Skip to main content

Correlation: B.Com. II Notes

 Correlation

In previous blog we discussed about the measure central tendency and Dispersion to use to study the variable. the correlation is a statistical concept that allows us to measure and understand the relationship between two or more variables. it provided a valuable information about that variables. e.g. price and demand of commodity, income and expenditure of family, height and weight of group of persons. their we use the relation   between this two variables. in above examples we see the one variable increases other variable is also changes in same or opposite direction. 

definition: Correlation is statistical tool which study the relationship between two or more variables. for analysis of correlation various method and techniques are used. 

example: i. Demand and supply of product                                                                                                                       ii. price and demand                                                                                                                                           iii. Income and expenditure

Use Of Correlation:

the correlation analysis is widely used in economic, business and other fields. 

i. To Predict: if we know the relation between two variables, we can estimate the value of them when value of other is known. e.g. we know the correlation between height and weight then we calculate the weight for known value of height.

ii. To control: The correlation also enables us to control our activity. e.g. we know the correlation between fertilizer and crop, then we control the yield and life of the crop, (use of more fertilizer is hazard to crop. 

iii. To Plan: the knowledge of correlation help to planning. e.g. if we know the relation between the rainfall and yield of crop, then we know the rainfall we calculate the yield of crop, depend on the yield of crop we  plan for import and export.

Types of Correlation:

i. Positive and Negative Correlation

ii. Linear and non-liner Correlation.

iii, simple, Multiple and  Partial Correlation.

Positive correlation: If both the variables changes in same direction i.e. if one variable increases other variable also increases or if the one variable decreases then other variable also decreases, the correlation is said to be positive correlation. e.g. i. height and weight mean height is increases weight also increases.

Negative Correlation: if the both variables are changes in opposite direction i.e. if one variable increases other variable decreases or if the one variable decreases then other variable also increases. e.g. price and demand mean the price increases demand is decreases.

the difference between the positive and negative correlation depends on the direction of change of two variables.

when change in one variable is not affected on other variable it is no correlation between two variables.

Linear Correlation:  this type of correlation is based on the nature of the graph of two variables. if the graph is straight line the correlation is Linear correlation.

if the graph is not a straight line but is curve is called Non-Linear Correlation.

Simple Correlation: we study only two variable say price and demand it is simple correlation.

Multiple Correlation: we study more than two variables is called multiple correlation.

Partial Correlation: we study the more than two variables but correlation is studied between two variables only, and the effect of the other variable is assumed as constant.

Methods of studying Correlation:

I. Graphical method : Scatter Diagram

II. Mathematical Method:  

i. Karl Pearson Coefficient of correlation 'r'

                                                   ii. Spearman's Rank Correlation Coefficient 'R'

i. Scatter diagram: scatter diagram is a graph showing correlation between two variable. the N pair of values (x1,y1), (x2,y2) .....(xn,yn) of two variables x and y are plotted on the graph or XY plane we get the points or dot's on graph and they are generally scattered so it is called scatter diagram or dot diagram.

the scatter point show the direction and degree of correlation between x and y or any two variable the direction of correlation is denoted as + or -sign and degree by r. 

from the  scattered diagram we interpret the degree and direction of correlation. 

Interpretation of scatter diagram :

i. if the graph shows all the point lie on a rising straight line towards the right, the correlation is perfect positive and r = +1 e.g. correlation between age and height of children , their is strong correlation between  age and height , as children grow older they get taller and this relationship is quite linear in their growth phases.  we see the graph 





ii. if the graph shows all the point lie in falling straight line towards the right, the correlation s perfect negative and r=-1.e.g. the correlation between price and demand if price increases demand decreases.


iii. if all the points lies in narrow strip which rise towards right.  Then the correlation is high degree positive correlation, i.e. 08<r>1.





iv. if all the points lie in narrow strip which rise towards left, then the correlation is high –ve correlation it is -1<r>-0.8





v. if all point rise on broad line or strip rising towards the right the correlation is low degree positive correlation.





vi. if all point rise on broad line or strip rising towards the left the correlation is low degree positive correlation.



vii. if all point are scattered without any pattern, then there is no correlation, i.e. r=0.



Merit and demerit:

Merit:

i.                    It is very easy to draw.

ii.                  It is easy to understand.

iii.                It gives quick idea about the correlation between variables.

Demerits:

i.                    It is not used for mathematical calculation.

ii.                  It no measure the degree of correlation.

iii.                It is not suitable when values are large.

Example 1. To draw a scatter diagram & indicate whether correlation is positive or negative.

Price:

17

18

19

21

25

26

30

Supply:

30

37

36

42

50

51

54

 

To draw scattered diagram we plot the price on x –axis and supply on y-axis. Then we get a scatted diagram.



From this diagram we see the the point are rising hence it is high positive correlation.

