Statistics
Exercise 13.2 • Variance & Standard Deviation
Key Formulas:
Variance ($\sigma^2$) for Ungrouped: $\frac{\sum (x_i – \bar{x})^2}{n}$
Variance for Grouped: $\frac{1}{N} \sum f_i (x_i – \bar{x})^2$
Shortcut Method Variance: $h^2 \left[ \frac{1}{N} \sum f_i y_i^2 – \left(\frac{1}{N} \sum f_i y_i\right)^2 \right]$ where $y_i = \frac{x_i – A}{h}$
Standard Deviation ($\sigma$) = $\sqrt{\text{Variance}}$
Variance ($\sigma^2$) for Ungrouped: $\frac{\sum (x_i – \bar{x})^2}{n}$
Variance for Grouped: $\frac{1}{N} \sum f_i (x_i – \bar{x})^2$
Shortcut Method Variance: $h^2 \left[ \frac{1}{N} \sum f_i y_i^2 – \left(\frac{1}{N} \sum f_i y_i\right)^2 \right]$ where $y_i = \frac{x_i – A}{h}$
Standard Deviation ($\sigma$) = $\sqrt{\text{Variance}}$
Q1
1. Find Mean and Variance: 6, 7, 10, 12, 13, 4, 8, 12
Mean
$\bar{x} = \frac{6+7+10+12+13+4+8+12}{8} = \frac{72}{8} = 9$.
Deviations
$(x_i – 9)$: -3, -2, 1, 3, 4, -5, -1, 3.
Sq Devs
9, 4, 1, 9, 16, 25, 1, 9. Sum = 74.
Variance
$\sigma^2 = \frac{74}{8} = 9.25$.
Mean = 9, Variance = 9.25
Q2
2. Find Mean and Variance of first n natural numbers.
Mean
Sum $\sum x = \frac{n(n+1)}{2}$. Mean $\bar{x} = \frac{n+1}{2}$.
Variance
Formula: $\sigma^2 = \frac{1}{n}\sum x^2 – (\bar{x})^2$.
$\sum x^2 = \frac{n(n+1)(2n+1)}{6}$.
$\sum x^2 = \frac{n(n+1)(2n+1)}{6}$.
Substitute
$\sigma^2 = \frac{(n+1)(2n+1)}{6} – \left(\frac{n+1}{2}\right)^2 = \frac{n+1}{2} [\frac{2n+1}{3} – \frac{n+1}{2}]$.
Simplify
$\frac{n+1}{2} [\frac{4n+2-3n-3}{6}] = \frac{n+1}{2} \cdot \frac{n-1}{6} = \frac{n^2-1}{12}$.
Mean = $\frac{n+1}{2}$, Variance = $\frac{n^2-1}{12}$
Q3
3. Find Mean and Variance of first 10 multiples of 3.
Data
3, 6, 9, 12, 15, 18, 21, 24, 27, 30. ($n=10$).
Mean
Sum = $3(1+2+…+10) = 3(55) = 165$. $\bar{x} = 16.5$.
Sq Sum
$\sum x^2 = 3^2(1^2+…+10^2) = 9(385) = 3465$.
Variance
$\frac{3465}{10} – (16.5)^2 = 346.5 – 272.25 = 74.25$.
Mean = 16.5, Variance = 74.25
Q4
4. Find Mean and Variance (Discrete Frequency).
| $x_i$ | $f_i$ | $f_ix_i$ | $d_i = x_i – \bar{x}$ | $f_id_i^2$ |
|---|---|---|---|---|
| 6 | 2 | 12 | -13 | 338 |
| 10 | 4 | 40 | -9 | 324 |
| 14 | 7 | 98 | -5 | 175 |
| 18 | 12 | 216 | -1 | 12 |
| 24 | 8 | 192 | 5 | 200 |
| 28 | 4 | 112 | 9 | 324 |
| 30 | 3 | 90 | 11 | 363 |
| 40 | 760 | 1736 |
Mean
$\bar{x} = \frac{760}{40} = 19$.
Variance
$\sigma^2 = \frac{1736}{40} = 43.4$.
Mean = 19, Variance = 43.4
Q5
5. Find Mean and Variance (Discrete Frequency).
| $x_i$ | $f_i$ | $f_ix_i$ | $d_i = x_i – 100$ | $f_id_i^2$ |
|---|---|---|---|---|
| 92 | 3 | 276 | -8 | 192 |
| 93 | 2 | 186 | -7 | 98 |
| 97 | 3 | 291 | -3 | 27 |
| 98 | 2 | 196 | -2 | 8 |
| 102 | 6 | 612 | 2 | 24 |
| 104 | 3 | 312 | 4 | 48 |
| 109 | 3 | 327 | 9 | 243 |
| 22 | 2200 | 640 |
Mean
$\bar{x} = \frac{2200}{22} = 100$.
Variance
$\sigma^2 = \frac{640}{22} = 29.09$.
