Karl Pearson's coefficient, also known as Pearson correlation coefficient, is a measure of the strength and direction of the linear relationship between two variables. It is defined from ungrouped data, which means that the data is presented in the form of individual observations rather than in pre-defined categories or groups.

Explanation:

- Definition of Karl Pearson's coefficient: The Karl Pearson's coefficient, denoted as r, is a number between -1 and 1 that measures the degree of association between two variables X and Y. It is calculated as the covariance between X and Y divided by the product of their standard deviations.
- Example of ungrouped data: Suppose we have the following data on the height (in inches) and weight (in pounds) of five individuals:

| Height | Weight |
|--------|--------|
| 60 | 120 |
| 65 | 150 |
| 70 | 180 |
| 63 | 135 |
| 68 | 165 |

- Calculating Karl Pearson's coefficient: To calculate the Karl Pearson's coefficient between height and weight, we first need to compute the mean and standard deviation of each variable:

| Variable | Mean | Standard Deviation |
|----------|------|--------------------|
| Height | 65.2 | 3.7 |
| Weight | 150 | 23.4 |

Then, we can compute the covariance between height and weight using the formula:

cov(X,Y) = Σ(Xi - X)(Yi - Y) / (n - 1)

where Xi and Yi are the values of height and weight for each individual, X and Y are their respective means, and n is the sample size. Plugging in the values, we get:

cov(Height, Weight) = (60-65.2)(120-150) + (65-65.2)(150-150) + (70-65.2)(180-150) + (63-65.2)(135-150) + (68-65.2)(165-150) / (5-1)

cov(Height, Weight) = -96.8

Finally, we can compute the Karl Pearson's coefficient as:

r = cov(X,Y) / (σX * σY)

where σX and σY are the standard deviations of X and Y, respectively. Plugging in the values, we get:

r = -96.8 / (3.7 * 23.4)

r = -0.72

- Interpretation of Karl Pearson's coefficient: The negative value of r indicates that there is a negative linear relationship between height and weight, meaning that as height increases, weight tends to decrease. The magnitude of r, which is 0.72 in this case, indicates that the relationship is moderately strong.