CORRELATION COEFFICIENT COMPUTATION

Pearson r is employed when the distribution is bivariate, continuous and normal ( continuous and normal. However the scores of the individuals concerned in each variable are approximately so).The Spearman rho is employed when the distribution is bivariate The contingency coefficient and its associates are employed when the data are frequency ranked in order of magnitude. The resulting ranks are used.

counts of individuals that belong to cross or joint categories of two contingency. Often the

data are presented in contingency tables.

CORRELATION COEFFICIENTS -COMPUTATION. A.

Pearson product moment correlation coefficient (Pearson r). Below are scores in a twenty itemed multiple choice test in each of a unit in Math and Chem. by 12 students (Scores are denoted X and Y respectively).

S/No.

5 11 8 7 14 10

10 16 10 8

15 12 13

Calculate the Pearson r for the above.

x=X-X y=Y-x S

}

deviation from the mean of X and Y scores.

ΣΧ

120

ΣΥ

144

Step 1: Calculate the deviations from the mean for both X and Y:

First, calculate the mean for both X and Y: Mean of X (X̄) = ΣX / n = 120 / 12 = 10 Mean of Y (Ȳ) = ΣY / n = 144 / 12 = 12

Now calculate the deviations from the mean for both X and Y: Deviation of X (dX) = X – X̄ Deviation of Y (dY) = Y – Ȳ

Here are the deviations for each pair (Xi, Yi):

i	X	Y	dX	dY
1	15	18	5	6
2	9	10	-1	-2
3	12	16	2	4
4	13	10	3	-2
5	6	8	-4	-4
6	10	10	0	-2
7	5	6	-5	-6
8	11	12	1	0
9	8	14	-2	2
10	7	15	-3	3
11	14	12	4	0
12	10	13	0	1

Step 2: Calculate the Pearson correlation coefficient (Pearson r):

Pearson r = Σ(dX * dY) / √(Σ(dX^2) * Σ(dY^2))

Calculate Σ(dX * dY): Σ(dX * dY) = (5 * 6) + (-1 * -2) + (2 * 4) + (3 * -2) + (-4 * -4) + (0 * -2) + (-5 * -6) + (1 * 0) + (-2 * 2) + (-3 * 3) + (4 * 0) + (0 * 1) = 152

Calculate Σ(dX^2) and Σ(dY^2): Σ(dX^2) = 5^2 + (-1)^2 + 2^2 + 3^2 + (-4)^2 + 0^2 + (-5)^2 + 1^2 + (-2)^2 + (-3)^2 + 4^2 + 0^2 = 77 Σ(dY^2) = 6^2 + (-2)^2 + 4^2 + (-2)^2 + (-4)^2 + (-2)^2 + (-6)^2 + 0^2 + 2^2 + 3^2 + 0^2 + 1^2 = 98

Now calculate Pearson r: Pearson r = Σ(dX * dY) / √(Σ(dX^2) * Σ(dY^2)) Pearson r = 152 / √(77 * 98) Pearson r ≈ 0.439

So, the Pearson correlation coefficient for the given data is approximately 0.439. This indicates a positive correlation between the Math and Chemistry scores of the students.

Summary: The passage introduces correlation analysis, a method used to measure the strength and direction of association between two variables. Correlation coefficients can range from +1 to -1, with higher values indicating a stronger relationship. The direction of the relationship is indicated by the sign of the coefficient: a positive sign indicates a positive relationship, while a negative sign indicates a negative relationship. Four common types of correlations are mentioned: Pearson correlation, Kendall rank correlation, Spearman correlation, and Point-Biserial correlation.

Explanation:

Correlation Purpose: Correlation analysis is used to determine how closely two variables are related to each other. It quantifies the strength and direction of the linear relationship between variables.
Correlation Coefficient: The passage mentions that the correlation coefficient can range between +1 and -1. A coefficient of +1 signifies a perfect positive correlation, meaning that as one variable increases, the other also increases proportionally. Conversely, a coefficient of -1 represents a perfect negative correlation, where one variable increases as the other decreases.
Strength of Relationship: The closer the correlation coefficient is to +1 or -1, the stronger the relationship between the variables. As the coefficient approaches 0, the relationship weakens.
Types of Correlations:
- Pearson Correlation: This is the most widely used correlation statistic and is suited for linear relationships. For instance, it’s applied to measure the association between two stocks in the stock market. It involves calculating the Pearson r correlation coefficient.
- Kendall Rank Correlation: Used for measuring associations in ranked data, it’s less affected by outliers.
- Spearman Correlation: Also used with ranked data, it’s sensitive to non-linear relationships.
- Point-Biserial Correlation: Similar to Pearson correlation but used when one variable is dichotomous (binary).
Pearson Correlation Formula: The formula to calculate Pearson r correlation is provided in the passage:
- rxy: Pearson r correlation coefficient between x and y
- n: Number of observations
- xi: Value of x for the ith observation
- yi: Value of y for the ith observation
Applications: Pearson correlation can address research questions involving the relationship between variables, such as age and height, temperature and ice cream sales, and job satisfaction and income.

In summary, correlation analysis, particularly Pearson correlation, provides valuable insights into how variables are related. It’s widely used in various fields to understand the strength and direction of associations

Evaluation

1. Correlation analysis measures the ____________ between two variables.
a) Causation
b) Association
c) Dissimilarity

2. A correlation coefficient ranges between ____________.
a) 0 and 2
b) -1 and 1
c) 0 and 1

3. A correlation coefficient of +0.8 indicates a ____________ relationship between variables.
a) Weak positive
b) Strong positive
c) Weak negative

4. When the correlation coefficient approaches 0, the relationship between variables becomes ____________.
a) Stronger
b) Weaker
c) Non-linear

5. Pearson correlation is suitable for ____________ relationships.
a) Non-linear
b) Linear
c) Categorical

6. Kendall rank correlation is less influenced by ____________.
a) Linear relationships
b) Outliers
c) Positive relationships

7. In the Pearson correlation formula, “xi” represents the value of ____________ for the ith observation.
a) y
b) x
c) n

8. The Pearson correlation formula involves dividing the sum of cross-product deviations by the ____________ of the sum of squared deviations.
a) Product
b) Difference
c) Square root

9. A researcher investigates the relationship between hours of study and exam scores using Pearson correlation. This question examines the relationship’s ____________.
a) Direction
b) Strength
c) Linearity

10. Pearson correlation is applied to determine the connection between ____________ and ____________.
a) Nominal data, ordinal data
b) Ranked data, categorical data
c) Linearly related variables

11. A positive sign of the correlation coefficient suggests a ____________ relationship between variables.
a) Negative
b) Positive
c) No

12. If the correlation coefficient is -0.5, the relationship is ____________ and ____________.
a) Positive, strong
b) Negative, strong
c) Non-linear, weak

13. Outliers can have a ____________ impact on correlation coefficients.
a) Significant
b) Minimal
c) Linear

14. Point-Biserial correlation is used when one variable is ____________.
a) Continuous
b) Dichotomous
c) Ordinal

15. The formula for Point-Biserial correlation is similar to the Pearson correlation formula, but with a ____________ variable.
a) Linear
b) Binary
c) Non-linear

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MEASURES OF VARIABILITY OR DISPERSION, STANDARD SCORES (Z-SCORES AND T-SCORES) AND THE NORMAL CURVE

MEASURES OF CENTRAL TENDENCY AND LOCATION: MEAN, MODE, MEDIAN AND GRAPHICAL LOCATION OF MODE, MEDIAN, QUARTILES, DECILES AND PERCENTILES