Laboratory Statistics

Handbook of Formulas and Terms

Anders Kallner

• Publisher: Elsevier
• Year: 2013
• eBook ISBN:
  9780124169739
• Topic: Physical Sciences
• Subtopic: Probability & Statistics

Laboratory Statistics: Handbook of Formulas and Terms presents common strategies for comparing and evaluating numerical laboratory data. In particular, the text deals with the type of data and problems that laboratory scientists and students in analytical chemistry, clinical chemistry, epidemiology, and clinical research face on a daily basis. This book takes the mystery out of statistics and provides simple, hands-on instructions in the format of everyday formulas. As far as possible, spreadsheet shortcuts and functions are included, along with many simple worked examples. This book is a must-have guide to applied statistics in the lab that will result in improved experimental design and analysis. - Comprehensive coverage of simple statistical concepts familiarizes the reader with formatted statistical expression - Simple, worked examples make formulas easy to use in real life - Spreadsheet functions demonstrate how to find immediate solutions to common problems - In-depth indexing and frequent use of synonyms facilitate the quick location of appropriate procedures

Formulas

Abstract

The most common procedures and concepts in everyday statistics in the laboratory are described and explained by formulas in their context. A few routines in basic mathematics are briefly discussed. Besides characteristics of distributions and estimators, analysis of variance is addressed. An often raised question is the comparison of measurement procedures and parametric and nonparametric applications which are discussed and expanded to regression, correlation, and graphical representations. Particular attention is paid to performance characteristics in terms of Bayes’ theorem and agreement between categorical data. As often as possible formulas are presented in different formats, thus highlighting how they relate to other evaluation procedures.

Key words

Comprehensive clear formulas; Definitions of statistical concepts; Connections between of formulas and concepts; Guidance to spread-sheet programming Worked examples

Basics

Logarithms and Exponents

The logarithm of a given number and a given base is the power to which the base must be raised to get the number.

If b is the base and a the given number, the logarithm is x. In many applications, the notation “log” refers to 10-logarithms (Briggs), i.e., the base 10 and ln refers to e-logarithms or “natural” logarithms with e = 2.7183 as the base:

(1)

(2)

and

(3)

(4)

(5)

(6)

(7)

(8)

Microsoft EXCEL® commands: Natural logarithm: LN(a); antilog: EXP(LN(a)) (cf. 2).

10-logarithms (Briggs) LOG(a); antilog: 10^LOG(a) (cf. 3).

Value of e = e¹: EXP(1) = 2.7183; e^b: EXP(b).

Examples

Let a = 5, b = 10, c = 3, and n = 2, then

Since e = 2.7183 and ^e log(5) = ln(5) = 1.61; anti ln(1.61) = 5 = e^1.61 = 2.7183^1.61

Calculation of the logarithms, natural or 10-logaritms is directly available in spreadsheet programs. If mathematical tables or calculators are used, logarithms are conventionally expressed with four decimals to achieve sufficient precision for everyday use. Table values can be interpolated.

Derivation—Calculus

The derivative of a function at a given input value describes the best linear approximation of the function near that input value, i.e., the slope of the tangent in that point. Therefore, if the “first derivative” is set to zero and solved, the maximum(s) and/or minimum(s) of the function will be obtained. In higher dimensions, second, third, etc. derivatives can be calculated and if a second derivative is set to zero, the inflexion point of the original function is identified. The derivative of a function f(x) is written dy/dx, y′, or f′(x) and interpreted as the “derivative of y with respect to x.”

The partial derivative of a function of several variables is its derivative with respect to one of those variables while the others are held constant.

The partial derivative is written ∂ y/∂ x.

Examples

The first derivative of a third degree function

is dy/dx = y′ = f′(x) = x² − 10x − 11 with maximum and minimum at x = 5 ± 6, i.e., x₁ = − 1 and x₂ = + 11, respectively. The second derivative is d²y/dx² = y″ = f″(x) = 2x − 10 and the inflexion point of the original function is x = − 5.

Draw the three functions and confirm the maximum, minimum, and inflexion point!

If y = n × x^k + constant, then a derivative will, in general terms, be

(9)

For a detailed discussion of derivative rules, derivatives, and partial derivatives, the reader is referred to special literature.

Trigonometry

Trigonometric Functions

In a right-angle triangle, i.e., a triangle with one angle equal to 90 °, i.e., one side perpendicular to another side, the sides surrounding the right angle are called cathetus (a and b in Figure 1A) and the opposite ...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Acknowledgment
• Introduction
• Some Notes on Nomenclature
• Formulas
• Some Metrological Concepts
• Further Reading
• Index