Karl Pearson's Coefficient of correlation:

    Karl Pearson's gives the mathematical method to measure the correlation between two variables. based on following assumptions. 

i. there is linear relationship between two variables.

ii. there is cause and effect relationship between two variables.

definition: Karl Pearson's Coefficient of correlation between two variables X and Y, denoted as corr(X,Y) or "r" and it is given as      

\(r = \frac{{\text{Cov}(X, Y)}}{{\sigma_X \sigma_Y}}\)

Where:

  • \( \text{Cov}(X, Y) \) is the covariance of \(X\) and \(Y\), calculated as: \[ \text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) \]
  • \( \sigma_X \) is the standard deviation of \(X\).
  • \( \sigma_Y \) is the standard deviation of \(Y\).

The correlation coefficient formula can also be written as:

\[ r = \frac{\sum_{} xy}{\sqrt{x^2} \cdot \sqrt{y^2}} \]

Where:

  • \( x = (x - \bar{x}) \)
  • \( y = (y - \bar{y}) \)

Properties of \( r \):

  • 1. \( r \) always lies between -1 to +1.
  • 2. \( r \) is unaffected by a change of origin and scale.
  • 3. \( r \) is equal to the square root of the product of two regression coefficients, i.e., \( r = \sqrt{b_{xy} \cdot b_{yx}} \).
  • 4. \( r \) is free from unit.

Interpretation of \( r \):

  • If \( r = +1 \), there is a perfect positive correlation between two variables.
  • If \( r = -1 \), there is a perfect negative correlation between two variables.
  • If \( r = 0 \), there is no correlation between two variables.
  • If \( 0.8 < r \leq 1 \), there is a high degree of positive correlation between two variables.
  • If \( -1 \leq r \leq -0.8 \), there is a high degree of negative correlation between two variables.
  • If \( 0.4 \leq r \leq 1 \), there is a low degree of positive correlation between two variables.
  • If \( -0.4 \leq r \leq -1 \), there is a low degree of negative correlation between two variables.
  • The type of correlation depends on the sign of \( r \): if \( r \) has a negative sign, it means negative correlation, and if \( r \) has a positive sign, it indicates positive correlation.

Merits and Demerits:

Merits:

  • \( r \) gives the numerical value of correlation.
  • It is also useful for estimation.

Demerits:

  • The value of \( r \) is not affected by extreme values, so it may not provide proper results in some cases.
  • Sometimes, it is difficult to calculate.

Comments

Post a Comment

Popular posts from this blog

MCQ'S based on Basic Statistics (For B. Com. II Business Statistics)

    (MCQ Based on Probability, Index Number, Time Series   and Statistical Quality Control Sem - IV)                                                            1.The control chart were developed by ……         A) Karl Pearson B) R.A. fisher C) W.A. Shewhart D) B. Benjamin   2.the mean = 4 and variance = 2 for binomial r.v. x then value of n is….. A) 7 B) 10 C) 8 D)9   3.the mean = 3 and variance = 2 for binomial r.v. x then value of n is….. A) 7 B) 10 C) 8 D)9 4. If sampl...
Post a Comment
Read more

Basic Concepts of Probability and Binomial Distribution , Poisson Distribution.

 Probability:  Basic concepts of Probability:  Probability is a way to measure hoe likely something is to happen. Probability is number between 0 and 1, where probability is 0 means is not happen at all and probability is 1 means it will be definitely happen, e.g. if we tossed coin there is a 50% chance to get head and 50% chance to get tail, it can be represented in probability as 0.5 for each outcome to get head and tail. Probability is used to help us taking decision and predicting the likelihood of the event in many areas, that are science, finance and Statistics.  Now we learn the some basic concepts that used in Probability:  i) Random Experiment OR Trail: A Random Experiment is an process that get one or more possible outcomes. examples of random experiment include tossing a coin, rolling a die, drawing  a card from pack of card etc. using this we specify the possible outcomes known as sample pace.  ii)Outcome: An outcome is a result of experi...
1 comment
Read more

Statistical Inference: Basic Terms and Definitions.

  📚📖 Statistical Inference: Basic Terms. The theory of estimation is of paramount importance in statistics for several reasons. Firstly, it allows researchers to make informed inferences about population characteristics based on limited sample data. Since it is often impractical or impossible to measure an entire population, estimation provides a framework to generalize findings from a sample to the larger population. By employing various estimation methods, statisticians can estimate population parameters such as means, proportions, and variances, providing valuable insights into the population's characteristics. Second, the theory of estimating aids in quantifying the estimates' inherent uncertainty. Measures like standard errors, confidence intervals, and p-values are included with estimators to provide  an idea of how accurate and reliable the estimates are. The range of possible values for the population characteristics and the degree of confidence attached to those est...
Post a Comment
Read more

