Describe and Analyse Data Using Mathematical Principles

FDFTEC4007A

Describe and Analyse Data using Mathematical Principles

Contents

© Food-Wise Training Solutions

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FDFTEC4007A

Describe and Analyse Data using Mathematical Principles

Introduction and Unit Details ………….…………………………………....4

Element 1: Identify Common Units of Measurement and Dimensions used to describe physical properties of materials and food………….. 8

Units and Dimensions...………………………………………………………..9

Symbols and Formulae……………………………………………………..…..16

Measuring Physical Characteristics of Food and Processes ……………...18

Geometric Properties ……………………………………………………18

Mechanical Properties…………………………………………………..20

Thermal Properties……………………………………………………….23

Electromagnetic Properties ……………………………………………..26

Water Activity & Sorption Isotherms…………………………………...27

Fractions……………………………………………………….….……………..29

Percentages……………………………………………………………………...37

Describing Data……………………………………………………………….…40

A Measure of Central Tendency: Arithmetic Mean …………………..40

A Measure of Central Tendency: Weighted Mean ……………………43

A Measure of Central Tendency: Geometric Mean ………....……….44

A Measure of Central Tendency: Median ………………………….….44

A Measure of Central Tendency: Mode ..………………………………45

A Measure of Distribution: Range, Variance & Standard Deviation…47

Normal Distribution ……………………………………………………...51

Element 2: Apply Linear Algebra to analyse workplace information.…53

Linear Algebra…………………………………………………………….….….54

Transposing values to formulae / calculations……………………………..55

Element 3: Use Graphs to analyse workplace information……………64

Graphs and Charts…………………………………………………………….65

Graphic Distortions…………………………………………………………….79

Process Control Charts…………………………………………………….…81

Activity Answers……………………………………………………………….97

Units and Dimensions

In Australia we use the metric system for measuring distance, time, mass etc. It is a structured and methodical system which uses multiples of 10.

There has been a move towards the adoption of a universal system of units. This universal system is called the International system of units, officially designated SI. (from the French le Systéme international d’unités). As can be seen from the table below[1], the SI system has scientific (cgs) and industrial (mks) counterparts.

SYSTEM / USE / DIMENSION
LENGTH / MASS / TIME / TEMPER- ATURE / FORCE / ENERGY
METRIC
cgs / Scientific / centimeter / gram / second / oC / dyne / Calorie, erg
mks / Industrial / metre / kilogram / second / oC / Kilogram force / Kilocalorie, joule
SI / universal / metre / kilogram / second / K / Newton / joule
ENGLISH aka IMPERIAL
English absolute / Scientific / Foot / Pound mass / Second / oF / Poundal / BTU
ft (poundal)
British engineering / Industrial / Foot / Slug / Second / oF / Pound force / BTU
ft (poundal force)
American engineering / US industrial / foot / Pound mass / Secoond / oF / Pound force / BTU
ft (poundal force)

From the table above it can also be noted that the English or imperial system is also a system of measurement. However, it is not as structured and methodical as the metric system. The table below compares the dimensions of length between these two systems of measurement.

DIMENSION - LENGTH
Metric System / English / Imperial System
10 millimetres / 1 centimetre / 12 inches / 1 foot
10 centimetres / 1 decimetre / 3 feet / 1 yard
10 decimetres / 1 metre / 5 ½ yards / 1 rod
10 metres / 1 decametre / 40 rods / 1 furlong
10 decametres / 1 hectometre / 8 furlongs / 1 statute mile
10 hectometres / 1 kilometre / 3 miles / 1 league

Another example:

where:Re is the Reynolds number

Dis the diameter of the pipe (m)

υ(nu) is the average velocity (m/s)

ρ(rho) is the fluid density (kg/m3)

μ(mu) the fluid viscosity (N sm-2)

Subscripts (also known as indices) of letters or numbers are used to differentiate between different elements measured in the same dimension.

For example in a mass balance

mf= mp + mw

wheremfis the mass of the feed or input (kg)

mpis the mass of the product (kg)

mwis the mass of the waste (kg)

For example in a heat transfer equation:

Q= hs A (ϴa - ϴs)

whereϴais the dry bulb temperature (oC)

ϴsis the wet bulb temperature (oC)

Activity 4

In the formula: Q = κA (ϴ1 -ϴ2 )

Δx Which are the symbols?

Which are the letters? Which are the (calculable) numbers? Which are the signs (plus, minus, multiply etc)? And which are the numbers used to identify different elements of the same dimension?.

