Topic 3: Statistical Data

What is Statistical Data?

Statistics is a collection of mathematical techniques that help to analyze and present data. 

Statistics is also used in associated tasks such as designing experiments and surveys and planning the collection and analysis of data. 

Data is the numerical information collected.

Representing about Statistic Data, they are:

•          Data
•          Discrete and Continuous
•         Tally
•          Bar chart
•         Frequency diagram
•          Frequency polygon
•         Histogram
•          Frequency
•          Qualitative and Quantitative

Data

•          Data is another word for information.
•         One way to collect data is using a survey or a questionnaire.
•          When data is collected there are lots of ways to represent it using different charts, tables and statistics.
•          There are different types of data.

Discrete and Continuous

•           Discrete data can be counted. They can take particular values.

       e.g. Number of children, number of trees in a garden.

•         Continuous data results when measuring things like length, time and mass. It cannot be measured exactly.

 e.g. The time taken to run 100m. It could be 9s or 9.8s or 9.81s. It can also be measured more accurately.

 Tally

Raw Data

Sally carried out a survey of shoe sizes. Her results are below:

•  4 5 6 4 4 3 2 4

•  5 3 7 4 5 6 6 4

•  3 7 2 5 4 3 4 5

–  This is the raw data. It is not organized in any way.

–  Is it discrete or continuous data?

–  The first thing to do to begin to analyses the data is to organize it into a tally chart.

Using Tally Marks

Shoe size	Tally	Frequency
2	ll	2
3	llll	4
4	llll  lll	8
5	ll ll	5
6	lll	3
7	ll	2
	Total	24

*Each tally mark represents one piece of data. Groups of 5 are represented as llll

*Frequency gives the total count of each size

Bar chart

·         The data on shoe sizes is discrete data.

·         Therefore, you can draw a bar chart.

Example of Bar Chart:

     Bar charts are one of the most commonly used types of graph and are used to display and compare the number, frequency or other measure (e.g. mean) for different discrete categories or groups.

         The graph is constructed such that the heights or lengths of the different bars are proportional to the size of the category they represent.

         The x-axis (the horizontal axis) represents the different categories it has no scale.

         The y-axis (the vertical axis) does have a scale and this indicates the units of measurement.

         The bars can be drawn either vertically or horizontally depending upon the number of categories and length or complexity of the category labels.

       There are various ways in which bar charts can be constructed and this makes them a very flexible chart type.

         For example, if there is more than one set of values for each category then grouped or component bar charts can be used to display the data.

         Further details about each of these different types of bar chart can be found in the associated study guide Bar Charts.

Frequency diagrams and polygons

This frequency diagram shows the heights of 200 people:

•          You can construct a frequency polygon by joining the midpoints of the tops of the bars.
•          Frequency polygons are particularly useful for comparing different sets of data on the same diagram. 

Constructing a frequency polygon

Diagram showing data 

i

*        Midpoints are marked on each bar and joined together

SEE THIS VIDEO! (learn something from this video).....

SOLVE THIS QUESTIONS?

     From the following information, distinguish whether it is (i) quantitative or qualitative, (ii) discrete or continuous data.       

                                                                                                                         

a)      The number of students in Cosmopolitan college in the year of 2012

b)      The size of shoes from National Diploma in Computer Studies for intake 2.

c)      The height of people in Brunei

d)      The color of car passing by my house

e)      The weight of babies

f)       The body shop perfume fragrances

g)      BDTVEC grading system (e.g. grade A, grade B, etc)

h)      The number of Brunei population in the year of 2011

i)       The brands of car sold in Brunei

j)       The number of voters that vote for Hairi as a president for Student Committee for CCCT in the year of 2012.

Topic 2: Arithmetic and Geometric Progression

Arithmetic Progression

Arithmetic Progression (AP) is difference between one term and the next is a constant, other words is, just add same value.

Example of Arithmetic Progression:

Example 1:

AP is to find common different is subtract:

   First (1) term – Second (2) term

   Second (2) term – First (1) term =1

So,       1+1 = 2

            2+1 = 3

            3+1 = 4

            4+1 = 5

The constant by adding (+1) to next no.

