Measures of Central Tendency

Generally, the term average is related to a measure of central tendency.

It is nothing but a value to represent a series of data/observations.


There are 3 measures of central tendency :

1.Mean

2.Median

3.Mode

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Before we get into a detailed description, lets know about the relationship among these three measures.There are 2 cases where the relationship can be explained:
     
Case 1. When the distribution of data is symmetrical. In this case all the three measures are equal. In other words ;

                                            MEAN = MEDIAN = MODE

Case 2. When the distribution of data is asymmetrical. In this case the measures aren't equal. So, an equation is given ;

                                          MODE = 3MEDIAN - 2MEAN  

And this equation is known as the empirical relationship between the mean, the median and the mode.

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Now let's go into a detailed description about the measures.

Mean is the result you get by adding two or more amounts together and dividing the total by the number of amounts .

It is denoted by  .

$$
{\displaystyle {\bar {x}}}

          There are 5 main types of mean :

          1.Arithmetic mean : It is the quantity obtained by summing two or more numbers or variables and then dividing by the number of numbers or variables. It is very much similar to the normal mean .

Formulae :

For individual series :

              = ∑x / N  

For discrete series :

              = ∑fx / N

For continuous series :

           = ∑fm / N    

                     where ;m is the mid value.

           
           2.Combined mean : It is the arithmetic mean of the means of two or more series.

Formula :

            c =  [(1)(N1) + (2)(N2 )] / N1 + N2

           

            3.Harmonic mean : It is the reciprocal of arithmetic average of the reciprocal of values of items in the variable.

Formulae :           

For individual series :

            = N / ∑(1/x)

For discrete series :

         = N / ∑f(1/x)

For continuous series :

           = N / ∑f(1/m)

                            where ;m is the mid value.

          
            4.Geometric mean : It is defined as the Nth root of the product of N number of observations in a series of data.

Formula :           

          GM =  N√x1 * x2 * x3 * ..........xn
                   where; n is the number of observations.

         

            5.Weighted mean: When calculating the arithmetic mean, the importance of all the items are considered to be equal. However, there may be situations in which all the items under considerations are not of equal importance. So we use the weighted mean formula to calculate the mean along with the importance or weightage of each observation in the data.

Formula:
           = ∑Wx / ∑W

Now lets look into the merits and the demerits of the mean ;

Merits :
1.It is simple and easy to apply.
2.It is based on all the observations in the data.

Demerits :
1.It is affected by extreme observations.
2.If any one of the variables are missing, mean cannot be computed.

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Median is the middle value of the series, when the data is arranged in an ascending or a descending order. It is denoted by M.
Before you apply the formula, make sure that the data is arranged in an ascending order.

Formulae:
For individual series:
         M = [(N+1)/2]th item
For discrete series:
         M = Size of [(N+1)/2]th item

For continuous series:

        We find the median class first .
        Median class = Class that has the item of the size (N/2)th item
        M = L1 + {[ (N/2) - cf ] / f } * i
        where; L1 = the lower limit of the median class 
                    cf  = the cumulative frequency of class preceding median class
                      f  = Frequency of median class.
                      i  = the difference of upper limit and the lower limit.

Now lets look into the merits and the demerits of the median ;

Merits :
1.It can be computed in case of frequency distribution with open ended classes.
2.It can be determined graphically.

Demerits :
1.It is not based on all the observations of the data.
2.It is affected by the fluctuation of sampling.

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Mode is the value which occurs the most frequently in a series. It is denoted by Z.

Formulae:
For individual series :
     The terms are arranged in any order. Ascending or Descending. If each term of the               series is occurring once, then there is no mode, otherwise the value that occurs                     maximum times is the Mode.
For discrete series :
     Here, the variable(x) which has the highest frequency will be the mode.
For continuous series :
     Here, we use a rigid formula;
     Z = L1 + ( f1 - f0 ) / [ 2(f1) - f0 - f2 ] * i
        where ; L1 = the lower limit of the modal class
                      f1 = the highest frequency 
                      f0 = the preceding frequency of f1
                      f2 = the successive frequency of f1
                       i  = the difference of upper limit and the lower limit.

