Accounting in the books of Hire-vendor
Hire Vendor: There is only one method of recording the entries in the books of hire-vendor. Irrespective of the fact whether the entries in the books of hire-purchaser are passed under the Asset Accrual Method or under the Total asset value Method.But the accounting entries in the books of hire-vendor are always passed under the total Asset Method. These entries are as under:-
(i)On delivery of goods to the hire-purchaser at the time of agreement:
Hire – purchaser A/c Dr. Cash Price
To Hire – Sales A/c.
(ii)On receipt of cash at the time of agreement (down payment), if any:
Cash/Bank A/c. Dr. (Amt. of down payment)
To Hire-Purchaser
(iii)On interest being due:
Hire – Purchaser A/c Dr. Amt. of Interest
To Interest A/c.
(iv)On receipt of instalment:
Cash/bank A/c. (Amt. of Instalment)
To Hire – Purchaser
(v)On Transfer of Balance of Hire-Sales A/c. to Trading A/c. (at the end of first year only):
Hire – Sales A/c Dr. Cash Price
To Trading A/c.
(vi)On Transfer of amount of interest to P/L A/c:
Interest A/c. Dr. (Balance of Intt. A/c.)
To P/L A/c.
Note: In solving a numerical problem, before recording the entries, the amount of interest included in various instalments will be separately calculated as already explained.
Posting in Ledger Accounts:
After passing entries in the journall of hire – vendor the following accounts will be opened in the ledger of hire – vendor and the postings will be made accordingly.
(i) Hire – Purchaser A/c.
(ii) Hire – Sales A/c. (only in first year)
(iii) Interest A/c.
Calculate the amount of annual instalment, and show the Journal entries and necessary ledger accounts in the books of Moti Ltd. for three years. The present value of Annuity of Rupees one for three years at 5% is 2.72325.
Worked out examples-3:
On 1st April,2005 X Company Ltd. purchased a machine from Y Machines Ltd. on hire-purchase basis, the cash price being Rs. 55,850 Rs. 15,000 was paid on the signing of the contract and the balance in three annual instalments of Rs. 15,000 each on 31st March each year. Interest is charged at 5% per annum. Depreciation was written off at rate of 10% per annum on the diminishing balance system.
Give journal entries in the books of X Company Ltd. whose accounting year ends on 31st March each year, under Asset Accrual Method.
Solution:
(a) under Asset Accrual Method
Journal Entries in the Books of X Co. Ltd. Date Particulars LF Dr.(Rs.) Cr.(Rs.)
2005
April 1 Machinery A/c …………….Dr.
To Bank A/c
(Being down payment made at the time of delivery)
15,000
15,000
2006
March 31 Machinery A/c ……………………………………...Dr.
Interest A/c ....................Dr.
To Y Machine Ltd.
(Being the first instalment due).
12,957
2,043
15,000
" Y MachinesLtd. …………………Dr.
To Bank
(Being the amount paid in first instalment)
15,000
15,000
" Depreciation A/c …………………Dr.
To Machinery A/c
(Being the depreciation charged)
5,585
5,585
" Profit & Loss A/c …………………Dr.
To Interest A/c
ToDepreciation A/c
(Being the amount transferred)
7,628
2,043
5,585
2007
March 31 Machinery A/c ……………………………………...Dr.
Interest A/c ....................Dr.
To Y Machine Ltd.
(Being the second instalment due).
13,605
1,395
15,000
" Y Machines Ltd. …………………Dr.
To Bank
(Being the amount paid in second instalment)
15,000
15,000
" Depreciation A/c …………………Dr.
To Machinery A/c
(Being the depreciation charged)
5,027
5,027
" Profit & Loss A/c …………………Dr.
To Interest A/c
ToDepreciation A/c
(Being the amount transferred)
6,422
1,395
5,027
2008
March 31 Machinery A/c ……………………………………...Dr.
Interest A/c ....................Dr.
To Y Machine Ltd.
(Being the third instalment due).
14,288
712
15,000
" Y Machines Ltd. …………………Dr.
To Bank
(Being the amount paid in third instalment)
15,000
15,000
" Depreciation A/c …………………Dr.
To Machinery A/c
(Being the depreciation charged)
4,524
4,524
" Profit & Loss A/c …………………Dr.