Example 2:

Find the sum of the first 10 terms for the series that we met above: $\displaystyle{4},{7},{10},{13},\ldots$

a= 4

$\displaystyle{d}={3}$

$\displaystyle{n}={10}$

$\displaystyle{S}_{{n}}=\frac{n}{{2}}{\left[{2}{a}_{{1}}+{\left({n}-{1}\right)}{d}\right]}=\frac{10}{{2}}{\left[{2}\times{4}+{9}\times{3}\right]}={175}$

$S$n =  2/n [2a+(n-1)d]

=10/2 [2×4+9×3]

=175

Example 3:

Find the sum of the first $\displaystyle{1000}$ odd numbers.

In this case, we have $\displaystyle{a}_{{1}}={1}$, $\displaystyle{d}={2}$ and $\displaystyle{n}={1000}.$

So, using the formula:

$\displaystyle{S}_{{n}}=\frac{n}{{2}}{\left[{2}{a}_{{1}}+{\left({n}-{1}\right)}{d}\right]}$

$S$n =  2/n [2a+(n-1)d]

the sum 

$\displaystyle{1}+{3}+{5}+\ldots$ for 1000 steps is given by:

$\displaystyle{S}_{{1000}}=\frac{1000}{{2}}{\left[{2}{\left({1}\right)}+{\left({1000}-{1}\right)}{\left({2}\right)}\right]}={1}\ {000}\ {000}$[2(1)+(1000−1)(2)]=1 000 000

AP have 2 types formula:

            1] to find any term

           n th term T_n
          = a + (n-1)d


           2] to find sum term

          S_n = n/2 [2a + (n-1)d] or S_n = n/2 [a + 1]

SEE THIS VIDEO! (learn something from this video).....

SOLVE THIS QUESTIONS?

1.      Find the number of terms in the following Arithmetic sequence

T_n = a + (n-1) d

a)     2,5,8…299

a=2

d=3

b)     2,4,6…246

a=2

d=2

2.      In an AP, the 24^th term is 122 and common different is 3, find the 5 series

122= a + (23)3

                                122=a +69…continue to answer…

3.      The first 20 terms of the of series 24⁺³,27,30,33..

S_n= n/2 [2a+(n-1)d]

a=24

d=3

Geometric Progression

Geometric Progression (GP) Sequence of numbers where the ratio between any two adjacent numbers is constant.

Example of Geometric Progression:

Example 1:

2,4,8,16,32,64,128,256…

This sequence has a factor of 2 between each number.

Each term (except the first term) is found by multiplying the previous term by 2. 

In General we write a Geometric Sequence like this:

{a, ar, ar², ar³, ... }

where:

• a is the first term, and
• r is the factor between the terms (called the "common ratio")

Example 2:

{1,2,4,8,...}

The sequence starts at 1 and doubles each time, so

• a=1 (the first term)
• r=2 (the "common ratio" between terms is a doubling)

And we get:

{a, ar, ar², ar³, ... }

= {1, 1×2, 1×2², 1×2³, ... }

= {1, 2, 4, 8, ... }

But be careful, r should not be 0:

               *When r=0, we get the sequence {a,0,0,...} which is not geometric

Example 3: 

x_n = ar^(n-1)

(We use "n-1" because ar⁰ is for the 1st term)

10, 30, 90, 270, 810, 2430, ...

This sequence has a factor of 3 between each number.

The values of a and r are:

• a = 10 (the first term)
• r = 3 (the "common ratio")

x_n = 10 × 3^(n-1)

So, the 4th term is:

x₄ = 10×3^(4-1) = 10×3³ = 10×27 = 270

And the 10th term is:

x₁₀ = 10×3^(10-1) = 10×3⁹ = 10×19683 = 196830

Formula of Geometric Progression:

        1) nth term T_n = ar^n-1

       2) S_n = a(1-rⁿ)/1-r or S_n = a(rⁿ – 1)/r-1

        3)  S_∞ = a/1-r

      

SOLVE THIS QUESTIONS?

1.      What is the eleventh term of Geometric Sequence

3,10,12,24…?

Tn=ar ^{n-1 (11-1=10)}

a=3

r=2

2.      What is the sum of the first eight terms of the geometric sequence 5, 15, 45, ... ?

S_n = a(1-rⁿ)/1-r

5, 15, 45, ... 
This sequence has a factor of 3 between each pair of numbers.

The values of a, r and n are:
• a = 5 (the first term) 
• r = 3 (the "common ratio")
• n = 8 (we want to sum the first 8 terms)