Introduction:
The Karl Pearson coefficient of skewness is a measure of asymmetry in a distribution. It determines whether the data is skewed to the left or right, or if it is symmetric. It is calculated using the formula:

Pearson's coefficient of skewness = 3 * (Mean - Median) / Standard Deviation

To calculate the Karl Pearson coefficient of skewness for the given income data, we need to follow the steps below:

Step 1: Calculate the Mean:
The mean is the average of the data points. We can calculate it by summing up all the values and dividing it by the total number of observations.

Mean = (15 * 200 + 33 * 300 + 63 * 400 + 83 * 500 + 100 * 600) / (15 + 33 + 63 + 83 + 100)

Step 2: Calculate the Median:
The median is the middle value of the data when arranged in ascending order. In this case, we have 5 different income ranges, so the median will be the value in the middle.

Median = 400

Step 3: Calculate the Standard Deviation:
The standard deviation measures the dispersion or spread of the data points. It can be calculated using the formula:

Standard Deviation = sqrt((sum of (Xi - Mean)^2) / n)

Where Xi represents each data point, Mean is the calculated mean from step 1, and n is the total number of observations.

Step 4: Calculate the Karl Pearson Coefficient of Skewness:
Using the formula mentioned earlier, we can calculate the Karl Pearson coefficient of skewness.

Karl Pearson Coefficient of Skewness = 3 * (Mean - Median) / Standard Deviation

Now, let's calculate the values:

Mean:
Mean = (15 * 200 + 33 * 300 + 63 * 400 + 83 * 500 + 100 * 600) / (15 + 33 + 63 + 83 + 100)

Mean = 446.7

Standard Deviation:
Standard Deviation = sqrt((15 * (200 - 446.7)^2 + 33 * (300 - 446.7)^2 + 63 * (400 - 446.7)^2 + 83 * (500 - 446.7)^2 + 100 * (600 - 446.7)^2) / (15 + 33 + 63 + 83 + 100))

Standard Deviation = 114.24

Karl Pearson Coefficient of Skewness:
Karl Pearson Coefficient of Skewness = 3 * (446.7 - 400) / 114.24

Karl Pearson Coefficient of Skewness = 1.407

Conclusion:
The Karl Pearson coefficient of skewness for the given income data is 1.407. Since the coefficient is positive, it indicates that the data is skewed to the right or positively skewed. This means that the income distribution has a tail towards higher values.