Standard Deviation And Absolute Deviation Independent Practice ... - Free Printable

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Let me go through each problem step by step and solve them for you.

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Problem 1:

Data: 10, 16, 18, 15, 15, 10, 23

#### Step 1: Find the Mean
$$
\text{Mean} = \frac{10 + 16 + 18 + 15 + 15 + 10 + 23}{7} = \frac{107}{7} \approx 15.29
$$

#### Step 2: Find Absolute Deviations from the Mean
| Data | |x - mean| |
|------|----------|
| 10 | |10 - 15.29| = 5.29 |
| 16 | |16 - 15.29| = 0.71 |
| 18 | |18 - 15.29| = 2.71 |
| 15 | |15 - 15.29| = 0.29 |
| 15 | |15 - 15.29| = 0.29 |
| 10 | |10 - 15.29| = 5.29 |
| 23 | |23 - 15.29| = 7.71 |

#### Step 3: Mean Absolute Deviation (MAD)
$$
\text{MAD} = \frac{5.29 + 0.71 + 2.71 + 0.29 + 0.29 + 5.29 + 7.71}{7} = \frac{22.00}{7} \approx 3.14
$$

✔ Answer: ≈ 3.14

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Problem 2:

Data: 41, 50, 38, 45, 35, 51, 32

#### Step 1: Mean
$$
\text{Mean} = \frac{41+50+38+45+35+51+32}{7} = \frac{292}{7} \approx 41.71
$$

#### Step 2: Absolute Deviations
| x | |x - 41.71| |
|---|-----------|
| 41 | 0.71 |
| 50 | 8.29 |
| 38 | 3.71 |
| 45 | 3.29 |
| 35 | 6.71 |
| 51 | 9.29 |
| 32 | 9.71 |

Sum = 0.71 + 8.29 + 3.71 + 3.29 + 6.71 + 9.29 + 9.71 = 41.51

#### MAD:
$$
\frac{41.51}{7} \approx 5.93
$$

✔ Answer: ≈ 5.93

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Problem 3:

Data: 10, 12, 18, 25, 25, 11, 22

#### Mean:
$$
\frac{10+12+18+25+25+11+22}{7} = \frac{123}{7} \approx 17.57
$$

#### Absolute Deviations:
| x | |x - 17.57| |
|---|-----------|
| 10 | 7.57 |
| 12 | 5.57 |
| 18 | 0.43 |
| 25 | 7.43 |
| 25 | 7.43 |
| 11 | 6.57 |
| 22 | 4.43 |

Sum = 7.57 + 5.57 + 0.43 + 7.43 + 7.43 + 6.57 + 4.43 = 40.00

MAD = $ \frac{40}{7} \approx 5.71 $

✔ Answer: ≈ 5.71

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Problem 4:

Data: 22, 33, 44, 45, 46, 48, 50, 51, 51, 22

#### Mean:
$$
\frac{22+33+44+45+46+48+50+51+51+22}{10} = \frac{412}{10} = 41.2
$$

#### Absolute Deviations:
| x | |x - 41.2| |
|---|---------|
| 22 | 19.2 |
| 33 | 8.2 |
| 44 | 2.8 |
| 45 | 3.8 |
| 46 | 4.8 |
| 48 | 6.8 |
| 50 | 8.8 |
| 51 | 9.8 |
| 51 | 9.8 |
| 22 | 19.2 |

Sum = 19.2+8.2+2.8+3.8+4.8+6.8+8.8+9.8+9.8+19.2 = 93.4

MAD = $ \frac{93.4}{10} = 9.34 $

✔ Answer: 9.34

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Problem 5:

Data: 10, 60, 80, 50, 50, 10, 20

#### Mean:
$$
\frac{10+60+80+50+50+10+20}{7} = \frac{280}{7} = 40
$$

#### Absolute Deviations:
| x | |x - 40| |
|---|-------|
| 10 | 30 |
| 60 | 20 |
| 80 | 40 |
| 50 | 10 |
| 50 | 10 |
| 10 | 30 |
| 20 | 20 |

Sum = 30+20+40+10+10+30+20 = 160

MAD = $ \frac{160}{7} \approx 22.86 $

✔ Answer: ≈ 22.86

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Problem 6:

Weights: 56, 64, 78, 78, 84, 70kg, 65kg, 89kg
(That's 8 weights)

#### Step 1: Mean
$$
\frac{56+64+78+78+84+70+65+89}{8} = \frac{584}{8} = 73
$$

#### Step 2: Standard Deviation

We use formula:
$$
\sigma = \sqrt{ \frac{\sum (x_i - \mu)^2}{n} }
$$

Compute squared deviations:

| x | x - μ | (x - μ)² |
|---|-------|----------|
| 56 | -17 | 289 |
| 64 | -9 | 81 |
| 78 | 5 | 25 |
| 78 | 5 | 25 |
| 84 | 11 | 121 |
| 70 | -3 | 9 |
| 65 | -8 | 64 |
| 89 | 16 | 256 |