To Interest A/c
To Depreciation A/c
(Being the amount transferred)
5,236
712
4,524
Wednesday, 4 January 2012
Hire-puchase system is a special system of purchase and sale of goods. Under this system purchaser pays the price of the goods in instalments. The instalments may be annual, six monthly, quarterly, monthly fortnightly etc. Under this system the goods are delivered to the purchaser at the time of agreement before the payment of instalments but the title on the goods is transferred after the payment of all instalments as per the hire-purchase agreement. The special feature of a hire-purchase transaction is that the payment of every instalment is treated as the payment of hire charges by the purchaser to the hire vendor till the payment of the last instalment.. After the payment of the last instalment, the amount of various instalments paid is appropriated towards the payment of the price of the goods sold and the ownership or the goods is transferred to the purchaser. Thus hire-purchase means a transaction where the goods are sold by vendor to the purchaser under the following conditions :
the goods will be delivered to the purchaser at the time of agreement.
the purchaser has a right to use the goods delivered.
the price of the goods will be paid in instalments.
every instalment will be treated to be the hire charges of the goods which is being used by the purchaser.
if all instalments are paid as per the terms of agreement , the title of the goods is transferred by vendor to the purchaser.
if there is a default in the payment of any of the instalments, the vendor will take away the goods from the possession of the purchaser without refunding him any amount received earlier in the form of various instalments.
Before discussing the characteristics of hire-purchase system, we must know what is a hire purchase agreement and what are the contents of a hire-purchase agreement. Hire-purchase agreement means a contract between the hire vendor and the hire purchaser regarding the sale of goods under certain conditions. Usually every hire-purchase agreement shall contain the following terms:
the cash price of the goods, cash price means the price at which goods may be purchased against cash payment.
the hire-purchase price, hire purchase price means the total amount which is payable by the hire-purchaser under the agreement.
the date on which the hire-purchase agreement will commence.
the description of the goods that will be delivered to the hire-purchaser at the commencement of the agreement.
the number of instalments to be paid by the hire-purchaser along with the amount of each instalment and the date of payment of each instalment.
the down payment if any, the down payment means the amount which is required to be paid by hire-purchaser to the hire vendor at the time of commencement of hire-purchase agreement.
the rate interest charged by the hire vendor (optional).
Characteristics of Hire-Purchase System
The characteristics of hire-purchase system are as under
Hire-purchase is a credit purchase.
The price under hire-purchase system is paid in instalments.
The goods are delivered in the possession of the purchaser at the time of commencement of the agreement.
Hire vendor continues to be the owner of the goods till the payment of last instalment.
The hire-purchaser has a right to use the goods as a bailer.
The hire-purchaser has a right to terminate the agreement at any time in the capacity of a hirer.
The hire-purchaser becomes the owner of the goods after the payment of all instalments as per the agreement.
If there is a default in the payment of any instalment, the hire vendor will take away the goods from the possession of the purchaser without refunding him any amount.
Difference between Hire-purchase system and Instalment payment system
Instalment Payment System is system of purchase and sale of goods in which title of goods is immediately transferred to the purchaser at the time of sale of goods and the sale price of the goods is paid in instalments. In the event of default in payment of any instalment, the seller has no right to take back goods from the possession of the purchaser. He can file a suit for the recovery of the outstanding balance of the price of goods sold. The followings are the differences between Hire-purchase system and Instalment payment system:
In Hire-purchase system, the transfer of ownership takes place after the payment of all instalments while in case of Instalment payment system, the ownership is transferred immediately at the time of agreement.
In Hire-purchase system, the hire-purchase agreement is like a contract of hire though later on it may become a purchase after the payment of last instalment while in Instalment payment system, the agreement is like a contract of credit purchase.
In case of default in payment , in Hire-purchase system the vendor has a right to back goods from the possession of the hire-purchaser while in case of Instalment payment system, the vendor has no right to take back the goods from the possession of the purchaser; he can simply sue for the balance due.
In Hire-purchase system, if the purchaser sells the goods to a third party before the payment of last instalment, the third party does not get a better title on the goods purchased. But in case of Instalment payment system, the third party gets a better title on the goods purchased.
In Hire-purchase system the provisions of the Hire-purchase Act apply to the transaction while in case of Instalment payment system, the provisions of Sale of Goods Act apply to the transaction.
