The English statistician Karl Pearson​ (1857-1936) introduced a formula for the skewness of a distribution.

                                               Upper P equals StartFraction 3 left parenthesis x overbar minus median right parenthesis Over s EndFractionP=3x−medians

Most distributions have an index of skewness between 

minus−3 

and 3. When P​ > 0 the data are skewed right. When P​ < 0 the data are skewed left. When P​ = 0 the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each.

Question content area bottom

Part 1

​(a)**** The coefficient of skewness for 

x overbarx 

​= 

1717​, 

s​ = 

2.32.3​, 

median​ = 

1818 

is P​ = 

enter your response here. 

​(Round to the nearest hundredth as​ needed.)

To calculate the coefficient of skewness \( P \), we use the formula introduced by Karl Pearson:

\[
P = \frac{3(\bar{x} - \text{median})}{s}
\]

### Given dataTo calculate the coefficient of skewness \( P \), we use the formula introduced by Karl Pearson:

\[
P = \frac{3(\bar{x} - \text{median})}{s}
\]

### Given 
- Mean \( \bar{x} = 17 \)
- Standard deviation \( s = 2.3 \)
- Median \( \text{median} = 18 \)

### Step-by-step calculation:

1. Subtract the median from the mean:
   \[
   \bar{x} - \text{median} = 17 - 18 = -1
   \]

2. Multiply the result by 3:
   \[
   3(\bar{x} - \text{median}) = 3(-1) = -3
   \]

3. Divide the result by the standard deviation \( s \):
   \[
   P = \frac{-3}{2.3} \approx -1.30
   \]

### Conclusion:
The coefficient of skewness is approximately \( P = -1.30 \).

Since \( P < 0 \), this indicates that the data are **skewed left**.

---

Would you like further clarification on this, or any details on the steps?

Here are 5 follow-up questions to expand on this:
1. What does a skewness value of 0 signify in a data distribution?
2. How does skewness affect the interpretation of a dataset's mean and median?
3. Can skewness alone determine the normality of a dataset? Why or why not?
4. What are some real-life examples of left-skewed distributions?
5. How would increasing the standard deviation \( s \) affect the skewness value?

**Tip**: Skewness helps assess the symmetry of a dataset. A strong positive or negative skew indicates more extreme values on one side of the distribution.

The English statistician Karl Pearson (1857-1936) introduced a formula for the skewness of a distribution.

P = (3(x̄ - median)) / s

Most distributions have an index of skewness between -3 and 3. When P > 0 the data are skewed right. When P < 0 the data are skewed left. When P = 0 the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each.

(a) The coefficient of skewness for x̄ = 17, s = 2.3, median = 18 is P = enter your response here. (Round to the nearest hundredth as needed.)

Learn how to calculate the coefficient of skewness using Karl Pearson's formula with given values for mean, standard deviation, and median. This example includes step-by-step instructions for calculating skewness and understanding the shape of the distribution. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation