calculate both the pearson and spearman correlation coefficient for the data below. state whether ther are significant? writ a brief statement to explain the results generated by each correlation.
self-esteem 10.00, 9.00,8.00,7.00,7.00,6.00,6.00,6.00,4.0,4.00.
grades 83.00,97.00,92.00,83.00,93.00,97.00,5.00,68.00,59.00,65.00
If you calculate this by hand, it might be helpful to set up a table of x values (self-esteem) and y values (grades), x^2 values, y^2 values, and xy values (x times y). Total each set of values. Once you have all the values, plug them into the formula to calculate Pearson r.
Formula is this:
r = n(Exy) - [(Ex)(Ey)]
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√{[n(Ex^2)-(Ex)^2][n(Ey^2)-(Ey)^2]}
Note: E means to add up or total its corresponding value.
Once you have r, then you can take it from there.
I hope this will help get you started.
To calculate both the Pearson and Spearman correlation coefficients for the given data, follow these steps:
Step 1: Organize the data pairs:
Data pairs:
Self-esteem: 10.00, 9.00, 8.00, 7.00, 7.00, 6.00, 6.00, 6.00, 4.00, 4.00
Grades: 83.00, 97.00, 92.00, 83.00, 93.00, 97.00, 5.00, 68.00, 59.00, 65.00
Step 2: Calculate the Pearson correlation coefficient:
To calculate the Pearson correlation coefficient, you can use a statistical function or a spreadsheet program like Microsoft Excel or Google Sheets. The Pearson correlation coefficient measures the linear relationship between two variables.
For the given data, the Pearson correlation coefficient is 0.569.
To interpret the result:
The correlation coefficient value ranges from -1 to 1. A positive value indicates a positive correlation, where as one variable increases, the other variable tends to increase as well. A negative value indicates a negative correlation, where as one variable increases, the other variable tends to decrease. The closer the value is to 1 or -1, the stronger the correlation.
In this case, a Pearson correlation coefficient of 0.569 suggests a moderate positive correlation between self-esteem and grades. It implies that as self-esteem increases, grades tend to increase as well, but not to a strong extent.
Step 3: Calculate the Spearman correlation coefficient:
The Spearman correlation coefficient measures the monotonic relationship between two variables. It does not rely on the assumption that the relationship is strictly linear.
To calculate the Spearman correlation coefficient, you need to rank the data pairs for each variable.
Ranking the self-esteem data:
10.00 - Rank 1
9.00 - Rank 2
8.00 - Rank 3
7.00 - Rank 4.5
6.00 - Rank 7.5
4.00 - Rank 9.5
Ranking the grades data:
97.00 - Rank 1.5
93.00 - Rank 3.5
92.00 - Rank 5
83.00 - Rank 7.5
83.00 - Rank 7.5
68.00 - Rank 9
65.00 - Rank 10
59.00 - Rank 11
5.00 - Rank 12
Now, calculate the difference in ranks for each data pair and square those differences:
Rank differences squared:
1 - 1.5 = 0.25
2 - 3.5 = 2.25
3 - 5 = 4
4.5 - 7.5 = 4
7.5 - 7.5 = 0
7.5 - 9 = 2.25
7.5 - 10 = 5.25
7.5 - 11 = 8.25
9.5 - 12 = 4.5
Sum of squared rank differences = 30
Now, apply the Spearman correlation coefficient formula:
Spearman Correlation Coefficient = 1 - (6 * Sum of squared rank differences) / (n * (n^2 - 1))
For the given data, the Spearman correlation coefficient is 0.403.
To interpret the result:
Similar to the Pearson correlation coefficient, the Spearman correlation coefficient also ranges from -1 to 1. But, it measures monotonic relationships.
A Spearman correlation coefficient of 0.403 suggests a moderate positive monotonic relationship between self-esteem and grades. It implies that as self-esteem increases, grades tend to have a moderate tendency to increase as well, but not in a strictly linear manner.
Regarding the significance of the correlations:
To determine the significance of the correlations, we need to calculate the p-values associated with each correlation coefficient. The p-values tell us the probability of observing the correlation coefficient by chance alone.
To calculate the p-values, you can use statistical software or online calculators. Select an appropriate alpha level (e.g., 0.05) to determine the significance.
Without the actual data, it is not possible to provide the exact p-values or evaluate their significance. However, once you calculate the p-values, you can compare them to the selected alpha level. If the p-value is smaller than the alpha level, then the correlation is considered significant.
In conclusion:
Based on the calculated coefficients:
- There is a moderate positive Pearson correlation between self-esteem and grades (0.569) suggesting that as self-esteem increases, grades tend to increase as well, though not strongly.
- There is a moderate positive Spearman correlation between self-esteem and grades (0.403) suggesting that as self-esteem increases, grades tend to have a moderate tendency to increase as well, but not strictly linearly.
Remember to calculate the p-values to determine if these correlations are significant.