Difference between Skewess and Kurtosis, ANOVA

 

🎯ANOVA

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🏫 Example Data

Three teaching methods with student scores:

• Group A (Lecture): 85, 80, 78
• Group B (Online): 90, 88, 92
• Group C (Discussion): 75, 70, 72

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🔎 Step 1: Calculate Group Means

• Mean A = (85+80+78)/3 = 81
• Mean B = (90+88+92)/3 = 90
• Mean C = (75+70+72)/3 = 72.3

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📊 Step 2: Calculate Overall Mean

[ \text{Overall Mean} = \frac{85+80+78+90+88+92+75+70+72}{9} = \frac{730}{9} \approx 81.1 ]

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🧮 Step 3: Between‑Group Variance (SSB)

Formula:
[ SSB = \sum n_i (\bar{X}_i - \bar{X})^2 ]

• For A: (3 \times (81 - 81.1)^2 = 3 \times (−0.1)^2 = 0.03)
• For B: (3 \times (90 - 81.1)^2 = 3 \times (8.9)^2 = 237.63)
• For C: (3 \times (72.3 - 81.1)^2 = 3 \times (−8.8)^2 = 232.32)

[ SSB = 0.03 + 237.63 + 232.32 = 469.98 ]

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📊 Step 4: Within‑Group Variance (SSW)

Formula:
[ SSW = \sum (X_{ij} - \bar{X}_i)^2 ]

• Group A: (85−81)^2 + (80−81)^2 + (78−81)^2 = 16 + 1 + 9 = 26
• Group B: (90−90)^2 + (88−90)^2 + (92−90)^2 = 0 + 4 + 4 = 8
• Group C: (75−72.3)^2 + (70−72.3)^2 + (72−72.3)^2 = 7.29 + 5.29 + 0.09 = 12.67

[ SSW = 26 + 8 + 12.67 = 46.67 ]

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📊 Step 5: Degrees of Freedom

• Between groups: (df_{between} = k - 1 = 3 - 1 = 2)
• Within groups: (df_{within} = N - k = 9 - 3 = 6)

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📊 Step 6: Mean Squares

• (MSB = SSB / df_{between} = 469.98 / 2 = 234.99)
• (MSW = SSW / df_{within} = 46.67 / 6 = 7.78)

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📊 Step 7: F‑Statistic

[ F = \frac{MSB}{MSW} = \frac{234.99}{7.78} \approx 30.2 ]

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✅ Step 8: Decision

• Compare F (≈30.2) with critical F from table (for df=2,6 at α=0.05, critical ≈5.14).
• Since 30.2 > 5.14 → Reject H₀.
• Conclusion: Teaching method does affect scores....

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🎯 Use of Skewness

• Skewness tells you if your data is tilted to one side.

• Real‑world use:

  • In finance, if returns on an investment are positively skewed, it means there are chances of very high profits but most returns are small.

  • In business, if customer income data is skewed, it shows whether most customers are low‑income or high‑income, which helps in pricing decisions. 

  🎯 Use of Kurtosis

• Kurtosis tells you if your data has many extreme values (outliers).

• Real‑world use:

  • In quality control, high kurtosis means more products are far from the average quality — a warning sign.

  • In risk management, high kurtosis in stock returns means more chances of extreme losses or gains.