FISHER

Calculates values for the Fisher transformation.

Syntax:

FISHER(r)

returns the value of the Fisher transformation at r, (-1 < r < 1).This function calculates:

$\frac{1}{2}ln(\frac{1+r}{1-r})$

Example:

FISHER(0)

returns 0.

To compare the two correlation coefficients ($r_A=0.75\text{ and }{r}_B=0.55$), the analyst must first transform them using the FISHER() function.

Step 1: Apply the FISHER() function to each correlation coefficient.

• For Product Line A ($r_A=0.75$):

$z_A^{\prime }=\text{FISHER}(0.75)=\frac{1}{2}\text{ln}(\frac{1+0.75}{1-0.75})=\frac{1}{2}\text{ln}(\frac{1.75}{0.25})=\frac{1}{2}\text{ln}(7)\approx 0.973$

• For Product Line B ($r_B=0.55$):

$z_B^{\prime }=\text{FISHER}(0.55)=\frac{1}{2}\text{ln}(\frac{1+0.55}{1-0.55})=\frac{1}{2}\text{ln}(\frac{1.55}{0.45})=\frac{1}{2}\text{ln}(3.444)\approx 0.618$

The transformed values, $z_A^{\prime }=0.973$ and $z_B^{\prime }=0.618$, are now approximately normally distributed, allowing for a valid statistical comparison.

Step 2: Calculate the test statistic (z-score) to compare the two transformed values.

The formula for the z-statistic to compare two independent correlations is:

$Z=\frac{z_A^{\prime }-z_B^{\prime }}{\sqrt{\frac{1}{n_A-3}+\frac{1}{n_B-3}}}$

Using the values from our table:

$Z=\frac{0.973-0.618}{\sqrt{\frac{1}{25-3}+\frac{1}{30-3}}}=\frac{0.355}{\sqrt{\frac{1}{22}+\frac{1}{27}}}=\frac{0.355}{\sqrt{0.0455+0.0370}}=\frac{0.355}{\sqrt{0.0825}}\approx \frac{0.355}{0.2872}\approx 1.236$

Step 3: Compare the calculated Z value to a critical value.

At a common significance level of α=0.05 for a two-tailed test, the critical z-score is $\pm 1.96$. Our calculated Z value of 1.236 is less than 1.96, so it is not statistically significant.

Conclusion:

The marketing analyst concludes that the observed difference in correlation between social media engagement and sales for Product A and Product B is not statistically significant. Despite the higher correlation for Product A, the data does not provide enough evidence to say with confidence that the relationship is truly stronger. The observed difference could be due to random chance. This highlights why the FISHER() function is an essential tool for drawing accurate conclusions about correlation coefficients.