1.6 Accounting In the books of Hire-purchaser
There are two methods of accounting in the books of Hire-purchaser. Their detailed description is as under:-
Asset Accrual Method:
Under this method it is considered that the hire-purchaser is the owner of the asset up to the value of the cash price paid by him in the from of down payment or the cash price paid included in various instalments. The following journal entries are recorded under this method.
(i)On taking the delivery of asset:
No entry is recorded.
(ii)On making the down payment (if any)
Asset A/c Dr. (Amount of down payment)
To Cash/Bank A/c.
(iii)On becoming the instalment due
Asset a/c. Dr (Balancing figure)
Intt. A/c. Dr. (Amt. of Intt.)
To Hire-Vendor A/c. (Amt. of Instalment)
(iv)On payment of instalment:
Hire-Vendor A/c Dr. (Amt. of Instalment)
To Cash/Bank A/c.
(v)On charging the Depreciation:
Depreciation A/c Dr. (Amt. of Depreciation)
To Asset A/c.
(vi)On Transfer of interest and depreciation to P/L A/c:
P/L A/c. (Total amt.)
To Interest A/c (Bal. of Intt. A/c.)
To Depreciation A/c. (Bal. of Dep. A/c.)
Under Total Assets Value Method:
Under this method of accounting in the books of hire-purchaser, is done on the assumption that the ownership of the asset is also transferred to the purchaser with the delivery of goods. The following journal entries are recorded under this method.
(i)On taking the delivery of assets at the time of agreement:
Asset A/c Dr. (Cash price of Asset)
To Hire vendor A/c.
(ii)On making the down-payment (if any):
Hire-Vendor....... A/c. Dr. (Amount of down payment)
To Cash/Bank A/c
(iii)On becoming the instalment due:
Interest A/c. Dr. (Amount of interest)
To Hire-Vendor A/c
(iv)On payment of instalment:
Hire-Vendor a/c Dr. (Amount of instalment)
To Cash/Bank A/c
(v)On charging the depreciations:
Depreciation A/c. Dr. (Amount of depreciation)
To Asset A/c.
(vi)On Transfer of interest and depreciation to P/L A/c:
P/L A/c. Dr. (Total)
To Interest A/c. (Bal. of Intt. A/c.)
To Depreciation A/c. (Bal. of Dep. A/c.)
Posting in Ledger Accounts: After passing journal entries under any of the methods discussed above, the following ledger accounts are opened in the ledger and the postings are made accordingly.
(i) Asset A/c. (e.g. Trucks A/c, Machinery A/c. etc.)
(ii) Vendor's A/c.
(iii) Interest A/c.
(iv) Depreciation A/c.
Note: Before recording the entries the amounts of interest and depreciation will be calculated in two separate tables showing the calculations of interest and depreciation.
Calculation of Interest
The total payment made under hire-purchase system is more than cash price. In fact, this excess of payment over the cash price is interest. It is very essential to calculate interest because the amount paid for interest is charged to revenue and the asset is capitalized at cash price. Thus normally all instalments will include a part of cash price and a part of interest on the outstanding balance. However the amount paid at the time of agreement (down payment) will not include any interest. The calculation of interest is made under two conditions:
(a) When interest is included in amount of instalment: Where the hire-purchase price i.e. payment made in the form of down payment and all instalments is more than the cash price, it is regarded that the interest is included in instalments. It is explained in the following example.
Worked out Example-1 (Calculation of Interest)
On Ist April,2005 Mr. X purchased from M/s Y & Co. one 'Motor Truck' under hire-purchase system, Rs. 5,000 being paid on delivery and the balance in five annual instalments of Rs. 7,500 each payable on 31st March each year. The cash price of the motor truck is Rs. 37,500 and vendors charge interest at the rate of 5 per cent per annum on yearly balances. Find out the amounts of principal and interest included in each instalment.
(b) When interest is not included in instalments: Where the total amount paid in the form of down payment and all instalments is exactly equal to the cash price, it is regarded that the interest is not included in instalments. It means that interest is payable in addition to the agreed amount of instalment. It is explained in the following example.
