Solve the problem.
-Skewness can be measured by Pearson's index of skewness: If $\mathrm { I } \geq 1.00$ or $\mathrm { I } \leq - 1.00$, the data can be considered significantly skewed. Find Pearson's index of skewness for the test scores below. $\begin{array} { r r r r r r } 68 & 80 & 37 & 94 & 72 & 42 \ 75 & 89 & 84 & 73 & 89 & \end{array}$
A) $0.32$
B) $- 0.34$
C) $- 0.32$
D) $- 0.27$

Question

Solve the problem.
-Skewness can be measured by Pearson's index of skewness:  If $\mathrm { I } \geq 1.00$ or $\mathrm { I } \leq - 1.00$, the data can be considered significantly skewed. Find Pearson's index of skewness for the test scores below. $\begin{array} { r r r r r r } 68 & 80 & 37 & 94 & 72 & 42 \ 75 & 89 & 84 & 73 & 89 & \end{array}$
A) $0.32$ 
B) $- 0.34$ 
C) $- 0.32$ 
D) $- 0.27$

Quizplus · Accepted Answer

First, we need to find the mean, median, and standard deviation of the data set.

Mean: 
We can use a calculator or add up all the values and divide by the number of values:
$$\frac{68 + 80 + 37 + 94 + 72 + 42 + 75 + 89 + 84 + 73 + 89}{11} = 73.3636 \approx 73.4$$
Median: 
We need to put the data set in order from least to greatest:
$$37, 42, 68, 72, 73, 75, 80, 84, 89, 89, 94$$
The median is the middle value (or the mean of the middle two values, if there are an even number of values):
$$\text{median} = 75$$
Standard Deviation: 
We can use a calculator or these steps to find the standard deviation:
1. Find the difference between each score and the mean.
2. Square each difference.
3. Add up all the squared differences.
4. Divide the sum by the number of scores minus 1.
5. Take the square root of the result.

Here are the calculations:

$$\begin{array} { c c c c c c c c c c c } x & 68 & 80 & 37 & 94 & 72 & 42 & 75 & 89 & 84 & 73 & 89 \ x - \bar{x} & -5.4 & 6.6 & -36.4 & 20.6 & -1.4 & -31.4 & 1.6 & 15.6 & 10.6 & -0.4 & 15.6 \ (x - \bar{x}) ^ { 2 } & 29.16 & 43.56 & 1324.96 & 424.36 & 1.96 & 985.96 & 2.56 & 243.36 & 112.36 & 0.16 & 243.36 \ \end{array}$$

$$\sum (x - \bar{x}) ^ { 2 } = 3402.16$$

$$s = \sqrt {\frac { \sum ( x - \bar { x } ) ^ { 2 } } { n - 1 } } = \sqrt {\frac { 3402.16 } { 10 } } \approx 23.12$$

Now we can use Pearson's index of skewness formula:

$$\mathrm { I } = \frac { 3 ( \bar { x } - \text{median} ) } { s }$$

$$\mathrm { I } = \frac { 3 ( 73.4 - 75 ) } { 23.12 } \approx -0.32$$

Since $\mathrm { I } \leq - 1.00$, the data can be considered significantly skewed, and the best choice is C, $-0.32$ as it falls within the acceptable range for significant skewness.

Solve the Problem I≥1.00\mathrm { I } \geq 1.00I≥1.00 Or I≤−1.00\mathrm { I } \leq - 1.00I≤−1.00

Solve the Problem $\mathrm { I } \geq 1.00$ Or $\mathrm { I } \leq - 1.00$