Index Numbers Assignment help




an Index is a statistical measure of changes in a representative group of individual data points. These data may be derived from any number of sources, including company performance, prices, productivity, and employment. Economic indices (index, plural) track economic health from different perspectives. Influential global financial indices such as the Global Dow, and the NASDAQ Composite track the performance of selected large and powerful companies in order to evaluate and predict economic trends. The Dow Jones Industrial Average and the S&P 500 primarily track U.S. markets, though some legacy international companies are included.The Consumer Price Index tracks the variation in prices for different consumer goods and services over time in a constant geographical location, and is integral to calculations used to adjust salaries, bond interest rates, and tax thresholds for inflation. The GDP Deflator Index, or real GDP, measures the level of prices of all new, domestically produced, final goods and services in an economy. Market performance indices include the labour market index/job index and proprietary stock market index investment instruments offered by brokerage houses.

Study of Numerical Data:

            Numerical data are measured by numerical value. Numerical data are analyzed by using the statistics methods. The results of the numerical data are given in the form of tables, charts, and graphs. Numerical data are produced at great expensive value. It also used in different fields. Most of these data will be analyzed only for the single purpose for which they are originally collected, even though the data have the potential value for other uses. The answers of the questions are in numerical value. From the study of numerical data it is known that they are represented by digits and numbers.

Let us study about numerical data indexing and application of numerical data.

Studying Numerical Data Indexing

Definition:

          Study of numerical data indexing attempts to overcome the limited use of data's. It creates a new resource for numerical data. In numerical data we can use one or two values for the given event. But in the Numerical data indexing we can use more values for the given event. It is considered to be extension of subject indexing. It is at a minimum, it contains the numerical value. It is also  attempt for a higher level indexing. The Numerical data's have different values. For every questions the answers are represented in numerical form. Numbers are very effective study of numerical data indexing and is representing the data.

Studying Application of Numerical Data

           Numerical data are widely used  for representing the graphs, charts, and tables. The numerical data are also used in representing the histograms. They are used for finding the mode, mean, and median of the data. The tables are widely used in newspaper, books and journals. In which numerical data are initially stored and then analyzed. Tables are very effective for representing the numerical data. There are many graph types are used for representing the numerical data. The Graphs are classified as bar charts, pie charts, histograms, and scatter plots. The graphs are used for plotting the numerical data. Histogram is the special type of bar charts.

An index number is an economic data figure reflecting price or quantity compared with a standard or base value. The base usually equals 100 and the index number is usually expressed as 100 times the ratio to the base value. For example, if a commodity costs twice as much in 1970 as it did in 1960, its index number would be 200 relative to 1960. Index numbers are used especially to compare business activity, the cost of living, and employment. They enable economists to reduce unwieldy business data into easily understood terms. In economics, index numbers generally are time series summarising movements in a group of related variables. In some cases, however, index numbers may compare geographic areas at a point in time. An example is a country's purchasing power parity. The best-known index number is the consumer price index, which measures changes in retail prices paid by consumers. In addition, a cost-of-living index (COLI) is a price index number that measures relative cost of living over time. In contrast to a COLI based on the true but unknown utility function, a superlative index number is an index number that can be calculated. Thus, superlative index numbers are used to provide a fairly close approximation to the underlying cost-of-living index number in a wide range of circumstances.

Generally a polymer sample contains chains of varying lengths and therefore its molecular mass is always expressed as an average. In contrast, natural polymers such as proteins, contains chains of identical length and hence have definite molecular mass.

Number average molecular mass

Number average molecular mass is obtained by dividing the total molecular mass of the sample by total number of molecules.

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Ni = number of molecules

Mi = molecular mass

Mass average molecular mass

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Polydispersity Index (PDI)

Polydispersity index is the ratio of the weight and number average molecular masses.

For natural polymers PDI is unity (monodispersed)

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In synthetic polymers:

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img is determined by employing methods which depend upon the number of molecules present in the polymer sample viz. colligative property like osmotic pressure. On the other hand, methods such as light scattering and ultra centrifuge depend on the mass of the individual molecules and yield weight-average molecular masses.

Binomial Theorem for Positive Integral Index in Algebra:

Introduction to algebra binomial theorem:

An algebraic expression which contains two terms is called a binomial expression.

For example (a+b), (2x-3y), (x + 1/y ), (x + 3/x ) ,( 2/x - 1/x^2 ) etc. are binomial expressions.

The general form of binomial expression is (x +a) and the expansion of (x +a) n , n in   N  is called the binomial theorem in algebra.This theorem was first given by Sir Isaac Newton. It gives a formula for the expansion of the powers of a binomial expression.

In earlier classes, we have learnt that :

(x +a) 0 = 1

(x+a) 1 =x + a

(x+a) 2 = x 2 + 2ax + a 2

(x +a) 3 = x 3 +3x 2a + 3xa 2 + a 3

(x +a) 4 = x 4 + 4 x 3 a + 6 x 2 a 2 + 4 x a 3 + a 4

 We observe the coefficients in the above expansions follow a particular pattern as given below:

   Index of the binomial                                                           Coefficients of various terms

                 0                                                                                                        1

                 1                                                                                                 1             1

                 2                                                                                             1         2             1

                 3                                                                                       1        3           3               1

                 4                                                                                  1         4         6           4              1

    We observe that each row is bounded by 1 on both sides.Any entry except the first and last , in a row is the sum of the two entries in the preceding row, one on the immediate left and the other on the immediate right.This above pattern is called as Pascal's triangle.It has been checked that the abvoe pattern hold good for the coefficients in the expansions of the binomial expressions having index (exponent) greater than 3 in algebra.

Binomial Theorem for Positive Integral Index in Algebra:

Theorem: If x and a are real numbers , then for all n in  N ,

( x + a ) n = nC0 x n a 0 + nC1 x n-1 a 1 + nC2 x n-2 a 2 +........+nCr x n-ra r+.....+nC n-1 x1 a n-1 + nC n x 0 a n

In general we write

( x + a ) n = img

Some Important Conclusions from the Binomial Theorem of Algebra:

  • ( x + a ) n = img  . Since r can have values from 0 to n , the total number of terms in the expansion is (n+1).
  • The sum of indices of x and a in each term is n.
  • Since nCr = n C n-r for r =0,1,2,3,....,n  `rArr`  nC0 = nCn , nC1 = nCn-1, nC 2 = nCn-2,...
  • Replacing a by -a , we get 

(x - a) n = nC0 x n a 0 - nC1 x n-1 a 1 + nC2 x n-2 a 2 +.................+ (-1) r nCr x n-ra r+.....+(-1) n-1 nC n-1 x1 a n-1 + (-1) n nC n x 0 a n

    ( x - a ) n = img   .

   Thus the terms in the expansion of (x - a) n are alternatively positive and negative,the last term is positive or negative according as n is even or odd.

  • Putting x = 1 and a = x in the expansion (x +a) n , we get:

( 1 + x) n = n C 0 + nC1 x + nC2 x 2 +...nCr x r +...+nCnx n

(1 +x) n = img

  •  Putting a=1 in the expansion (x+a) n , we get  

 (x +1) = n C 0 x n + nC1 x n-1+ nC2n-2 +...nCrn-r +...+nCn

 (1 +x) n =

  • Putting x =1 and a = -x , we get

(1 -x ) nn C 0 - nC1 x + nC2 x 2 +... + ( - 1) r nCr x r +...+ (-1) n nCnx n

img

  • The coefficient of (r+1)th term in the expansion of (1+x) n is  n C r
  • The coefficient of x r in the expansion (1+x)n is  n C r 

 


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