Mean = 100, Variance = 29.09
Q6
6. Find Mean and Standard Deviation using Short-cut method.
| $x_i$ | $f_i$ | $y_i = x_i – 64$ | $f_iy_i$ | $f_iy_i^2$ |
|---|---|---|---|---|
| 60 | 2 | -4 | -8 | 32 |
| 61 | 1 | -3 | -3 | 9 |
| 62 | 12 | -2 | -24 | 48 |
| 63 | 29 | -1 | -29 | 29 |
| 64 | 25 | 0 | 0 | 0 |
| 65 | 12 | 1 | 12 | 12 |
| 66 | 10 | 2 | 20 | 40 |
| 67 | 4 | 3 | 12 | 36 |
| 68 | 5 | 4 | 20 | 80 |
| 100 | – | 0 | 286 |
Mean
$A + \frac{\sum f_iy_i}{N} = 64 + \frac{0}{100} = 64$.
Variance
$\frac{286}{100} – (0)^2 = 2.86$.
Std Dev
$\sqrt{2.86} \approx 1.69$.
Mean = 64, SD = 1.69
Q7
7. Find Mean and Variance (Grouped Data).
| Class | Mid ($x_i$) | $f_i$ | $y_i = \frac{x_i – 105}{30}$ | $f_iy_i$ | $f_iy_i^2$ |
|---|---|---|---|---|---|
| 0-30 | 15 | 2 | -3 | -6 | 18 |
| 30-60 | 45 | 3 | -2 | -6 | 12 |
| 60-90 | 75 | 5 | -1 | -5 | 5 |
| 90-120 | 105 | 10 | 0 | 0 | 0 |
| 120-150 | 135 | 3 | 1 | 3 | 3 |
| 150-180 | 165 | 5 | 2 | 10 | 20 |
| 180-210 | 195 | 2 | 3 | 6 | 18 |
| 30 | – | 2 | 76 |
Mean
$105 + \frac{2}{30} \times 30 = 107$.
Variance
$30^2 [\frac{76}{30} – (\frac{2}{30})^2] = 900 [\frac{76}{30} – \frac{4}{900}] = 900 \frac{76}{30} – 4 = 2280 – 4 = 2276$.
Mean = 107, Variance = 2276
Q8
8. Find Mean and Variance (Grouped Data).
| Class | Mid ($x_i$) | $f_i$ | $y_i = \frac{x_i – 25}{10}$ | $f_iy_i$ | $f_iy_i^2$ |
|---|---|---|---|---|---|
| 0-10 | 5 | 5 | -2 | -10 | 20 |
| 10-20 | 15 | 8 | -1 | -8 | 8 |
| 20-30 | 25 | 15 | 0 | 0 | 0 |
| 30-40 | 35 | 16 | 1 | 16 | 16 |
| 40-50 | 45 | 6 | 2 | 12 | 24 |
| 50 | – | 10 | 68 |
Mean
$25 + \frac{10}{50} \times 10 = 25 + 2 = 27$.
Variance
$10^2 [\frac{68}{50} – (\frac{10}{50})^2] = 100 [1.36 – 0.04] = 100(1.32) = 132$.
Mean = 27, Variance = 132
Q9
9. Mean, Variance and SD using Short-cut method (Height in cms).
| Class | Mid ($x_i$) | $f_i$ | $y_i = \frac{x_i – 92.5}{5}$ | $f_iy_i$ | $f_iy_i^2$ |
|---|---|---|---|---|---|
| 70-75 | 72.5 | 3 | -4 | -12 | 48 |
| 75-80 | 77.5 | 4 | -3 | -12 | 36 |
| 80-85 | 82.5 | 7 | -2 | -14 | 28 |
| 85-90 | 87.5 | 7 | -1 | -7 | 7 |
| 90-95 | 92.5 | 15 | 0 | 0 | 0 |
| 95-100 | 97.5 | 9 | 1 | 9 | 9 |
| 100-105 | 102.5 | 6 | 2 | 12 | 24 |
| 105-110 | 107.5 | 6 | 3 | 18 | 54 |
| 110-115 | 112.5 | 3 | 4 | 12 | 48 |
| 60 | – | 6 | 254 |
Mean
$92.5 + \frac{6}{60} \times 5 = 92.5 + 0.5 = 93$.
Variance
$5^2 [\frac{254}{60} – (\frac{6}{60})^2] = 25 [4.233 – 0.01] = 25(4.223) = 105.58$.
Std Dev
$\sqrt{105.58} = 10.27$.
Mean = 93, Var = 105.58, SD = 10.27
Q10
10. Mean and Variance (Diameters of Circles).
Note
Classes are inclusive. Midpoints: $\frac{33+36}{2}=34.5$, etc. Step $h=4$.
| Class | Mid ($x_i$) | $f_i$ | $y_i = \frac{x_i – 42.5}{4}$ | $f_iy_i$ | $f_iy_i^2$ |
|---|---|---|---|---|---|
| 33-36 | 34.5 | 15 | -2 | -30 | 60 |
| 37-40 | 38.5 | 17 | -1 | -17 | 17 |
| 41-44 | 42.5 | 21 | 0 | 0 | 0 |
| 45-48 | 46.5 | 22 | 1 | 22 | 22 |
| 49-52 | 50.5 | 25 | 2 | 50 | 100 |
| 100 | – | 25 | 199 |
Mean
$42.5 + \frac{25}{100} \times 4 = 42.5 + 1 = 43.5$.
Variance
$4^2 [\frac{199}{100} – (\frac{25}{100})^2] = 16 [1.99 – 0.0625] = 16(1.9275) = 30.84$.
Mean = 43.5, Variance = 30.84