COMPUTER MATHS

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QUALITATIVE MATHEMATICS

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MEASURES OF CENTRAL TENDENCY

Arithmetic Mean

This is the average of any given observation e.g 8,4,9 will be
8+4+9/3=7
For a grouped data mean is calculated as
Mean = Efx/Ef
E = sum of
f = frequency
x = midpoint
Other types of mean include;
Geometric mean;
This is the nth root of the products of numbers i.e
5,2,1,3
(5*2*1*3)*1/4=2.34
Harmonic mean;
This is the reciprocal of arithmetic mean of reciprocal of numbers of items.i.e
1/x+1/x+1/x)/3
Median
This the middle value of any given observation.When the data is arranged in ascending or descending order it is calculated as follows;
9,6,5,7,1
1,5,6,7,9
6 is the median because is the middle value.
For a grouped data mean is calculated using the following formula;
L+(C1/2N-CF)*F Where;
L=lower class limit.
N=sum of frequency column.
CF=cumulative frequency before median class.
F=frequency of median class.
C=median class size.
Mode.
This is the most frequent number in any given observation i.e
6,4,8,4. mode is = 4.
For a grouped data mode is calculated as follows;
L+(F1-F0)/(2F1-F0-F2) Where;
L = lower class limit of modal class.
F1 = highest frequency.
F0 = frequency before f1.
F2 = frequency after f1.
C = modal class size.
N/B;
Modal class is the class with the highest frequency.
Histogram.
This is a diagrammatic representation of information in order to determine the mode.(frequency against upper class limit)

Measures of Dispersion.

This analyses how data is spread from measures of location. i.e Arithmetic mean.
There are varies measures namely;
Range.
Variance and standard deviation.
Co-efficient of variance
Quartiles.
Range
This is the difference between the highest value and the lowest value.
Variance.
This is the mean squre deviation.It is calculated as follows;
Variance = E(X-MEAN)/N;N =number of items.
Standard deviation;
This is the squreroot of variance.
Co-efficient of variance;
This is the ratio between standard variation and the mean.It is a measure of risk to any business,
Quortiles.
This is a distribution which is divided into for equal parts.
lower quartile
L+(1/4N-CF)F)*F
Upper quartile

L+(3/4N-CF)/F)*C.
Quartile deviation
This is the difference between upper quartile and lower quartile.
Semi inter quartile range
This the quartile deviation divided by two
Ogive;
This is a cumulative frequency curve which is drawn to estimateQ1,Q2 and Q3 as follows.


Measures of skewness
This is evaluation and analysis of degree of symmetry.
There are three types of analysis namely;
Normal distribution
positive skewness
Negative skewness

Normal distribution(Gausian Distribution)

This arises when the mean and the mode are equal.
Properties of normal distribution
Its bell shaped
The three measures of central tendancy are equal.
Has a standard deviation of one.
Its explained a symbol of Z
positive skewness
It arises when mode is greater than mean.
negative skewness It occurs when mean is greater than mode.
In order to determine skewness the following formula's are used

Formula 1;
s.k=3(mean*median)/standard deviation.
Formula 2.
s.k=(Mean-mode)/standard deviation
Interpretation
If we use the above formula and you get a 0 then it means their is a normal distribution
If the answer is positive it means their is a positive skewness.
If the answer is negative it means their is a negative skewness.
CORRELATION
Its the existence of relationship between two variables or the degree of relationship between two variables.
TYPES OF CORRELATIONS
Positive correlation
An increase in one variable leads to an increase in the other variables i.e
Increase in price leads to an increase in quantity supplied
Increase in advertisement leads to an increase in sales
Negative Correlation
an increase in one variable leads to a decrease in the other variable i.e
An increase in price leads to a decrease in quantity demanded
Spurious Correlation
This is the relationship between many variables.It is also called Non-sense correlation .
The degree of relationship is determined by two co-efficients as follows.

Spear-sons Moment co-efficient of correlation.
R=N{xy-{x{y/([n{x*x-{(x)*(x)]*[n{y*y-{(y*y))*0.5 Where;
n=Number of items
x=In depended variables(x axis)
y=Depended variables(y axis)

spears man rank co-efficient of correlation.

