Paired t Test

  • How to carry out the paired t test
  • Extraneous variation often causes the two-sample t test to be insensitive
  • Extraneous variation can be removed if a paired structure exists
  • The paired t test is a one-sample t test on the mean difference
  • Data collection must be planned to create the necessary paired structure
  • Loss of degrees of freedom when differences are taken

Fisher's F Test

  • How to carry out one-sided F tests
  • How to carry out two-sided F tests
  • The F test compares standard deviations (σ), not means (μ)
  • Tables of critical values of F


  • How to carry out a one-way ANOVA
  • Testing multiple means
  • Multiple pair-wise comparisons result in a loss of control of α
  • The Bonferroni (over)correction
  • One-way ANOVA tables
  • Post-hoc, multiple-comparison testing
  • Duncan's multiple range test


  • How to carry out the studentized T test for the detection of outliers
  • How to carry out Dixon's Q test for the detection of outliers
  • Acceptable and unacceptable uses for outlier testing
  • Guidelines for outlier testing

Central Limit Theorem

  • The central limit theorem
  • Illustrations for several non-Gaussian distributions

Day 3 (two of three optional groups will be covered)

Group I, Detection Limits


  • The calibration relationship
  • Sensitivity defined
  • Non-constant sensitivity
  • Sensitivity depends on the choice of analyte
  • Sensitivity depends on the choice of units


  • Characteristics of universal detectors
  • Characteristics of selective detectors
  • The selectivity ratio
  • The effect of the selectivity ratio on quantitation

Limit Of Detection

  • Statistical concepts associated with LD, MDA, and LQ
  • The true mean of the blank (μb)
  • The standard deviation of the blank (σb)
  • The value of LD affects the false positive rate for a blank sample
  • The value of LD must be consistent with the application
  • Policies for reporting "not detected"

Minimum Detectable Amount

  • LDA — the "limit-of-detection amount"
  • The concept of "consistent detection"
  • False negatives
  • The MCDA — the "minimum-consistently-detectable amount"

Limit Of Quantitation

  • Percent relative standard deviation (%RSD) at the LDA
  • %RSD at the MCDA
  • LQA — the "limit-of-quantitation amount" — based on 10% RSD
  • Guidelines for reporting results to clients

Mandel Sensitivity

  • Conventional sensitivity S
  • Mandel sensitivity, SM
  • Utility of a common Mandel response scale for different measurement methods
  • The relationships of LDA, MCDA, and LQA with Mandel sensitivity
  • Diverse measurement methods can be compared using Mandel sensitivity

Group II, Statistical Process Control

Process Capability

  • Voice of the process
  • Voice of the customer
  • Standard specification limits zL and zU
  • Specified tolerance
  • Process capability ratio Cp
  • Centered process capability ratio Cpk
  • "Six-sigma quality"

Method Development

  • The need for a working group
  • Data quality objectives (DQOs)
  • Usefulness of control charts as a means of continuous validation
  • The need for among-laboratory agreement on standards
  • Importance of interlaboratory evaluation of methods

Control Charts

  • Construction details of control charts
  • Control limits are not specification limits
  • Caution against continually calculating control limits
  • Control charts work for both production processes and measurement processes

Lack Of Control

  • Illustrate the "3σ rule"
  • Illustrate the "run-of-eight rule"
  • Additional information can be extracted from control charts
  • Why you should not use "caution alarms" and "warning alarms"

Extremes Of Control

  • Two separate concepts: "within specifications" and "in statistical control"
  • Wheeler's ideal state
  • Threshold state A
  • Threshold state B
  • The brink of chaos
  • The state of chaos
  • Cautions about the effects of entropy creeping into processes

Interlaboratory Testing

  • The reference laboratory
  • Within-laboratory variation
  • Among-laboratory variation
  • Youden plots

Group III, Bioassays

Ratio Of Means

  • The effect of uncertainty in the numerator
  • The effect of uncertainty in the denominator
  • The exact equation for calculating the uncertainty of a ratio (Fieller's theorem)
  • Confidence intervals for a ratio


  • Definition of biological assays (bioassays)
  • The theory of bioassays
  • Comparative nature of bioassays
  • Direct assays, and an example of a direct potency assay
  • Requirements for pooling variances
  • Logarithmic transformation of data as a variance-stabilizing function
  • Alternative calculation of potency using log-transformed data
  • Finney's criticisms of direct assays

Slope Ratio Analysis

  • The condition of similarity
  • Examples showing where the condition of similarity is satisfied and where it is not
  • The common-intercept model
  • The separate-intercepts model
  • How to test for the adequacy of the common-intercept model
  • The slope ratio R and the confidence interval for ρ

Parallel Line Analysis

  • Drug effectiveness as a function of dose
  • The relationship between the fraction effective and the cumulative Gaussian distribution
  • Probits, and why their use gives a straight line relationship
  • Logits, and why their use gives a roughly straight line relationship
  • The theory of analytic dilution analysis
  • Why "parallelism" is necessary for the condition of similarity
  • The four-parameter model

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