Now lets look into the merits and the demerits of the mode ;

Merits :

1.It is not affected by extreme values.

   It can be obtained even if the extreme values are not known.

2.Mode can be located on the graph also.

Demerits :

1.It is not based upon all the observation. 
2.It is affected to a greater extent fluctuations of sampling. 

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If you wish to know the basics of statistics, use the following link :

https://basicmathematix.blogspot.in/2018/04/basics-of-statistics.html

Branches, sources of data, merits and demerits of statistics

Basic Statistics

We can understand the meaning of statistics from the following definitions :

According to Prof. Horace Secrist  “ Statistics is an aggregate of facts affected to a marked extent by the multiplicity of causes, numerically expressed, enumerated or estimated according to a reasonable standard of accuracy, collected in a systematic manner for a predetermined purpose and placed in relation to each other ".

According to Croxton and Cowden  " Statistics may be defined as the collection , presentation ,analysis and interpretation of numerical data ". 

Branches of statistics:

1. Descriptive statistics
It relates to the collection and presentation of data in different forms such as graphs, tables and diagrams to describe or represent the data in a simpler way. Companies use this branch of statistics to prepare annual reports, financial accounts, etc.

2. Inferential statistics
It is about the techniques used for the analysis of data, computing estimations and drawing conclusions from them. These estimations are drawn from the sampling and its testing. Future events forecasts are predicted and made from this type of statistics.  

Statistics deals with the collection of data.

Sources of data:

1. Primary source
They are also known as statistical source. They are obtained directly in the field. Census and surveys are used to obtain data here.

2. Secondary source
Also known as non-statistical source, are obtained from the already existing records(data collected primarily) from different governmental and non governmental organisations.

Now let's look into the merits and demerits of statistics :

Merits:

1.Allows to present facts in a definite form .
2.Facilitates comparison .
3.Facilitates predictions/forecasts .
4.Formulation of suitable policies .
5.Simplifies the data .

Demerits:

1.Does not deal with individual measurement.

2.Excludes qualitative information.

3.Result is only an average. We ignore the bigger picture .

4.Bias is possible.

Now let’s learn about averages, also known as the measures of central tendency.

First let’s know the meaning of average ;It is a value to represent a series of data.

Its functions are similar to the merits of statistics.

Essentials of a good average:

1. Should be simple to calculate.

2. Should be easy to understand.

3. Should be rigidly defined.

4.Should be based on all items of observations.

These are the main types of averages:

1.Mean

2.Median

3.Mode

Simple and Compound Interest - meaning and formulas

Simple and compound interest 

Simple interest is the money that is paid only on an original amount of money that has been borrowed or invested, and not on the extra money that the original amount earns.

Simple interest or SI = (P*T*R)/100

P is the principal (the money borrowed or lent out for a certain period) .  

R is the interest rate (the rate at which the interest has to be calculated) .

T is the time (the period for which the principal was borrowed) .

Amount is the summation of principal and simple interest .

A = P + SI

There's another way to calculate the amount .

A = P [ 1 + TR/100 ]

Let's calculate the simple interest for the following problem.

Principal amount = 10000
Time = 2 years
Rate of interest = 5% per annum

Sol. 

Let's get to the compound interest (CI) .

Compound interest is the  interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan .

To calculate the CI we need to first compute the total amount when the CI is accumulated with the principal .
So ,to calculate this amount we use the following formula :



After we ascertain the amount ,we can calculate the CI by using the following formula :


CI = A - P

The basic concept of CI is that interest is added back to the principal sum so that interest is earned on that added interest during the next compounding period .

To the right is another formula to calculate the CI(compound interest). 

Let's take a look at how this works.