This is used to establish relationship between two variables,especially when ranking of items when necessary.
The following formula is applied


r=1-(6{d*d/n(n8n-1))
D=Difference between ranks
n=Number of items

REGRESSION
This is the nature of relationship between variables.The nature of relationship is determined by regression line equation.(line of best fit)
This is the general form y=a+bx
y=Dependant variables
x=Independent variables
a=Constant(fixed cost)
b=gradient which represents variable cost per unit
Types of Regression
Simple linear Regression
It has one in depended variables(x) of general formula.
y=a+bx
Multiple Linear
It has many independent variables of general formula
Y=a+bx+bx+bx
The value of b and a in simple Linea regression is determine as follows;

b=n{xy-{x{y/n{x*x-{(x*x)
a={y-b{x/n.

Co-efficient of Determination(r*r)

In depended variables can be used to explain independent variables.it is used to show by how much dependent variables rely on independent variables.it is calculated by squaring co-efficient of correlation as follows;
r*r=(n{xy-{x{y/(n[{x*x-{[x*x]]*[n{[y*y]-{[y*y]])squire
Time Series
This is a mathematical technique used to predict the trend.This is the observation which has been their for a very long period period of time.
Components of Time series.Thi is the characteristics of the time series namely;
Secular Trend 'T'
Seasonal Variation 'S'
Random Variation 'R'
Cyclic Variation 'c'
Secular Trend;This is the observation seen over long period of time.i.e increase in prices of commodities
seasonal variation;This are the observation seen over long period of time.
Random variation;This are things which are beyond human control and they occur as a result of state of nature.
Cyclic variation;this is variation normally seen in business trade cycle.
During boom phase their is high sales,high profits and high employment levels.
During races their is losses,no sales and high unemployment level.

Methods of fitting a Trend line

Semi-average

Least square equation method

Moving average method
N/BThe line of best fit (Regression line equation) is used to predict line values
i.e Y=a+bx
b=n{x*y-{x{y/n{(x*x)-{(x*x)
a={y-b{x/n
Deseosonalization of time series

This is the removal of seasonal variation from the other components of time series.
Their are two major methods used for this technique;

Additive model

Multiplicative model
Additive model
Under this we sum up the four components of time series e.g
Secular + Random + Cyclic + Seasonal components.
Under additive model,the trend values are subtracted from the actual values in oder to remove seasonal variation
N/B Additive model is applied when the four components of the time series are independently from one another.
Multiplicative model.
Under this model the four components of time series are multiplied i.e
Seasonal*Random *Secular.
In order to determine seasonal variation using multiplicative model the actual values are divided by trend line values.,br> 

N/B 

 The model is applied when the four components of time series are depended of one another.

This are numbers that illustrate changes in prises

Simple price index is calculated as follows;
Simple price index=(p1/p0)*100%
Where;
p1=Price in the current year
p0=Price in the base year

weighted

The weight of a given commodity is combined with prices.
Their are three types of weighted index namely;

Lasypyres index number

paasches index number

Fishers ideal index number.
Lasypyres index
this uses the base year as the weight.
There are two types of Lasypyres index namely;

Lasypyres price index
={(p1q1/p0q0)*100%

lasypyres quality index;
={q1p0/q0p0)*100%
Where;
p1=Price in the current year.
p0=Price in the base year.
q1=Quantity in the current year.
q0=Quantity in the base year
Paasches index
This uses the current year as the weight.
it is of two types namely;

Paasches price index
={(p1q1/p0q1)*100%

Paasches quantity index
={(q1p1/q0p1)*100%
Fishers ideal index
This is quadratic mean between Lasypyres index and Paasches index as follows;

Fishers ideal price index;
(Paasches pirce index*Lasypyres price index)*0.5

Fishers ideal quantity index;
(Paasches Quantity index*Lasypyres quantity index)*0.5
Network Analysis

This is a mathematical technique used to plan and control the activities of any given project.
Rules for drawing Network analysis.

A network diagram should start from the left towards the right hand.

Every event must have a start and an end.

Dagler(not connected) activity not allowed in the network analysis.

Dummy(sample) can be introduced to simplify the connection between activities.

Lines should not intersect.
N/B>

Foreword pass; is the movement from the start to the end.The values are added and if in the event the root has more than one event,the highest number is chosen.
Backward
pass is the movement from the end to the start.The values are subtracted and the smallest number is considered.
Project Evaluation Review Technique.
The duration of each and every activity is determined as follows;
Duration=O +4M+p/6