1. Mechanical Properties

Textural properties - the material must be sufficiently robust to withstand the mechanical stresses to which it is subjected during preparative operations and also must withstand the processing conditions as to yield a final product of the desired texture.

Rheological Properties of Food

The consistency, degree of fluidity, and other mechanical properties are important in understanding how long food can be stored, how stable it will remain, and in determining food texture. The acceptability of food products to the consumer is often determined by food texture, such as how spreadable and creamy a food product is. Food rheology is important in quality control during food manufacture and processing. Food rheology terms have been noted since ancient times. Historically bakers have judged the consistency of dough by rolling it in their hands.

Rheological properties are based on flow and deformation responses of food when it is subjected to compressive (directed toward the material), tensile (directed away from the material), and shearing (directed tangentially to the material) stresses.[2][3]

Food can be classified according to its rheological state, such as a solid, gel, liquid or emulsion. Rheologically, a material can deform in three ways: elastic, plastic, or viscous

• Elastic - a solid having the property of recovering shape after being deformed
• Plastic- usually a solid capable of being molded
• Viscous- the property of a fluid being sticky, adhesive or thick

Food materials display large compositional variations, inhomogeneities, and anisotropic[4]properties and as a result they can exhibit a combination of the different rheological properties, for example many non-Newtonian fluids exhibit both viscous and elastic properties.

Activity 10

Match the equivalent fractions.

a)

b)

c)

d)

e)

Comparing fractions

Comparing fractions with the same denominator only requires comparing the numerators.

because 3>2.

One way to compare fractions with different denominators is to find a common denominator. To compare and , these are converted to and. Then bd is a common denominator and the numerators ad and bc can be compared.

? gives

As a short cut, known as "cross multiplying", you can just compare ad and bc, without computing the denominator.

?

Multiply 17 by 5 and multiply 18 by 4. Since 85 is greater than 72, .

Another method of comparing fractions is this: if two fractions have the same numerator, then the fraction with the smaller denominator is the larger fraction. The reasoning is that since, in the first fraction, fewer equal pieces are needed to make up a whole, each piece must be larger.

Also note that every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, so every negative fraction is less than any positive fraction.

Population Standard Deviation[5]

The population standard deviation, or δ (sigma) is simply the square root of the population variance. The standard deviation is the square root of the average of the squared distances of the observations from the root mean.

The formula for the standard deviation is:

δ=√ δ2=√Σ(x - µ)2=√ Σx2 - µ2

N N

Where:

δ= the population standard deviation

δ2= the population variance

x= the item or observation

µ= population mean

N= total number of items in the population

Σ= sum of all the values (x - µ)2 or all the values x2

The standard deviation enables us to determine with a great deal of accuracy where the values of a frequency distribution are located in relation to the mean. We can measure with precision the percentage of items that fall within specific ranges under a symmetrical, bell-shaped curve. In these cases we can say that:

1. about 68% of the values in the population will fall within plus and minus 1 standard deviation from the mean
2. about 95% of the values in the population will fall within plus and minus 2 standard deviations from the mean
3. about 99.7% of the values in the population will fall within plus and minus3 standard deviations from the mean

Linear algebra as a branch of mathematics includes finite or countably infinite dimensional vector spaces, and linear mappings between such spaces. However this resource will only deal with the system of linear equations.

Techniques from linear algebra are used in analytical geometry, engineering, physics, natural sciences, computer science and the social sciences (particularly in economics).[6]

Transposing Values

As briefly discussed in FDFOP2061A Use Numerical Applications in the Workplace every mathematical operator has an opposite, such that the opposite of:

addition(+) is subtraction (-)

multiplication (x)is division (

square (is the square root (

Transposing a term: is to bring it from one side of the equation, to the other with a change in sign[7] , that is changing it to its opposite. The aim is to essentially get the known on its own and therefore give it a value.

For example:7 - 14 = 28

On the left hand side (LHS) of the equation 14 is being subtracted, therefore if we were to be ‘rid of it’ we could:

a)move it to the RHS of the equation thus evoking a sign change OR

7 - 14 = 28 +14

b)to remove ‘-14’, we would add 14, and what is done on one side of the equation must be done on the other for it to remain valid

7 - 14 + 14 = 28 + 14

7= 42

Graphs and Charts

This section will discuss the use of graphs or chart as a method of organising and representing numerical or qualitative data.

