Question 1198353: The sum of fifteen observations, whose mode is 8, was found to be 150 with coefficient of variation of 20%
a) Calculate the pearsonian coefficient of skewness and give appropriate conclusion.
(b) Are smaller values more or less frequent than bigger values for this distribution?
(c) If a constant k was added on each observation, what will be the new pearsonian coefficient of skewness? Show your steps. What do you conclude from this?
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! **a) Calculate Pearson's Coefficient of Skewness**
1. **Find the Mean:**
* Mean (x̄) = Sum of observations / Number of observations
* Mean (x̄) = 150 / 15 = 10
2. **Find the Standard Deviation:**
* Coefficient of Variation (CV) = (Standard Deviation / Mean) * 100
* 20% = (Standard Deviation / 10) * 100
* Standard Deviation = (20/100) * 10 = 2
3. **Estimate the Median:**
* Since the mode is 8 and the data likely has some positive skew (as we'll see below), we can estimate the median to be slightly higher than the mode. Let's assume the median is 9.
4. **Calculate Pearson's Coefficient of Skewness:**
* Skewness = 3 * (Mean - Median) / Standard Deviation
* Skewness = 3 * (10 - 9) / 2
* Skewness = 1.5
**Conclusion:**
* The positive value of the skewness coefficient (1.5) indicates that the distribution is **positively skewed** or **right-skewed**. This means that the tail of the distribution extends further to the right, and there are more observations towards the lower end of the data.
**b) Are smaller values more or less frequent than bigger values?**
* Since the distribution is positively skewed, **smaller values are more frequent than bigger values**.
**c) Effect of Adding a Constant (k)**
* Adding a constant (k) to each observation will:
* **Shift the mean by k:** New Mean = Old Mean + k
* **Not change the standard deviation:** The spread of the data remains the same.
* **Shift the median by k:** New Median = Old Median + k
* **New Skewness:**
* Skewness (New) = 3 * (New Mean - New Median) / Standard Deviation
* Skewness (New) = 3 * [(Old Mean + k) - (Old Median + k)] / Standard Deviation
* Skewness (New) = 3 * (Old Mean - Old Median) / Standard Deviation
* Skewness (New) = Skewness (Old)
* **Conclusion:**
* Adding a constant to each observation **does not change the skewness** of the distribution.
Let me know if you have any further questions or would like to explore other concepts related to skewness and data analysis!
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