1. Calculate Sample Variance `(S^2)` from the following grouped data
Solution:
`x` `(1)` | Frequency `(f)` `(2)` | `f*x` `(3)=(2)xx(1)` | `f*x^2=(f*x)xx(x)` `(4)=(3)xx(1)` |
0 | 1 | 0 | 0 |
1 | 5 | 5 | 5 |
2 | 10 | 20 | 40 |
3 | 6 | 18 | 54 |
4 | 3 | 12 | 48 |
--- | --- | --- | --- |
| `n=25` | `sum f*x=55` | `sum f*x^2=147` |
Mean `bar x = (sum fx)/n`
`=55/25`
`=2.2`
Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`
`=(147 - (55)^2/25)/24`
`=(147 - 121)/24`
`=26/24`
`=1.0833`
2. Calculate Sample Variance `(S^2)` from the following grouped data
X | Frequency |
10 | 3 |
11 | 12 |
12 | 18 |
13 | 12 |
14 | 3 |
Solution:
`x` `(1)` | Frequency `(f)` `(2)` | `f*x` `(3)=(2)xx(1)` | `f*x^2=(f*x)xx(x)` `(4)=(3)xx(1)` |
10 | 3 | 30 | 300 |
11 | 12 | 132 | 1452 |
12 | 18 | 216 | 2592 |
13 | 12 | 156 | 2028 |
14 | 3 | 42 | 588 |
--- | --- | --- | --- |
| `n=48` | `sum f*x=576` | `sum f*x^2=6960` |
Mean `bar x = (sum fx)/n`
`=576/48`
`=12`
Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`
`=(6960 - (576)^2/48)/47`
`=(6960 - 6912)/47`
`=48/47`
`=1.0213`
3. Calculate Sample Variance `(S^2)` from the following grouped data
Class | Frequency |
2 - 4 | 3 |
4 - 6 | 4 |
6 - 8 | 2 |
8 - 10 | 1 |
Solution:
Class `(1)` | Frequency `(f)` `(2)` | Mid value `(x)` `(3)` | `f*x` `(4)=(2)xx(3)` | `f*x^2=(f*x)xx(x)` `(5)=(4)xx(3)` |
2-4 | 3 | 3 | 9 | 27 |
4-6 | 4 | 5 | 20 | 100 |
6-8 | 2 | 7 | 14 | 98 |
8-10 | 1 | 9 | 9 | 81 |
--- | --- | --- | --- | --- |
-- | `n = 10` | -- | `sum f*x=52` | `sum f*x^2=306` |
Mean `bar x = (sum fx)/n`
`=52/10`
`=5.2`
Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`
`=(306 - (52)^2/10)/9`
`=(306 - 270.4)/9`
`=35.6/9`
`=3.9556`
4. Calculate Sample Variance `(S^2)` from the following grouped data
Class | Frequency |
0 - 2 | 5 |
2 - 4 | 16 |
4 - 6 | 13 |
6 - 8 | 7 |
8 - 10 | 5 |
10 - 12 | 4 |
Solution:
Class `(1)` | Frequency `(f)` `(2)` | Mid value `(x)` `(3)` | `f*x` `(4)=(2)xx(3)` | `f*x^2=(f*x)xx(x)` `(5)=(4)xx(3)` |
0-2 | 5 | 1 | 5 | 5 |
2-4 | 16 | 3 | 48 | 144 |
4-6 | 13 | 5 | 65 | 325 |
6-8 | 7 | 7 | 49 | 343 |
8-10 | 5 | 9 | 45 | 405 |
10-12 | 4 | 11 | 44 | 484 |
--- | --- | --- | --- | --- |
-- | `n = 50` | -- | `sum f*x=256` | `sum f*x^2=1706` |
Mean `bar x = (sum fx)/n`
`=256/50`
`=5.12`
Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`
`=(1706 - (256)^2/50)/49`
`=(1706 - 1310.72)/49`
`=395.28/49`
`=8.0669`
5. Calculate Sample Variance `(S^2)` from the following grouped data
Class | Frequency |
10 - 20 | 15 |
20 - 30 | 25 |
30 - 40 | 20 |
40 - 50 | 12 |
50 - 60 | 8 |
60 - 70 | 5 |
70 - 80 | 3 |
Solution:
Class `(1)` | Frequency `(f)` `(2)` | Mid value `(x)` `(3)` | `d=(x-A)/h=(x-45)/10` `A=45,h=10` `(4)` | `f*d` `(5)=(2)xx(4)` | `f*d^2` `(6)=(5)xx(4)` |
10 - 20 | 15 | 15 | -3 | -45 | 135 |
20 - 30 | 25 | 25 | -2 | -50 | 100 |
30 - 40 | 20 | 35 | -1 | -20 | 20 |
40 - 50 | 12 | 45=A | 0 | 0 | 0 |
50 - 60 | 8 | 55 | 1 | 8 | 8 |
60 - 70 | 5 | 65 | 2 | 10 | 20 |
70 - 80 | 3 | 75 | 3 | 9 | 27 |
--- | --- | --- | --- | --- | --- |
| `n = 88` | ----- | ----- | `sum f*d=-88` | `sum f*d^2=310` |
Mean `bar x = A + (sum fd)/n * h`
`=45 + -88/88 * 10`
`=45 + -1 * 10`
`=45 + -10`
`=35`
Sample Variance `S^2 = ((sum f*d^2 - (sum f*d)^2/n)/(n-1)) * h^2`
`=((310 - (-88)^2/88)/87) * 10^2`
`=((310 - 88)/87) * 100`
`=222/87 * 100`
`=2.5517 * 100`
`=255.1724`
6. Calculate Sample Variance `(S^2)` from the following grouped data
Class | Frequency |
20 - 25 | 110 |
25 - 30 | 170 |
30 - 35 | 80 |
35 - 40 | 45 |
40 - 45 | 40 |
45 - 50 | 35 |
Solution:
Class `(1)` | Frequency `(f)` `(2)` | Mid value `(x)` `(3)` | `d=(x-A)/h=(x-37.5)/5` `A=37.5,h=5` `(4)` | `f*d` `(5)=(2)xx(4)` | `f*d^2` `(6)=(5)xx(4)` |
20 - 25 | 110 | 22.5 | -3 | -330 | 990 |
25 - 30 | 170 | 27.5 | -2 | -340 | 680 |
30 - 35 | 80 | 32.5 | -1 | -80 | 80 |
35 - 40 | 45 | 37.5=A | 0 | 0 | 0 |
40 - 45 | 40 | 42.5 | 1 | 40 | 40 |
45 - 50 | 35 | 47.5 | 2 | 70 | 140 |
--- | --- | --- | --- | --- | --- |
| `n = 480` | ----- | ----- | `sum f*d=-640` | `sum f*d^2=1930` |
Mean `bar x = A + (sum fd)/n * h`
`=37.5 + -640/480 * 5`
`=37.5 + -1.3333 * 5`
`=37.5 + -6.6667`
`=30.8333`
Sample Variance `S^2 = ((sum f*d^2 - (sum f*d)^2/n)/(n-1)) * h^2`
`=((1930 - (-640)^2/480)/479) * 5^2`
`=((1930 - 853.3333)/479) * 25`
`=1076.6667/479 * 25`
`=2.2477 * 25`
`=56.1935`
This material is intended as a summary. Use your textbook for detail explanation.
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