1. (1). A correlation measures the relationship (direction, form and strength or consistency)
between two variables.
(2). Pearson correlation r measures the degree of linear relationship.
(3). Coefficient of determination 𝒓𝟐 evaluates the strength of a relationship.
(4). Spearman correlation 𝒓𝒔 measures the consistency of direction in the relationship between
x and y – that is, the degree to which the relationship is one-directional, or monotonic.
2. (1). Special correlations are used to measure relationships between variables with specific
characteristics.
(2). Spearman correlation is used when both variables are ordinal scales.
(3). The point-biserial correlation is used to measure the strength of the relationship when one
of the two variables is dichotomous, (0, 1).
(4). The phi-coefficient is used when both variables are dichotomous.
3. Pearson correlation (For numerator, i.e., interval or ratio scale)
Step 1: ∑ 𝑥, ∑ 𝑦, ∑ 𝑥 2 , ∑ 𝑦 2 , ∑ 𝑥𝑦
Step 2: 𝑆𝑆𝑥 = ∑ 𝑥 2 −
(∑ 𝑥)2
𝑛
, 𝑆𝑆𝑦 = ∑ 𝑦 2 −
(∑ 𝑦)2
𝑛
Step 3: The sum of product of deviations: 𝑆𝑃 = ∑ 𝑥𝑦 −
Step 4: Pearson correlation: 𝑟 =
∑𝑥∑𝑦
𝑛
𝑆𝑃
√(𝑆𝑆𝑥 )(𝑆𝑆𝑦 )
4. Hypothesis Tests with the Pearson Correlation
Step 1: H0: 𝜌 = 0 (𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛. )
H1: 𝜌 ≠ 0 (𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛. )
Or: When there is a specific prediction about the direction of the correlation.
H0: 𝜌 ≤ 0 (𝑇ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑛𝑜𝑡 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒. )
H1: 𝜌 > 0 (𝑇ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒. )
Step 2: Pearson correlation
Step 3: 𝑟𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 by df=n-2 and Table B-6
Step 4: Statement
5. Coefficient of determination 𝑟 2 measures the proportion of variability in one variable that can
be determined from the relationship with the other variable.
6. Spearman Correlation (Both variables are measured on an ordinal scale of
measurement.)
Step 1: All values are computed with the ranks, not the original scores.
Original Data
x
y
Ranks
x
y
xy
∑ 𝑥𝑦
Compute ∑ 𝑥 , ∑ 𝑦 , ∑ 𝑥 2 , ∑ 𝑦 2 , ∑ 𝑥𝑦
Step 2: 𝑆𝑆𝑥 , 𝑆𝑆𝑦
Step 3: 𝑆𝑃
Step 4: 𝑟𝑠 =
𝑆𝑃
√(𝑆𝑆𝑥 )(𝑆𝑆𝑦 )
Step 5: Statement: The Spearman correlation indicates that the data show a strong/weak
negative/positive trend.
7. Special formula for the Spearman correlation
Note: It only can be used after the scores have been converted to ranks and only when there are
no ties among the ranks. If there are relatively few tied ranks, the formula still may be used, but
it loses accuracy as the number of ties increases.
Step 1: H0: 𝜌𝑠 = 0 (𝑇ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑧𝑒𝑟𝑜. )
H1: 𝜌𝑠 ≠ 0 (𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛. )
Step 2:
Original Data
x
y
Ranks
x
y
Difference
D=x–y
𝐷 2 = (𝑥 − 𝑦)2
∑𝐷 = 0
Step 3. 𝑟𝑠 = 1 −
∑ 𝐷2 =
6 ∑ 𝐷2
𝑛(𝑛2 −1)
n: the number of scores in each column
D: the difference between the x rank and y rank for each individual
Step 4: Statement
8. Point-biseral correlation
Step1: H0: 𝜌 = 0
H1: 𝜌 ≠ 0
Step 2: df= n-1
n is the total sample size
𝑟𝑃𝐵 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙, 𝑇𝑎𝑏𝑙𝑒 𝐵 − 6
Step 3: (1). Convert data to “0” or “1”.
Example:
Training No Training
2
9
4
3
1
0
5
1
8
2
⟹
x
2
4
1
5
8
9
3
0
1
2
y OR
0 1
0 1
0 1
0 1
0 1
1 0
1 0
1 0
1 0
1 0
𝑥2
4
16
1
25
64
81
9
0
1
4
𝑦2
0
0
0
0
0
1
1
1
1
1
∑ 𝑥 = ∑ 𝑦 = ∑ 𝑥2 = ∑ 𝑦2
(2) 𝑆𝑆𝑥 = ∑ 𝑥 2 − (∑ 𝑥)2 /𝑛
(3) 𝑆𝑆𝑦 = ∑ 𝑦 2 − (∑ 𝑦)2 /𝑛
(4) 𝑆𝑃 = ∑ 𝑥𝑦 −
∑𝑥∑𝑦
𝑛
xy
∑ 𝑥𝑦
(5) 𝑟𝑃𝐵 =
𝑆𝑃
√(𝑆𝑆𝑥 )(𝑆𝑆𝑦 )
Step 4: Make a statement.
9. Note:
𝑜𝑟𝑑𝑖𝑛𝑎𝑙
Non-numerical data →
ranks: Spearman
A relationship between two categories
Example:
Pearson
Adopted Children Birth Parents
Numerical data
Two opposite categories
Example:
Male
Point-Biserial
Female