Workedout Example-2 (Calculation of Interest): On April 1,2005, A Transport Company purchased a Motor Lorry from Motor Supply Co. Ltd. on hire-purchase basis, the cash price being Rs. 60,000. Rs. 15,000 on signing of the contract and balance in three annual instalments of Rs. 15,000 each on 31st March every year. In addition to it, interest at 5 per cent per annum was also payable to vendors on outstanding balances.
the goods will be delivered to the purchaser at the time of agreement.
the purchaser has a right to use the goods delivered.
the price of the goods will be paid in instalments.
every instalment will be treated to be the hire charges of the goods which is being used by the purchaser.
if all instalments are paid as per the terms of agreement , the title of the goods is transferred by vendor to the purchaser.
if there is a default in the payment of any of the instalments, the vendor will take away the goods from the possession of the purchaser without refunding him any amount received earlier in the form of various instalments.
Before discussing the characteristics of hire-purchase system, we must know what is a hire purchase agreement and what are the contents of a hire-purchase agreement. Hire-purchase agreement means a contract between the hire vendor and the hire purchaser regarding the sale of goods under certain conditions. Usually every hire-purchase agreement shall contain the following terms:
the cash price of the goods, cash price means the price at which goods may be purchased against cash payment.
the hire-purchase price, hire purchase price means the total amount which is payable by the hire-purchaser under the agreement.
the date on which the hire-purchase agreement will commence.
the description of the goods that will be delivered to the hire-purchaser at the commencement of the agreement.
the number of instalments to be paid by the hire-purchaser along with the amount of each instalment and the date of payment of each instalment.
the down payment if any, the down payment means the amount which is required to be paid by hire-purchaser to the hire vendor at the time of commencement of hire-purchase agreement.
the rate interest charged by the hire vendor (optional).
Characteristics of Hire-Purchase System
The characteristics of hire-purchase system are as under
Hire-purchase is a credit purchase.
The price under hire-purchase system is paid in instalments.
The goods are delivered in the possession of the purchaser at the time of commencement of the agreement.
Hire vendor continues to be the owner of the goods till the payment of last instalment.
The hire-purchaser has a right to use the goods as a bailer.
The hire-purchaser has a right to terminate the agreement at any time in the capacity of a hirer.
The hire-purchaser becomes the owner of the goods after the payment of all instalments as per the agreement.
If there is a default in the payment of any instalment, the hire vendor will take away the goods from the possession of the purchaser without refunding him any amount.
Difference between Hire-purchase system and Instalment payment system
Instalment Payment System is system of purchase and sale of goods in which title of goods is immediately transferred to the purchaser at the time of sale of goods and the sale price of the goods is paid in instalments. In the event of default in payment of any instalment, the seller has no right to take back goods from the possession of the purchaser. He can file a suit for the recovery of the outstanding balance of the price of goods sold. The followings are the differences between Hire-purchase system and Instalment payment system:
In Hire-purchase system, the transfer of ownership takes place after the payment of all instalments while in case of Instalment payment system, the ownership is transferred immediately at the time of agreement.
In Hire-purchase system, the hire-purchase agreement is like a contract of hire though later on it may become a purchase after the payment of last instalment while in Instalment payment system, the agreement is like a contract of credit purchase.
In case of default in payment , in Hire-purchase system the vendor has a right to back goods from the possession of the hire-purchaser while in case of Instalment payment system, the vendor has no right to take back the goods from the possession of the purchaser; he can simply sue for the balance due.
In Hire-purchase system, if the purchaser sells the goods to a third party before the payment of last instalment, the third party does not get a better title on the goods purchased. But in case of Instalment payment system, the third party gets a better title on the goods purchased.
In Hire-purchase system the provisions of the Hire-purchase Act apply to the transaction while in case of Instalment payment system, the provisions of Sale of Goods Act apply to the transaction.
1.6 Accounting In the books of Hire-purchaser
There are two methods of accounting in the books of Hire-purchaser. Their detailed description is as under:-
Asset Accrual Method:
Under this method it is considered that the hire-purchaser is the owner of the asset up to the value of the cash price paid by him in the from of down payment or the cash price paid included in various instalments. The following journal entries are recorded under this method.
(i)On taking the delivery of asset:
No entry is recorded.
(ii)On making the down payment (if any)
Asset A/c Dr. (Amount of down payment)
To Cash/Bank A/c.
(iii)On becoming the instalment due
Asset a/c. Dr (Balancing figure)
Intt. A/c. Dr. (Amt. of Intt.)
To Hire-Vendor A/c. (Amt. of Instalment)
(iv)On payment of instalment:
Hire-Vendor A/c Dr. (Amt. of Instalment)
To Cash/Bank A/c.
