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Exercise 8.2 · Q8
Q.

Compute variance and standard deviation for the following data:

X2468101214161820
F810107643426
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✓ Free question

Step 1: This is an ungrouped frequency distribution with N=∑fi=8+10+10+7+6+4+3+4+2+6=60N=\sum f_i = 8+10+10+7+6+4+3+4+2+6=60.

Step 2: Build fixif_ix_i: 2(8)=16,4(10)=40,6(10)=60,8(7)=56,10(6)=60,12(4)=48,14(3)=42,16(4)=64,18(2)=36,20(6)=1202(8)=16,4(10)=40,6(10)=60,8(7)=56,10(6)=60,12(4)=48,14(3)=42,16(4)=64,18(2)=36,20(6)=120; their sum ∑fixi=542\sum f_ix_i = 542, so xˉ=542/60≈9.033\bar{x}=542/60\approx9.033.

Step 3: Build fixi2f_ix_i^2: 32,160,360,448,600,576,588,1024,648,240032,160,360,448,600,576,588,1024,648,2400; their sum ∑fixi2=6836\sum f_ix_i^2 = 6836.

Step 4: Var(X)=683660−(9.033)2≈113.933−81.601=32.332\text{Var}(X) = \dfrac{6836}{60} - (9.033)^2 \approx 113.933 - 81.601 = 32.332.

Step 5: S.D.=32.332≈5.69\text{S.D.} = \sqrt{32.332} \approx 5.69.

✓Final answer

Variance ≈32.33\approx 32.33, S.D. ≈5.69\approx 5.69

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