Find Karl Pearson's Coefficient Of Skewness From The Given Data.a. ${ \begin{tabular}{|l|c|c|c|c|c|} \hline Wages & 100 & 110 & 120 & 130 & 140 \ \hline No. Of Workers & 5 & 6 & 8 & 6 & 4 \ \hline \end{tabular} }$b.

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Introduction

In statistics, the coefficient of skewness is a measure used to describe the asymmetry of a distribution. It helps us understand the shape of the distribution and whether it is symmetrical or skewed. Karl Pearson's coefficient of skewness is one of the methods used to calculate the skewness of a distribution. In this article, we will learn how to find Karl Pearson's coefficient of skewness from the given data.

What is Karl Pearson's Coefficient of Skewness?

Karl Pearson's coefficient of skewness is a measure of the asymmetry of a distribution. It is calculated using the following formula:

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

where:

  • Mean is the average value of the data
  • Median is the middle value of the data
  • Standard Deviation is a measure of the spread of the data

Step 1: Calculate the Mean

To calculate the mean, we need to add up all the values and divide by the number of values.

Wages No. of workers
100 5
110 6
120 8
130 6
140 4

Mean = (100 * 5 + 110 * 6 + 120 * 8 + 130 * 6 + 140 * 4) / (5 + 6 + 8 + 6 + 4) Mean = (500 + 660 + 960 + 780 + 560) / 29 Mean = 3560 / 29 Mean = 122.76

Step 2: Calculate the Median

To calculate the median, we need to arrange the data in order and find the middle value.

Wages No. of workers
100 5
110 6
120 8
130 6
140 4

The median is the middle value, which is 120.

Step 3: Calculate the Standard Deviation

To calculate the standard deviation, we need to calculate the variance first.

Variance = Σ (xi - μ)² / (n - 1)

where:

  • xi is each value in the data
  • μ is the mean
  • n is the number of values

Variance = ((100 - 122.76)² * 5 + (110 - 122.76)² * 6 + (120 - 122.76)² * 8 + (130 - 122.76)² * 6 + (140 - 122.76)² * 4) / (29 - 1) Variance = ((22.76)² * 5 + (12.76)² * 6 + (2.24)² * 8 + (7.24)² * 6 + (17.24)² * 4) / 28 Variance = (519.35 + 161.33 + 10.01 + 52.49 + 596.49) / 28 Variance = 1239.67 / 28 Variance = 44.25

Standard Deviation = √Variance Standard Deviation = √44.25 Standard Deviation = 6.65

Step 4: Calculate Karl Pearson's Coefficient of Skewness

Now that we have the mean, median, and standard deviation, we can calculate Karl Pearson's coefficient of skewness.

Karl Pearson's Coefficient of Skewness = (3 * (Mean - Median)) / Standard Deviation Karl Pearson's Coefficient of Skewness = (3 * (122.76 - 120)) / 6.65 Karl Pearson's Coefficient of Skewness = (3 * 2.76) / 6.65 Karl Pearson's Coefficient of Skewness = 8.28 / 6.65 Karl Pearson's Coefficient of Skewness = 1.24

Conclusion

In this article, we learned how to find Karl Pearson's coefficient of skewness from the given data. We calculated the mean, median, and standard deviation, and then used these values to calculate the coefficient of skewness. The coefficient of skewness is a measure of the asymmetry of a distribution, and it can be used to describe the shape of the distribution. We hope this article has been helpful in understanding how to calculate Karl Pearson's coefficient of skewness.

Discussion

The coefficient of skewness is an important measure in statistics, as it helps us understand the shape of a distribution. A symmetrical distribution has a coefficient of skewness of 0, while a skewed distribution has a non-zero coefficient of skewness. The coefficient of skewness can be used to identify outliers in a distribution, and it can also be used to compare the shape of different distributions.

Limitations

While Karl Pearson's coefficient of skewness is a useful measure, it has some limitations. It is sensitive to outliers, and it can be affected by the presence of outliers in the data. Additionally, it is not a robust measure, and it can be affected by small changes in the data.

Future Work

In future work, we can explore other methods of calculating the coefficient of skewness, such as the moment coefficient of skewness and the interquartile range coefficient of skewness. We can also explore the use of the coefficient of skewness in different fields, such as finance and economics.

References

  • Pearson, K. (1895). "Note on regression and inheritance in the case of two parents." Proceedings of the Royal Society of London, 58, 240-242.
  • Pearson, K. (1905). "Mathematical contributions to the theory of evolution. X. Supplement to a memoir on skew variation." Philosophical Transactions of the Royal Society of London, 200, 253-275.
  • Fisher, R. A. (1925). "Applications of 'Student's' distribution." Metron, 5(2), 90-104.
    Frequently Asked Questions (FAQs) about Karl Pearson's Coefficient of Skewness ====================================================================================

Q: What is Karl Pearson's coefficient of skewness?

A: Karl Pearson's coefficient of skewness is a measure of the asymmetry of a distribution. It is calculated using the formula: Karl Pearson's Coefficient of Skewness = (3 * (Mean - Median)) / Standard Deviation

Q: What is the purpose of Karl Pearson's coefficient of skewness?

A: The purpose of Karl Pearson's coefficient of skewness is to describe the shape of a distribution. It helps us understand whether a distribution is symmetrical or skewed.

Q: How is Karl Pearson's coefficient of skewness calculated?

A: Karl Pearson's coefficient of skewness is calculated using the following steps:

  1. Calculate the mean of the data.
  2. Calculate the median of the data.
  3. Calculate the standard deviation of the data.
  4. Use the formula: Karl Pearson's Coefficient of Skewness = (3 * (Mean - Median)) / Standard Deviation

Q: What is the range of values for Karl Pearson's coefficient of skewness?

A: The range of values for Karl Pearson's coefficient of skewness is from -3 to 3. A value of 0 indicates a symmetrical distribution, while a non-zero value indicates a skewed distribution.

Q: What are the limitations of Karl Pearson's coefficient of skewness?

A: The limitations of Karl Pearson's coefficient of skewness include:

  • It is sensitive to outliers.
  • It can be affected by the presence of outliers in the data.
  • It is not a robust measure.
  • It can be affected by small changes in the data.

Q: What are some alternative methods for calculating the coefficient of skewness?

A: Some alternative methods for calculating the coefficient of skewness include:

  • Moment coefficient of skewness
  • Interquartile range coefficient of skewness

Q: How is Karl Pearson's coefficient of skewness used in real-world applications?

A: Karl Pearson's coefficient of skewness is used in various real-world applications, including:

  • Finance: to analyze the distribution of stock prices and returns.
  • Economics: to analyze the distribution of income and wealth.
  • Medicine: to analyze the distribution of patient outcomes and health metrics.

Q: What are some common mistakes to avoid when calculating Karl Pearson's coefficient of skewness?

A: Some common mistakes to avoid when calculating Karl Pearson's coefficient of skewness include:

  • Not checking for outliers in the data.
  • Not using the correct formula.
  • Not calculating the standard deviation correctly.
  • Not interpreting the results correctly.

Q: How can I improve my understanding of Karl Pearson's coefficient of skewness?

A: To improve your understanding of Karl Pearson's coefficient of skewness, you can:

  • Practice calculating the coefficient of skewness using different datasets.
  • Read and understand the underlying theory and mathematics.
  • Use online resources and tutorials to learn more about the coefficient of skewness.
  • Join online communities and forums to discuss and learn from others.