Index Number

 Index Number      Introduction  We seen in measures of central tendency the data can be reduced to a single figure by calculating an average and two series can be compared by their averages. But the data are homogeneous then the average is meaningful. (Data is homogeneous means data in same type). If the two series of the price of commodity for two years. It is clear that we cannot compare the cost of living for two years by using simple average of the price of the commodities. For that type of problem we need type of average is called Index number. Index number firstly defined or developed to study the effect of price change on the cost of living. But now days the theory of index number is extended to the field of wholesale price, industrial production, agricultural production etc. Index number is like barometers to measure the change in change in economics activities.   An index may be defined as a " specialized  average designed to measure the...
2 comments
Read more

B. Com. -I Statistics Practical No. 1 Classification, tabulation and frequency distribution –I: Qualitative data.

  Shree GaneshA B. Com. Part – I: Semester – I OE–I    Semester – I (BASIC STATISTICS PRACTICAL-I) Practical: 60 Hrs. Marks: 50 (Credits: 02) Course Outcomes: After completion of this practical course, the student will be able to: i) apply sampling techniques in real life. ii) perform classification and tabulation of primary data. iii) represent the data by means of simple diagrams and graphs. iv) summarize data by computing measures of central tendency.   LIST OF PRACTICALS: 1. Classification, tabulation and frequency distribution –I: Qualitative data. 2. Classification, tabulation and frequency distribution –II : Quantitative data. 3. Diagrammatic representation of data by using Pie Diagram and Bar Diagrams. 4. Graphical representation of data by using Histogram, Frequency Polygon, Frequency Curve and     Locating Modal Value. 5. Graphical representation of data by using Ogive Curves and Locating Quartile Values....
Post a Comment
Read more

B. Com. I Practical No. 3 :Diagrammatic representation of data by using Pie Diagram and Bar Diagrams.

Practical No. 3 :Diagrammatic representation of data by using Pie Diagram and Bar Diagrams. Diagrammatic Presentation. We have observed the classification and tabulation method. We use this method to take a lot of information and make it fit into a small table. The reason we do this is to make the information more organized and easier to understand. Tabulation helps us arrange data neatly so that it's not messy and confusing. tabulation is a way to make big files of information look neat and tidy in a table.  but better and beautiful way to represent data using diagrams and graphs. the diagram and graph have some advantages because that used to visualise the data. that helps to understand and give information easily to any common man or any one, following are the some  advantages of diagram and graph.  I. Advantages i. Data Representation: Diagrams and graphs are excellent for presenting data visually, making trends, comparisons, and statistical information easier to...
Post a Comment
Read more

Method of Moment & Maximum Likelihood Estimator: Method, Properties and Examples.

 Statistical Inference I: Method Of Moment:   One of the oldest method of finding estimator is Method of Moment, it was discovered by Karl Pearson in 1884.  Method of Moment Estimator Let X1, X2, ........Xn be a random sample from a population with probability density function (pdf) f(x, θ) or probability mass function (pmf) p(x) with parameters θ1, θ2,……..θk. If μ r ' (r-th raw moment about the origin) then μ r ' = ∫ -∞ ∞ x r f(x,θ) dx for r=1,2,3,….k .........Equation i In general, μ 1 ' , μ 2 ' ,…..μ k ' will be functions of parameters θ 1 , θ 2 ,……..θ k . Let X 1 , X 2 ,……X n be the random sample of size n from the population. The method of moments consists of solving "k" equations (in Equation i) for θ 1 , θ 2 ,……..θ k to obtain estimators for the parameters by equating μ 1 ' , μ 2 ' ,…..μ k ' with the corresponding sample moments m 1 ' , m 2 ' ,…..m k ' . Where m r ' = sample m...
Post a Comment
Read more

Time Series

 Time series  Introduction:-         We see the many variables are changes over period of time that are population (I.e. population are changes over time means population increase day by day), monthly demand of commodity, food production, agriculture production increases and that can be observed over period of times known as time series. Time series is defined as a set of observation arranged according to time is called time series. Or a time Series is a set of statistical observation arnging chronological order. ( Chronological order means it is arrangements of variable according to time) and it gives information about variable.  Also we draw the graph of time series to see the behaviour of variable over time. It can be used of forecasting. The analysis of time series is helpful to economist, business men, also for scientist etc. Because it used to forecasting the future, observing the past behaviour of that variable or items. Also planning for future...
1 comment
Read more

Non- Parametric Test: Run Test

Non- Parametric Test  A Non-Parametric tests is a one of the part of Statistical tests that non-parametric test does not assume any particular distribution for analyzing the variable. unlike the parametric test are based on the assumption like normality or other specific distribution  of the variable. Non-parametric test is based on the rank, order, signs, or other non-numerical data. we know both test parametric and non-parametric, but when use particular test? answer is that if the assumption of parametric test are violated such as data is not normally distributed or sample size is small. then we use Non-parametric test they can used to analyse categorical data  or ordinal data and data are obtained form in field like psychology, sociology and biology. For the analysis use the  some non-parametric test that are Wilcoxon signed-ranked test, mann-whiteny U test, sign test, Run test, Kruskal-wallis test. but the non-parametric test have lower statistical power than ...
4 comments
Read more