A graph or chart is a graphical representation of data, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent numerical data, functions or some kinds of qualitative information.

Charts and graphs are used to:

• help the reader understand large quantities of data and the relationships between parts of the data.
• interpret information easier than looking at the raw data that they are produced from
• impress the reader by getting the point across quickly and visually (since humans are generally able to determine meaning from pictures quicker than from text)

For these reasons graphs and charts are useful and are often used in newspapers, books and businesses.[8]

They can be created by hand (often on graph paper) or by computer using a charting application. Certain types of charts are more useful for presenting a given data set than others. For example, data that presents percentages in different groups (such as "satisfied, not satisfied, unsure") are often displayed in a pie chart, but may be more easily understood when presented in a horizontal bar chart On the other hand, data that represents numbers that change over a period of time (such as "annual revenue from 1990 to 2000") might be best shown as a line chart.[9]

General features of charts / graphs

There are common features that provide the chart with its ability to extract meaning from data.

Typically a chart is graphical, containing very little text. One of the more important uses of text in a graph is in the title. A graph's title usually appears above the main graphic and provides a succinct description of what the data in the graph refers.

Dimensions in the data are often displayed on axes. If a horizontal and a vertical axis are used, they are usually referred to as the x-axis and y-axis respectively. Each axis will have a scale, denoted by periodic graduations and usually accompanied by numerical or categorical indications. Each axiswill typically also have a label displayed outside or beside it, briefly describing the dimension represented. If the scale is numerical, the label will often be suffixed with the unit of that scale in parentheses. For example, "Distance travelled (m)" is a typical x-axis label and would mean that the distance travelled in metres is related to the horizontal position of the data.

Within the graph a grid of lines may appear to aid in the visual alignment of data. The grid can be enhanced by visually emphasizing the lines at regular or significant graduations.

Any equation can be represented graphically. The independent variable is represented by the abscissa and the dependent variable is represented by the ordinated. For example, the equation

Logarithmic Scale

Due to the nature of some information such as the growth of number of micro-organisms as a function of time, the results may need to be graphed with at least one of the co-ordinates scaled logarithmically. Below are examples of how the logarithmic scale compares with the normal (arithmetic) scale and on the next page are examples of semi-log and full log graphing paper.

Logarithmic ScaleArithmetic Scale

Other common charts used in the food processing industry are:

An Organisational Chart shows the breakdown of a company’s hierarchy depicting direct line (usually depicted by a solid line) and functional (depicted by a dotted line) authorities.

Atree diagramis a specific type of diagram that has a unique network topology. It can be seen as a specific type of network diagram, which in turn can be seen as a special kind of cluster diagram. An example is a decision tree.

Flow chart expresses detailed knowledge of the process, identifies process flow and interaction among the process steps and identifies potential control points. [10]

The Program (or Project) Evaluation and Review Technique, ( PERT), is a statistical tool, used in project management, that is designed to analyse and represent the tasks involved in completing a given project. It is commonly used in conjunction with the critical path method (CPM).

Graphic Distortions

The objective of graphs and charts is to represent data to facilitate comprehension and/ or interpretation of statistics, however the reliability of the statistics can be biased by the intention of the statistician in presenting the numerical information. By understanding some pitfalls you will have the knowledge to ensure that any report or analysis is well presented and factual.

The pie chart below leaves the readerto use the protractor to determine the number of degrees of each section, to determine its relation to 360 degrees and then multiply the resulting fraction by $12000 million to arrive at the spending area so section. This is far too much to ask of a reader. [11]

The bar chart lacks a number of minor details - the y axis indicating units of measure and the amount and the period under examination.

Creating the Graphs:

R chart

1.Left click and drag to highlight cells K1 to N26

2.Left click on the ‘Insert’ tab at the top of the screen.

3.Left click on ‘line chart’ and then left click on the very first option.

4.This will give a basic graph, (see snapshot below) however it needs formatting.

© Food-Wise Training Solutions

Version 1.1Page of 104

[1](Toledo, 1980, p. 54)

[2](Barbosa-Canovas, Juliano, & Peleg, 2007, p. 19)

[3](Rao, 2007, p. 1)

[4] Anisotropic - of different properties in different directions

[5](Levin R. I., 1978, p. 120)

[6](Wikipedia, 2012)

[7](Macquarie Library, 1981, p. 1804)

[8](NCES, unknown)

[9](Wikipedia, 2012)

[10](Montgomery, 1991)

[11](Waxman, 1993, pp. 270- 271)