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: 10LOG(a) (cf. 3).

Value of e = e1: EXP(1) = 2.7183; eb: 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 = e1.61 = 2.71831.61

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 × xk + 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 side the hypotenuse (c). The relation between these sides is expressed by the Pythagoras' theorem :

Figure 1 (A) Right-angle triangle. (B) The unit circle.

The proportions or "image" of any triangle are determined by the angles (A = BAC, B = ABC, and C = ACB). The angles can be defined by the trigonometric functions referring to a right-angle triangle (Figure 1A):

Provided the angle is known and expressed in radians EXCEL provides numerical values of these quantities SIN(A), COS(A), and TAN(A). The cotangent for an angle is the inverse of its tangent and is not available as a separate function in EXCEL.

Radian is defined as the angle AOB in the circle (Figure 1B) where the arc AB is equal to the radius OB. Since the circumference is 2 × radius × pi(p) corresponding to 360 °, an angle of 1 radian will correspond to 360/(2 × p) or 57.3 °.

EXCEL provides conversions between degrees and radians: RADIANS (angle in degrees) and DEGREES (angle in radians), respectively. Therefore, to express the sine of 30 °, the function would be SIN(RADIANS(30)) = SIN(0.52) = 0.5. The reverse of the trigonometric functions is arcsine, arccosine, and arctangent, respectively. In EXCEL, the functions are ASIN(A), ACOS(A) and ATAN(A). Thus, to convert a sine of 0.5 to degrees, the function would be DEGREES(ASIN(0.5)).

Scales-Types of Data

Data can be expressed on four types or scales of data: nominal, ordinal, interval , and ratio .

Data on a nominal scale may be numbers or any other information that describes a property. There is no size relation between the entities.

Data expressed on an ordinal scale are of different sizes and can thus be ordered or ranked. The scale may be arbitrary and the intervals between numbers unequal. Data expressed on an ordinal scale can be measured and are thus quantities. Not all statistical procedures can be applied to ordinal data. Examples may be "good," "excellent," and "superior," or +1, +2, +3 etc. with no defined difference between the results.

Data with equal intervals between numbers are of two kinds and can be expressed on an interval scale and a ratio scale. The ratio scale is characterized by-apart from equally sized units-a natural zero value, whereas the interval scale may have an arbitrarily defined zero. A commonly cited quantity that is expressed on an interval scale is temperature expressed as degrees Celsius or Fahrenheit whereas if expressed in Kelvin a ratio scale is used. Consequently, 40 K is twice as much as 20  °C, whereas 40  °C is not twice as much as 20  °C. However, there are as many degrees between 40 and 20  °C as between 20 and 0  °C.

Distributions of Data

Histogram

A histogram displays the number of data points in each of defined categories or intervals-often called "bins." It is a rough representation of the frequency probability distribution of data. The resolution and details of the distribution depend largely on the size and number of bins. Usually, the bin sizes are made equal in the interesting interval but that is not always the case. Designing a histogram manually is easy, but tedious and EXCEL offers two different possibilities. The simper is to activate the "Data analysis" function which is an add-in to the program and found under the Data tab. This is straightforward and allows an individual design of the bins as an option to those calculated by the program. The routine has the disadvantage of not allowing modifications interactively. A fully flexible procedure is obtained by the "frequency" function. This is an "array" function.

In short, define the desired bins, mark a set of empty cells, one cell more than the number of bins and write in the first cell = FREQUENCY(A1:AN1, B1:BN2) and press Control + Shift + Enter. The array is then created in the marked cells which are filled with the copied formula and subsequently with the number of items in each bin. The array...