(v)On charging the Depreciation:
Depreciation A/c Dr. (Amt. of Depreciation)
To Asset A/c.
(vi)On Transfer of interest and depreciation to P/L A/c:
P/L A/c. (Total amt.)
To Interest A/c (Bal. of Intt. A/c.)
To Depreciation A/c. (Bal. of Dep. A/c.)
Under Total Assets Value Method:
Under this method of accounting in the books of hire-purchaser, is done on the assumption that the ownership of the asset is also transferred to the purchaser with the delivery of goods. The following journal entries are recorded under this method.
(i)On taking the delivery of assets at the time of agreement:
Asset A/c Dr. (Cash price of Asset)
To Hire vendor A/c.
(ii)On making the down-payment (if any):
Hire-Vendor....... A/c. Dr. (Amount of down payment)
To Cash/Bank A/c
(iii)On becoming the instalment due:
Interest A/c. Dr. (Amount of interest)
To Hire-Vendor A/c
(iv)On payment of instalment:
Hire-Vendor a/c Dr. (Amount of instalment)
To Cash/Bank A/c
(v)On charging the depreciations:
Depreciation A/c. Dr. (Amount of depreciation)
To Asset A/c.
(vi)On Transfer of interest and depreciation to P/L A/c:
P/L A/c. Dr. (Total)
To Interest A/c. (Bal. of Intt. A/c.)
To Depreciation A/c. (Bal. of Dep. A/c.)
Posting in Ledger Accounts: After passing journal entries under any of the methods discussed above, the following ledger accounts are opened in the ledger and the postings are made accordingly.
(i) Asset A/c. (e.g. Trucks A/c, Machinery A/c. etc.)
(ii) Vendor's A/c.
(iii) Interest A/c.
(iv) Depreciation A/c.
Note: Before recording the entries the amounts of interest and depreciation will be calculated in two separate tables showing the calculations of interest and depreciation.
Calculation of Interest
The total payment made under hire-purchase system is more than cash price. In fact, this excess of payment over the cash price is interest. It is very essential to calculate interest because the amount paid for interest is charged to revenue and the asset is capitalized at cash price. Thus normally all instalments will include a part of cash price and a part of interest on the outstanding balance. However the amount paid at the time of agreement (down payment) will not include any interest. The calculation of interest is made under two conditions:
(a) When interest is included in amount of instalment: Where the hire-purchase price i.e. payment made in the form of down payment and all instalments is more than the cash price, it is regarded that the interest is included in instalments. It is explained in the following example.
Worked out Example-1 (Calculation of Interest)
On Ist April,2005 Mr. X purchased from M/s Y & Co. one 'Motor Truck' under hire-purchase system, Rs. 5,000 being paid on delivery and the balance in five annual instalments of Rs. 7,500 each payable on 31st March each year. The cash price of the motor truck is Rs. 37,500 and vendors charge interest at the rate of 5 per cent per annum on yearly balances. Find out the amounts of principal and interest included in each instalment.
(b) When interest is not included in instalments: Where the total amount paid in the form of down payment and all instalments is exactly equal to the cash price, it is regarded that the interest is not included in instalments. It means that interest is payable in addition to the agreed amount of instalment. It is explained in the following example.
Workedout Example-2 (Calculation of Interest): On April 1,2005, A Transport Company purchased a Motor Lorry from Motor Supply Co. Ltd. on hire-purchase basis, the cash price being Rs. 60,000. Rs. 15,000 on signing of the contract and balance in three annual instalments of Rs. 15,000 each on 31st March every year. In addition to it, interest at 5 per cent per annum was also payable to vendors on outstanding balances.
Friday, 30 December 2011
Solved Example on Probable Error
Example: If the value of coefficient of correlation between two series is +.9 and its probable error is .0128, what would be the value of n?
Solution: P.E. (r) = .6745 (1-r2)/√n from the given data
0.0128 = .6745 (1-(.9)2)/√n
0.0128 = (.6745×(1 - .81))/√n=(.128155)/√3
.0128 = (.128155)/√n
.0128 = .128155 or √n = (.128155)/0.128 = 10
n = (10)2 = 100
Solution: P.E. (r) = .6745 (1-r2)/√n from the given data
0.0128 = .6745 (1-(.9)2)/√n
0.0128 = (.6745×(1 - .81))/√n=(.128155)/√3
.0128 = (.128155)/√n
.0128 = .128155 or √n = (.128155)/0.128 = 10
n = (10)2 = 100
Mathematical Properties of Coefficient of Correlation
The following are the important mathematical properties of the Coefficient of Correlation or r. .
