Find Karl Pearson skewness from the following population 7, 5, 9, 7, 8, 6.

Steps to Solve

1. Calculate the Mean

To start, we need to find the mean of the numbers. The mean is calculated by summing all the numbers and dividing by the total count of numbers.

Mean ($\mu$) is given by:

$$ \mu = \frac{\sum_{i=1}^{n} x_i}{n} $$

For our data set: $7, 5, 9, 7, 8, 6$

The sum is:

$$ 7 + 5 + 9 + 7 + 8 + 6 = 42 $$

The count ($n$) is 6, so:

$$ \mu = \frac{42}{6} = 7 $$

2. Calculate the Standard Deviation

Next, we need to find the standard deviation, which measures the dispersion of the data set.

The formula for standard deviation ($\sigma$) is:

$$ \sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}} $$

We calculate each squared deviation:

• $(7 - 7)^2 = 0$
• $(5 - 7)^2 = 4$
• $(9 - 7)^2 = 4$
• $(7 - 7)^2 = 0$
• $(8 - 7)^2 = 1$
• $(6 - 7)^2 = 1$

Sum of squared deviations:

$$ 0 + 4 + 4 + 0 + 1 + 1 = 10 $$

Now, plugging this into the formula:

$$ \sigma = \sqrt{\frac{10}{6}} \approx 1.29 $$

3. Calculate the Skewness

Now, we calculate Karl Pearson's skewness using the formula:

$$ Skewness = \frac{3(\mu - Median)}{\sigma} $$

First, we need to find the median. The median of the sorted data set ($5, 6, 7, 7, 8, 9$) is the average of the two middle numbers (7 and 7), thus:

$$ Median = 7 $$

Now applying the values:

$$ Skewness = \frac{3(7 - 7)}{1.29} = \frac{0}{1.29} = 0 $$

4. Interpret the Result

The skewness value of $0$ indicates that the data set is symmetrical. In other words, there's no skewness toward the left or right.