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BTEC NATIONALS LEVEL 3, MATHS UNIT 4, STATISTICAL METHODS

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  • Mathematics for Engineering Technicians 299

    UNIT

    4

    Statistical Methods

    Your view of statistics has probably been formed from what you read in the papers, or what you see on the television. Survey use to show which political party is going to win the election, why men grow moustaches, if smoking damages your health, the average cost of housing by area, and all sorts of other interesting data! So statistics is used to analyse the results of such surveys and when used correctly, it attempts to eliminate the bias that often appears when collecting data on controversial issues.

    Statistics is concerned with collecting, sorting and analysing numerical facts, which originate from several observations. These facts are collated and summarized, then presented as tables, charts or diagrams, etc.

    In this brief introduction to statistics, we look at two speci c areas. First, we consider the collection and presentation of data in its various forms. Then we look at how we measure such data, concentrating on nding average values.

    If you study statistics beyond this course, you will be introduced to the methods used to make predictions based on numerical data and the probability that your predictions are correct. At this stage in your learning, however, we will only be considering the areas of data handling and measurement of central tendency (averages), mentioned above.

    TYK 4.10

    1. A parallelogram has an area of 60 cm 2 , if its perpendicular height is 10 cm, what is the length of one of the parallel sides?

    2. Figure 4.43 shows the cross-section of a template, what is its area?

    3. An annulus has an inside diameter of 0.75 m and an external diameter of 0.9 m, determine its area.

    4. Find the volume of a circular cone of height 6 cm and base radius 5 cm.

    5. Find the area of the curved surface of a cone (not including base) whose base radius is 3 cm and whose vertical height is 4 cm. Hint : you need rst to nd the slant height.

    6. If the area of a circle is 78.54 mm 2 , nd its diameter to 2 signi cant gures.

    7. A cylinder of base radius 5 cm has a volume of 1 L (1000 cm 3 ), nd its height.

    8. A pipe of thickness 5 mm has an external diameter of 120 mm, nd the volume of 2.4 m of pipe material.

    9. A batch of 2000 ball bearings are each to have a diameter of 5 mm. Determine the volume of metal needed for the manufacture of the whole batch.

    10. Determine the volume and total surface area of a spherical shell having an internal diameter of 6 cm and external diameter of 8 cm.

    50 cm

    30 cm

    60 c

    m

    3 cm

    4 cm

    Figure 4.43 Figure for question 2 in TYK 4.10

    Test

    your know

    ledg

    e

    TYK

  • Mathematics for Engineering Technicians300UN

    IT 4

    Data manipulation

    In almost all scienti c, engineering and business journals, newspapers and Government reports, statistical information is presented in the form of charts, tables and diagrams, as mentioned above. We now look at a small selection of these presentation methods, including the necessary manipulation of the data to produce them.

    Charts Suppose , as the result of a survey, we are presented with the following statistical data ( Table 4.4 ).

    Num

    ber

    empl

    oyed

    1000

    800

    600

    400

    200

    0

    Priv

    ate

    busi

    ness

    Pub

    lic b

    usin

    ess

    Agr

    icul

    ture

    Eng

    inee

    ring

    Tran

    spor

    t Man

    ufac

    ture

    Leis

    ure

    indu

    stry

    Edu

    catio

    n

    Hea

    lth

    Oth

    ers

    Category of employment

    Figure 4.44 Bar chart representing number employed by category

    KEY POINT

    Statistics is concerned with collecting, sorting and analysing numerical facts

    Table 4.4 Results of a survey

    Major category of employment Number employed

    Private business 750

    Public business 900

    Agriculture 200

    Engineering 300

    Transport 425

    Manufacture 325

    Leisure Industry 700

    Education 775

    Health 500

    Other 125

    Now, ignoring for the moment the accuracy of this data, let us look at typical ways of presenting this information in the form of charts, in particular the bar chart and the pie chart .

    Bar chart In its simplest form, the bar chart may be used to represent data by drawing individual bars ( Figure 4.44 ) using the gures from the raw data (the data in the table).

  • Mathematics for Engineering Technicians 301

    UNIT

    4

    Now , the scale for the vertical axis, the number employed, is easily decided by considering the highest and lowest values in the table, 900 and 125, respectively. Therefore, we use a scale from 0 to 1000 employees. Along the horizontal axis, we represent each category by a bar of even width. We could just as easily have chosen to represent the data using column widths instead of column heights.

    Now the simple bar chart above tells us very little that we could not have determined from the table. So, another type of bar chart that enables us to make comparisons, the proportionate bar chart, may be used.

    In this type of chart, we use one bar , with the same width throughout its height, with horizontal sections marked-off in proportion to the whole. In our example, each section would represent the number of people employed in each category compared with the total number of people surveyed.

    In order to draw a proportionate bar chart for our employment survey, we rst need to total the number of people who took part in the survey. This total comes to 5000. Now, even with this type of chart we may represent the data either in proportion by height or in proportion by percentage. If we were to choose height, then we need to set our vertical scale at some convenient height, say, 10 cm. Then we would need to carry out 10 simple calculations to determine the height of each individual column.

    For example, given that the height of the total 10 cm represents 5000 people, then the height of the column for those employed in private

    business 7505000

    10 1 5

    . cm . This type of calculation is then repeated

    for each category of employment. The resulting bar chart is shown in Figure 4.45 .

    10 cm Others

    Health

    Education

    Leisure industry

    Manufacture

    Transport

    EngineeringAgriculture

    Public business

    Private business

    Figure 4.45 Proportionate bar chart graduated by height

  • Mathematics for Engineering Technicians302UN

    IT 4

    Example 4.49 Draw a proportionate bar chart for the employment survey shown in Table 4.4 using the percentage method.

    For this method all that is required is to fi nd the appropriate percentage of the total (5000) for each category of employment. Then, choosing a suitable height of column to represent 100%, mark on the appropriate percentage for each of the 10 employment categories. To save space, only the fi rst fi ve categories of employment have been calculated.

    1.

    private business

    7505000

    100 15

    %

    2.

    public business

    9005000

    100 18

    %

    3.

    agriculture

    2005000

    100 4

    %

    4.

    engineering

    3005000

    100 6

    %

    5.

    transport

    4255000

    100 8 5

    . %

    Similarly , manufacture 6.5%, leisure industry 14%, education 15.5%, health 10% and other categories 2.5%.

    Figure 4.46 shows the completed bar chart.

    Other categories of bar chart include horizontal bar charts , where for instance Figure 4.44 is turned through 90 in a clockwise direction. One last type may be used to depict data given in chronological (time) order. Thus, for example, the horizontal x -axis is used to represent, hours, days, years, etc., while the vertical axis shows the variation of the data with time.

    Example 4.50 Represent the following data on a chronological bar chart.

    Year Number employed in general engineering (thousands)

    2003 800

    2004 785

    2005 690

    2006 670

    2007 590

    Since we have not been asked to represent the data on any specifi c bar chart we will use the simplest, involving only the raw data. Then, the only concern is the scale we should use for the vertical axis.

  • Mathematics for Engineering Technicians 303

    UNIT

    4 To present a true representation, the scale should start from zero and extend to, say, 800 ( Figure 4.47 a ). If we wish to emphasize a trend , that is, the way the variable is rising or falling with time, we could use a very much exaggerated scale ( Figure 4.47 b ). This immediately emphasizes the downward trend since 1995. Note that this data is fi ctitious (made-up) and used here merely for emphasis!

    Pie chart In this type of chart the data is presented as a

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