The Coefficient of Correlation lies between —1 and +1. It cannot exceed unity.
Symbolically —1 ≤ r ≤ +1
Proof of the property is given below:
Let x and y denote the deviation of x and y series from their actual arithmetic average and ax and ay be their standard deviations respectively. Then,
The Coefficient Of Correlation.JPG
But (∑x2)/σ_(x2 ) = n because σ_(x2 ) = (∑x2)/n ∴ (∑x2)/σ_(x2 ) = (∑x2)/(∑x2 ) × n = n
Similarly (∑x2)/σ_(x2 ) = n and
(2∑xy)/σ_(x σ_y ) = 2nr because r = (2∑xy)/σ_(x σ_y )
As such ∑ ((∑x)/σ_x +(∑y)/σ_y )2 = n + n + 2nr = 2n + 2nr = 2n (1 + r)
But ∑ (x/σ_x +y/σ_y )2 is the sum of square of real quantities and as such
cannot be negative. At best it can be 0.
Now 2n (1+r) ≥ 0
Therefore r cannot be less than —1 or —1≤ r.similarly by expanding
∑ (x/σ_x +y/σ_y )2 it can be proved that this value would be 2n(1—r) and hence, r cannot be greater than + 1 or r≤ +1.
Hence—1 ≤ r ≤ - 1.
Sometimes it appears that the values of the various variables so obtained are inter-related. It is likely that such relationship may be obtained in two series relating to the heights and weights of a group of persons. It may be observed that weights increase with increase in heights- so that tall people are heavier than short sized people. Similarly, if the data are collected about the prices of a commodity and the quantities sold at different prices, two series would be obtained. One variable would be the various prices of the commodity and the other variable would be the quantities sold at these prices. In two such series we are again likely to find some relationship. With increase in the price of the commodity the quantity sold is bound to decrease. We can thus conclude that there is some relationship between price and demand. Such relationships can be found in many types of series, for example, prices and supply, heights and weights of persons, prices of sugar and sugarcane, ages of husbands and wives, etc.
The Coefficient of Correlation lies between —1 and +1. It cannot exceed unity.
Symbolically —1 ≤ r ≤ +1
Proof of the property is given below:
Let x and y denote the deviation of x and y series from their actual arithmetic average and ax and ay be their standard deviations respectively. Then,
The Coefficient Of Correlation.JPG
But (∑x2)/σ_(x2 ) = n because σ_(x2 ) = (∑x2)/n ∴ (∑x2)/σ_(x2 ) = (∑x2)/(∑x2 ) × n = n
Similarly (∑x2)/σ_(x2 ) = n and
(2∑xy)/σ_(x σ_y ) = 2nr because r = (2∑xy)/σ_(x σ_y )
As such ∑ ((∑x)/σ_x +(∑y)/σ_y )2 = n + n + 2nr = 2n + 2nr = 2n (1 + r)
But ∑ (x/σ_x +y/σ_y )2 is the sum of square of real quantities and as such
cannot be negative. At best it can be 0.
Now 2n (1+r) ≥ 0
Therefore r cannot be less than —1 or —1≤ r.similarly by expanding
∑ (x/σ_x +y/σ_y )2 it can be proved that this value would be 2n(1—r) and hence, r cannot be greater than + 1 or r≤ +1.
Hence—1 ≤ r ≤ - 1.
Sometimes it appears that the values of the various variables so obtained are inter-related. It is likely that such relationship may be obtained in two series relating to the heights and weights of a group of persons. It may be observed that weights increase with increase in heights- so that tall people are heavier than short sized people. Similarly, if the data are collected about the prices of a commodity and the quantities sold at different prices, two series would be obtained. One variable would be the various prices of the commodity and the other variable would be the quantities sold at these prices. In two such series we are again likely to find some relationship. With increase in the price of the commodity the quantity sold is bound to decrease. We can thus conclude that there is some relationship between price and demand. Such relationships can be found in many types of series, for example, prices and supply, heights and weights of persons, prices of sugar and sugarcane, ages of husbands and wives, etc.
correlation
A computer while calculating correlation coefficient between two variables X and 7 from 25 pairs of observations obtained the following results:
N=25, ∑X=125, ∑X2 = 650,
∑Y=100, 272=460, ∑XY=508.
