The skewness is calculated using the formula:
Skewness = 1 n ∑ i = 1 n ( x i − μ ^ ) 3 [ 1 n ∑ i = 1 n ( x i − μ ^ ) 2 ] 3 2 \text{Skewness} = \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \hat{\mu})^3}{\left[\frac{1}{n} \sum_{i=1}^{n} (x_i - \hat{\mu})^2\right]^{\frac{3}{2}}} Skewness = [ n 1 ∑ i = 1 n ( x i − μ ^ ) 2 ] 2 3 n 1 ∑ i = 1 n ( x i − μ ^ ) 3 Given:
n = 200
∑ i = 1 n ( x i − μ ^ ) 2 = 774 , 759.90 \sum_{i=1}^{n} (x_i - \hat{\mu})^2 = 774,759.90 ∑ i = 1 n ( x i − μ ^ ) 2 = 774 , 759.90 ∑ i = 1 n ( x i − μ ^ ) 3 = − 13 , 476.784 \sum_{i=1}^{n} (x_i - \hat{\mu})^3 = -13,476.784 ∑ i = 1 n ( x i − μ ^ ) 3 = − 13 , 476.784
Step-by-step calculation:
Calculate the numerator: 1 200 × ( − 13 , 476.784 ) = − 67.38392 ‘ \frac{1}{200} \times (-13,476.784) = -67.38392` 200 1 × ( − 13 , 476.784 ) = − 67.38392‘
First, compute 1 200 × 774 , 759.90 = 3 , 873.7995 \frac{1}{200} \times 774,759.90 = 3,873.7995 200 1 × 774 , 759.90 = 3 , 873.7995
Then raise to the power of 3/2: ( 3 , 873.7995 ) 3 / 2 = ( 3 , 873.7995 ) 1.5 (3,873.7995)^{3/2} = (3,873.7995)^{1.5} ( 3 , 873.7995 ) 3/2 = ( 3 , 873.7995 ) 1.5
Since $3,873.7995^{1.5} = 3,873.7995 \times \sqrt{3,873.7995}$
3 , 873.7995 ≈ 62.239 \sqrt{3,873.7995} \approx 62.239 3 , 873.7995 ≈ 62.239 So $3,873.7995 \times 62.239 \approx 241,000$3. Final calculation: − 67.38392 241 , 000 ≈ − 0.0002795 \frac{-67.38392}{241,000} \approx -0.0002795 241 , 000 − 67.38392 ≈ − 0.0002795
The value of the skewness is slightly negative and not far from 0, implying the data is symmetrical.
