Answer :
The Sk1 value is negative (-0.68), it indicates that the data is negatively skewed. This means that the left tail of the distribution is longer or stretched out compared to the right tail. In other words, the majority of the data points are concentrated on the right side of the distribution, and there are a few extreme values on the left side.
the Pearson's First Coefficient of skewness is a measure of the skewness or asymmetry in a dataset. It tells us whether the data is symmetric, positively skewed, or negatively skewed. To estimate and interpret the coefficient of skewness, we can follow these steps:
Step 1: Calculate the mean (μ) and the standard deviation (σ) of the given dataset.
For the given data:
91, 62, 54, 72, 76, 84, 38, 76, 70, 84, 59, 82, 76, 74, 52, 76, 85, 76
The mean (μ) can be calculated by adding up all the values and dividing by the total number of values:
μ = (91 + 62 + 54 + 72 + 76 + 84 + 38 + 76 + 70 + 84 + 59 + 82 + 76 + 74 + 52 + 76 + 85 + 76) / 18 = 73.67
the standard deviation (σ), we can use the formula:
σ = √(Σ(xi - μ)² / n)
where Σ represents the sum, xi is each individual value, μ is the mean, and n is the total number of values.
Step 2: Calculate the median (Me) of the dataset.
the median, arrange the data in ascending order:
38, 52, 54, 59, 62, 70, 72, 74, 76, 76, 76, 76, 82, 84, 84, 85, 91
Since there are 18 values, the median will be the average of the 9th and 10th values:
Me = (76 + 76) / 2 = 76
Step 3: Calculate Pearson's First Coefficient of skewness (Sk1) using the formula:
Sk1 = 3 * (μ - Me) / σ
Substitute the values into the formula:
Sk1 = 3 * (73.67 - 76) / σ
Step 4: Calculate the standard deviation of the dataset (σ).
Using the formula mentioned in Step 1, we can calculate σ. The sum of squared differences from the mean can be calculated as:
Σ(xi - μ)² = (91 - 73.67)² + (62 - 73.67)² + (54 - 73.67)² + (72 - 73.67)² + (76 - 73.67)² + (84 - 73.67)² + (38 - 73.67)² + (76 - 73.67)² + (70 - 73.67)² + (84 - 73.67)² + (59 - 73.67)² + (82 - 73.67)² + (76 - 73.67)² + (74 - 73.67)² + (52 - 73.67)² + (76 - 73.67)² + (85 - 73.67)² + (76 - 73.67)²
Σ(xi - μ)² = 1930.33
Using the formula for the standard deviation from Step 1:
σ = √(Σ(xi - μ)² / n) = √(1930.33 / 18) = √107.24 = 10.36
Step 5: Substitute the values into the Sk1 formula:
Sk1 = 3 * (73.67 - 76) / 10.36 = -0.68
The estimated Pearson's First Coefficient of skewness for the given data is -0.68.
Since the Sk1 value is negative (-0.68), it indicates that the data is negatively skewed. This means that the left tail of the distribution is longer or stretched out compared to the right tail. In other words, the majority of the data points are concentrated on the right side of the distribution, and there are a few extreme values on the left side.
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