It was, however, discovered at the time of checking that two pairs of observation were not correctly copied. They were taken as (6, 14) and (8, 6) while the correct values were (8, 12) and (6. 8) prove that the correct value of the correlation coefficient should be 2/3. (I.C.W.A., Final, 1977)
Solution: Corrected ∑X = 125-6-8+8+6= 125
Corrected ∑Y = 100—14—6+12+8 = 100
Corrected ∑X2= 650—62— 82+82+ 62 = 950
Corrected ∑Y2 = 460—142—62+122+82 = 436
Corrected ∑XY= 508—(6 x 14)—(8 x 6) + (8 x 12) +(6 x 8)=520
Now the Corrected value of the Coefficient of Correlation or
Corrected r = (N∑XY-(∑X)(∑Y))/(√(N∑X2- (∑X)2 ) √(N∑X2- (∑X)2 ))
= ((25×520)- (125×100))/(√(25×650-(125)2 ) √(25×436-(100)2 ))
= 500/√(625×900) = 500/(25×30) = 500/750 = 2/3
N=25, ∑X=125, ∑X2 = 650,
∑Y=100, 272=460, ∑XY=508.
It was, however, discovered at the time of checking that two pairs of observation were not correctly copied. They were taken as (6, 14) and (8, 6) while the correct values were (8, 12) and (6. 8) prove that the correct value of the correlation coefficient should be 2/3. (I.C.W.A., Final, 1977)
Solution: Corrected ∑X = 125-6-8+8+6= 125
Corrected ∑Y = 100—14—6+12+8 = 100
Corrected ∑X2= 650—62— 82+82+ 62 = 950
Corrected ∑Y2 = 460—142—62+122+82 = 436
Corrected ∑XY= 508—(6 x 14)—(8 x 6) + (8 x 12) +(6 x 8)=520
Now the Corrected value of the Coefficient of Correlation or
Corrected r = (N∑XY-(∑X)(∑Y))/(√(N∑X2- (∑X)2 ) √(N∑X2- (∑X)2 ))
= ((25×520)- (125×100))/(√(25×650-(125)2 ) √(25×436-(100)2 ))
= 500/√(625×900) = 500/(25×30) = 500/750 = 2/3
Calculation of Pearson's Coefficient of Correlation
Direct Method No. 1
The steps involved are as follows:
(1) Find the means of the two series (X and y1)
(2) Find the deviations of each item of a series from its mean (x and y). Here x = (X—X) and = (Y- Y)
(3) Square these deviations and total them (∑x2 and ∑y2).
(4) Multiply the respective deviations of the two series and total them (∑xy)
(5) Substitute the above values in the following formula:
r = (∑xy)/(√(n&(∑x2)/n )× √((∑y2)/n))=(∑xy)/(nσ1 σ2 )
r = (∑xy)/√(∑x2×∑y2 )
Solved Example :Calculate the coefficient of correlation from the following data by the Spearman's Rank Differences method:
Prices of
Prices of
Prices of
Prices of
Tea ($)
Coffee ($)
Tea ($)
Coffee ($)
75
120
60
110
88
134
80
140
95
150
81
142
70
115
50
100
Solution: Calculation of Coefficient of Rank Correlation
Prices of Tea
(X)
R1
Prices of
Coffee
(Y)
R2
R1 R2
(d)
d2
75
4
120
4
0
0
83
7
134
5
+2
4
95
8
150
8
0
0
70
3
115
3
0
0
60
2
110
2
0
0
80
5
140
6
—1
1
81
6
142
7
—1
1
50
1
100
1
0
0
n = 8
n = 8
0
∑d2 = 6
The steps involved are as follows:
(1) Find the means of the two series (X and y1)
(2) Find the deviations of each item of a series from its mean (x and y). Here x = (X—X) and = (Y- Y)
(3) Square these deviations and total them (∑x2 and ∑y2).