Sum of squares = 289 + 81 + 25 + 25 + 121 + 9 + 64 + 256 = 950

Variance = $ \frac{950}{8} = 118.75 $

Standard Deviation = $ \sqrt{118.75} \approx 10.90 $

✔ Answer: ≈ 10.90 kg

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Problem 7:

Heights (cm): 54, 48, 35, 50, 20, 25, 40, 24, 38, 22

#### Step 1: Mean
$$
\frac{54+48+35+50+20+25+40+24+38+22}{10} = \frac{356}{10} = 35.6
$$

#### Step 2: Variance and SD

Squared deviations:

| x | x - 35.6 | (x - 35.6)² |
|---|--------|------------|
| 54 | 18.4 | 338.56 |
| 48 | 12.4 | 153.76 |
| 35 | -0.6 | 0.36 |
| 50 | 14.4 | 207.36 |
| 20 | -15.6 | 243.36 |
| 25 | -10.6 | 112.36 |
| 40 | 4.4 | 19.36 |
| 24 | -11.6 | 134.56 |
| 38 | 2.4 | 5.76 |
| 22 | -13.6 | 184.96 |

Sum = 338.56 + 153.76 + 0.36 + 207.36 + 243.36 + 112.36 + 19.36 + 134.56 + 5.76 + 184.96 = 1500.8

Variance = $ \frac{1500.8}{10} = 150.08 $

SD = $ \sqrt{150.08} \approx 12.25 $

✔ Answer: ≈ 12.25 cm

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Problem 8:

Temperatures: 34, 26, 21, 21, 25, 26, 27

#### Step 1: Mean
$$
\frac{34+26+21+21+25+26+27}{7} = \frac{180}{7} \approx 25.71
$$

#### Step 2: Squared Deviations

| x | x - 25.71 | (x - 25.71)² |
|---|----------|-------------|
| 34 | 8.29 | 68.72 |
| 26 | 0.29 | 0.08 |
| 21 | -4.71 | 22.18 |
| 21 | -4.71 | 22.18 |
| 25 | -0.71 | 0.50 |
| 26 | 0.29 | 0.08 |
| 27 | 1.29 | 1.66 |

Sum = 68.72 + 0.08 + 22.18 + 22.18 + 0.50 + 0.08 + 1.66 = 115.40

Variance = $ \frac{115.40}{7} \approx 16.49 $

SD = $ \sqrt{16.49} \approx 4.06 $

✔ Answer: ≈ 4.06°F

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Problem 9:

Data: 70, 71, 72, 73, 74, 75, 76, 77, 78, 80

This is nearly consecutive numbers.

#### Step 1: Mean
$$
\frac{70+71+72+73+74+75+76+77+78+80}{10} = \frac{756}{10} = 75.6
$$

#### Step 2: MAD

| x | |x - 75.6| |
|---|---------|
| 70 | 5.6 |
| 71 | 4.6 |
| 72 | 3.6 |
| 73 | 2.6 |
| 74 | 1.6 |
| 75 | 0.6 |
| 76 | 0.4 |
| 77 | 1.4 |
| 78 | 2.4 |
| 80 | 4.4 |

Sum = 5.6+4.6+3.6+2.6+1.6+0.6+0.4+1.4+2.4+4.4 = 32.2

MAD = $ \frac{32.2}{10} = 3.22 $

✔ Answer: 3.22

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Problem 10:

Data: 76, 87, 82, 25, 54, 15, 24

#### Step 1: Mean
$$
\frac{76+87+82+25+54+15+24}{7} = \frac{443}{7} \approx 63.29
$$

#### Step 2: MAD

| x | |x - 63.29| |
|---|----------|
| 76 | 12.71 |
| 87 | 23.71 |
| 82 | 18.71 |
| 25 | 38.29 |
| 54 | 9.29 |
| 15 | 48.29 |
| 24 | 39.29 |

Sum = 12.71 + 23.71 + 18.71 + 38.29 + 9.29 + 48.29 + 39.29 = 190.00

MAD = $ \frac{190}{7} \approx 27.14 $

✔ Answer: ≈ 27.14

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## ✔ Final Answers Summary:

1. 3.14
2. 5.93
3. 5.71
4. 9.34
5. 22.86
6. 10.90 kg
7. 12.25 cm
8. 4.06°F
9. 3.22
10. 27.14

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