A man deposits Rs 1000 in a savings account at an interest rate of 10% per annum. At the end of one year he will get Rs 100 interest on his deposit. However, unless he takes out his Rs 100 in cash, it will be added to his original Rs 1000. Thus, if he leaves his money in his account, in the next year the bank will be paying him interest on his original Rs 1000 plus the Rs 100 interest,i.e.,Rs 1100. In the third year the interest will once again be added to the new principal of Rs 1100, and so on for as long as the money is left in the account.

This kind of interest is known as compound interest.


Now let's calculate the compound interest for the following problem.

P = 1000
R = 10%p.a.
T = 5 years

Sol. 

Transpose of a Matrix ,matrix's determinant and matrice's properties

The matrix obtained by interchanging the rows into columns or vice versa is known as Transpose of a matrix .
                                                                                         
 In the diagram given above we can see how the rows become columns .The 1st row becomes the first column ,the second row becomes the second column and the third row becomes the third column .

The value that can be computed from the elements of a square matrix is known as the Determinant .

To the right is the format and an example for computing the determinant for a 2*2 matrix .


To the left is the format to compute the determinant for a 3*3 matrix .
If the matrix is known as 'A', then the determinant of the matrix A is denoted det(A), det A or |A| .

An example to understand the calculation of the determinant of a 2 * 2 matrix.

Let’s find the determinant of the matrix given below:

Below is the calculation of the determinant of a 3 * 3 matrix.

Now let's look into the properties of a matrix.

Here below are the important properties of a matrix :

Property 1 : A + B  =  B + A 

Property 2 : A + (B + C)  =  (A + B) + C 

Property 3 : A(BC)  =  (AB)C   

Property 4 : A(B + C)  =  AB + AC 

Property 5 : (A + B)C  =  AC + BC

Matrix and it's uses and types

Firstly ,what is a matrix?

A Matrix is a rectangular array of numbers arranged in rows and
columns .

Matrices can contain complex numbers but they are designed to keep the data as simple and small as possible .The image to the right has two examples of matrices.


Now let's go into the uses of matrix.       

Use of matrix in the business field.

Consider a company with several outlets selling several different products, a matrix provides a concise way of keeping track of stock.

Outlet	Milo	Milk	Bar soap	Bath gel
Ghana	110	130	170	180
Nigeria	290	105	145	125
Togo	170	90	140	160
Benin	145	115	200	195

Reading across a row of the matrix, the firm(Nestle) can determine the level of stock in any of its outlets.

By reading down the column, the firm can determine the stock of any of it’s products.

Use of matrix in graph theory.

The adjacency matrix of a finite graph is a basic notion of the graph theory.

Use of matrix in the field of computer science.

It is used in computer graphics; the first model of quantum mechanics also known as matrix mechanics.

To our left ,we can see the format of a matrix .
'm' represents the rows and 'n' the columns .
'i' represents the number of rows and 'j' the number of columns .

This number of rows and columns that a matrix has is known as the order of matrix .
                                                                                                                                                                                             
Now lets get to types of matrix .

The following are the types of matrices which are commonly used to represent data :

1. Row matrix : This matrix has only one row and 'n' number of columns .

2. Column matrix : This matrix has only one column and 'm' number of rows .

3. Zero matrix or the null matrix : This one is called as such because the elements it would contain are equal to zero (0) .


4. Square matrix : The square matrix has equal number of rows and columns ,pretty obvious why it is named 'square matrix' .It simply means that 'm' = 'n' .



5. Diagonal matrix : It is a square matrix in which the non diagonal elements are equal to zero .

                 
6. Scalar matrix : It is a diagonal matrix in which all the diagonal elements are equal .


7. Unit matrix or the identity matrix : A square matrix with ones (1) on the diagonal and zeros (0) elsewhere .This matrix is to be denoted by 'I' .

8. Upper Triangular Matrix : A matrix in which the elements below the main diagonal are equal to zero '0' .

9. Lower Triangular Matrix : A matrix in which the elements above the main diagonal are equal to zero '0' .