How to find deviation in Spearman's correlation?

To find the deviation in Spearman's correlation, you can follow these steps:

1. Calculate the rank of each pair of data points in both variables. If you have two variables X and Y, rank the data points in each variable separately from smallest to largest.

2. Calculate the difference between the ranks of each pair of data points in both variables. This will give you the deviation for each pair of data points.

3. Square each deviation to get rid of negative values and to emphasize larger deviations.

4. Sum up all the squared deviations.

5. Use the formula for Spearman's correlation to calculate the correlation coefficient. The formula is:

ρ = 1 - (6 * Σd^2) / (n * (n^2 - 1))

where:
ρ = Spearman's correlation coefficient
Σd^2 = sum of squared deviations
n = number of data points

6. The correlation coefficient will range from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.

Standard Error of the Spearman's Correlation Coefficient (r_s)

The standard error of r_s measures the sampling error associated with the correlation coefficient. It provides an estimate of how much the correlation coefficient would vary if the sampling process were repeated.

Formula:

```
SE(r_s) = sqrt(1 - r_s^2) / sqrt(N - 2)
```

where:

- r_s is the Spearman's correlation coefficient
- N is the sample size

Interpretation:

The standard error indicates the expected range of variability in r_s across different samples of the same size. A smaller standard error implies a more stable estimate of the correlation coefficient.

Confidence Interval:

The standard error can be used to construct a confidence interval for the true correlation coefficient in the population. A 95% confidence interval is given by:

```
r_s +/- 1.96 - SE(r_s)
```

Hypothesis Testing:

The standard error can also be used for hypothesis testing. To test the hypothesis that the true correlation coefficient is zero, the z-score is calculated as:

```
z = r_s / SE(r_s)
```

If the absolute value of z is greater than 1.96, the null hypothesis can be rejected at the 5% significance level.

Additional Notes:

- The standard error of r_s assumes a normal distribution of ranks.
- The assumption of normality can be violated in small samples or with non-continuous data.
- If the assumption is violated, alternative methods for estimating the standard error may be more appropriate, such as bootstrapping or jackknifing.