(4) Multiply the respective deviations of the two series and total them (∑xy)
(5) Substitute the above values in the following formula:
r = (∑xy)/(√(n&(∑x2)/n )× √((∑y2)/n))=(∑xy)/(nσ1 σ2 )
r = (∑xy)/√(∑x2×∑y2 )
Solved Example :Calculate the coefficient of correlation from the following data by the Spearman's Rank Differences method:
Prices of
Prices of
Prices of
Prices of
Tea ($)
Coffee ($)
Tea ($)
Coffee ($)
75
120
60
110
88
134
80
140
95
150
81
142
70
115
50
100
Solution: Calculation of Coefficient of Rank Correlation
Prices of Tea
(X)
R1
Prices of
Coffee
(Y)
R2
R1 R2
(d)
d2
75
4
120
4
0
0
83
7
134
5
+2
4
95
8
150
8
0
0
70
3
115
3
0
0
60
2
110
2
0
0
80
5
140
6
—1
1
81
6
142
7
—1
1
50
1
100
1
0
0
n = 8
n = 8
0
∑d2 = 6
Calculation Of Coefficient
Karl Pearson, the great biologist and statistician, has given a formula for the calculation of coefficient of correlation. According to it the coefficient of correlation of two variables is obtained by dividing the sum of the products of the corresponding deviations of the various items of the two series from their respective means by the product of their standard deviations and the number of pairs of observations.
Thus, if x1, x2, x3 ...... xn are the deviations of various items of the first variable from mean value and y1, y2, y3 …… yn are the corresponding deviations of the second variable from its mean value, the sum of the products of these corresponding deviations would be ∑xy. If further, the standard deviations of the two variables are respectively and if n is the number of pairs of observations, Karl Pearson's coefficient of correlation represented by r would be
It is clear from the above formula that if ∑xy is positive, the coefficient of correlation would also be a positive figure indicating positive correlation between the two series. If, on the other hand, ∑xy is negative, coefficient of correlation would also be negative, indicating that the correlation between the two series is negative, ∑xy would be positive, if generally, positive and negative deviations in one series are associated with positive and negative deviations in the other series also. The value of ∑xy would be negative, if generally, the positive deviations of one variable are associated with the negative deviations in the other variable and vice versa. If positive and negative deviations of one variable are indifferently associated with the deviations of the other variable the value of ∑xy would be 0 or near it, indicating absence of correlation between the two series. The value of this coefficient of correlation is always between + 1 and —1. It cannot exceed unity.
The above formula of Karl Pearson is based on the study of co-variance between two series. The co-variance between two series is written as follows: Co-variance = (∑xy)/n
Where x and y stand for the deviations of the two series from their respective means.
To study correlation, the co-variance of the two series is divided by (he product of their standard deviations. Thus, covariance of the two series
r = (covariance of the two series)/√((variance of series 1)(variance of series 2))
= Coveriance/(σ1× σ2 ) = (∑xy)/(nσ1 σ2 )
This formula is known as the Product Moment Formula of Coefficient of Correlation.
Thus, if x1, x2, x3 ...... xn are the deviations of various items of the first variable from mean value and y1, y2, y3 …… yn are the corresponding deviations of the second variable from its mean value, the sum of the products of these corresponding deviations would be ∑xy. If further, the standard deviations of the two variables are respectively and if n is the number of pairs of observations, Karl Pearson's coefficient of correlation represented by r would be
It is clear from the above formula that if ∑xy is positive, the coefficient of correlation would also be a positive figure indicating positive correlation between the two series. If, on the other hand, ∑xy is negative, coefficient of correlation would also be negative, indicating that the correlation between the two series is negative, ∑xy would be positive, if generally, positive and negative deviations in one series are associated with positive and negative deviations in the other series also. The value of ∑xy would be negative, if generally, the positive deviations of one variable are associated with the negative deviations in the other variable and vice versa. If positive and negative deviations of one variable are indifferently associated with the deviations of the other variable the value of ∑xy would be 0 or near it, indicating absence of correlation between the two series. The value of this coefficient of correlation is always between + 1 and —1. It cannot exceed unity.
The above formula of Karl Pearson is based on the study of co-variance between two series. The co-variance between two series is written as follows: Co-variance = (∑xy)/n
Where x and y stand for the deviations of the two series from their respective means.
To study correlation, the co-variance of the two series is divided by (he product of their standard deviations. Thus, covariance of the two series
r = (covariance of the two series)/√((variance of series 1)(variance of series 2))
= Coveriance/(σ1× σ2 ) = (∑xy)/(nσ1 σ2 )
This formula is known as the Product Moment Formula of Coefficient of Correlation.
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