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1. Karl Pearsons coefficient Correlation
Karl Pearson has given a quantitative method of calculating
correlation.
It is an important and widely used method of studying correlation.
Karl Pearsons coefficient Correlation is generally written as “r”.

2. 𝑟 =
𝛴𝑥𝑦
𝑁𝜎𝑥𝜎𝑦
Formula
𝑟 = 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
𝑥 = x − ҧ
𝑥
𝑦 = Y − ഥ
𝒀
𝜎𝑥 = Standard deviation of X Series
𝜎y = Standard deviation of Y Series
𝑁 = 𝑁𝑢𝑚𝑏𝑒𝑟𝑜𝑓 𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠

3. Modified Formula
𝑟 =
σ 𝑥𝑦
𝛴𝑥2 × 𝛴𝑦2
𝒙 = 𝐱 − ഥ
𝒙
𝒚 = 𝐲 − ഥ
𝒚

4. X 2 3 4 5 6 7 8
Y 4 7 8 9 10 14 18
X 𝒙 𝒙𝟐 Y 𝒚 𝒚𝟐 xy
2 4
3 7
4 8
5 9
6 10
7 14
8 18
𝚺𝐗=
N=
ത
𝐱 =
𝚺𝐱= 𝜮𝒙𝟐 = 𝜮𝒀 =
N=
ഥ
𝒀 =
𝚺𝐲= 𝜮𝒚𝟐 = 𝚺𝐱𝐲=

5. Step Deviation Method Formula

6. Price 5 10 15 20 25
Demand 40 35 30 25 20
X 𝐝𝒙
(𝑋 − 𝐴𝑥)
𝐝𝑥′
(
𝐝𝑥
𝑐1
)
𝐝𝑥′2 Y 𝒅𝒚
(𝑋 − 𝐴𝑥)
𝐝𝒚′
(
𝐝𝒚
𝑐𝟐
)
𝐝𝒚′2
𝐝𝑥′
𝐝𝒚′
5 40
10 35
15 30
20 25
25 20
N=5 𝚺𝐝𝐱’= 𝜮𝒅𝒙′𝟐 = N=5 𝜮𝒅𝒚′𝟐 = 𝜮𝒅𝒚′𝟐 = 𝚺𝐝𝐱′𝐝𝐲′=

7. PROPERTIES OF CORRELATION COEFFICIENT
r has no unit. It is a pure number. It means units of measurement
are not parts of r.
A negative value of r indicates an inverse relation, and if r is
positive, the two variables move in the same direction.
If r = 0, the two variables are uncorrelated. There is no linear
relation between them. However, other types of relation may be
there.
If r 1 or r = -1, the correlation is perfect or proportionate. A high
value of r indicates strong linear relationship, +1 or -1.
The value of the correlation coefficient lies between minus one
and plus one, ., -1 ≤ r ≤ + 1. If the value of r lies outside this range,
it indicates error in calculation.

8. Marks in
Statistics
15 10 20 28 12 10 16 18
18Marks in
Economics
16 14 10 12 11 15 18 12
Calculate coefficient of rank correlation between the marks in Economics and
Statistics, as indicated by 8 answer books of each of the two examiners.

9. X 𝑹𝟏 𝒀 𝑹𝟐 D 𝑫𝟐
15 5 16 2 3 9
10 7.5 14 4 3.5 12.25
20 2 10 8 -6 36
28 1 12 5.5 -4.5 20.25
12 6 11 7 -1 1
10 7.5 15 3 4.5 20.25
16 4 18 1 3 9
18 3 12 5.5 -2.5 6.25
N=8 𝜮𝑫𝟐=114
𝟕 + 𝟖
𝟐
= 𝟕. 𝟓
𝟓 + 𝟔
𝟐
= 𝟓. 𝟓

10. Merits and Demerits of Rank Correlation
Merits
Spearman's rank correlation method is easier than Pearson's method of
correlation.
Rank correlation method is very convenient method when B the series
give only order of preference and not the actual values of the variables.
Rank correlation is a superior method of analysis in case of qualitative
distributions such as those relating to virtue, wisdom or ignorance.
Demerits
Rank correlation method cannot be used in case of group frequency
distributions.
It can handle only a limited number of observations. It is generally not
used when the number of observations exceeds 20.