9.4 - Standard - Deviation - Practice - Worksheet - IM - 3 | PDF ... - Free Printable

Let's solve Practice Problem #1 step by step to calculate the standard deviation of the test scores:

Given Test Scores:

22, 99, 102, 33, 57, 75, 100, 81, 62, 29

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Step 1: Find the Mean (Average)

Add all the scores and divide by the number of scores.

$$
\text{Mean} = \frac{22 + 99 + 102 + 33 + 57 + 75 + 100 + 81 + 62 + 29}{10}
$$

$$
= \frac{660}{10} = 66
$$

So,
Mean (x̄) = 66
n = 10

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Step 2: Fill in the Table

We'll compute:
- Difference from the mean: $ x - \bar{x} $
- Square of difference: $ (x - \bar{x})^2 $

| Test Score (x) | Difference from the mean (x - x̄) | (Difference from the mean)² |
|----------------|-----------------------------------|-------------------------------|
| 22 | 22 - 66 = -44 | (-44)² = 1936 |
| 99 | 99 - 66 = 33 | (33)² = 1089 |
| 102 | 102 - 66 = 36 | (36)² = 1296 |
| 33 | 33 - 66 = -33 | (-33)² = 1089 |
| 57 | 57 - 66 = -9 | (-9)² = 81 |
| 75 | 75 - 66 = 9 | (9)² = 81 |
| 100 | 100 - 66 = 34 | (34)² = 1156 |
| 81 | 81 - 66 = 15 | (15)² = 225 |
| 62 | 62 - 66 = -4 | (-4)² = 16 |
| 29 | 29 - 66 = -37 | (-37)² = 1369 |

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Step 3: Sum of Squared Differences

Now add up the last column:

$$
\sum(x - \bar{x})^2 = 1936 + 1089 + 1296 + 1089 + 81 + 81 + 1156 + 225 + 16 + 1369
$$

Let’s compute step-by-step:

- 1936 + 1089 = 3025
- 3025 + 1296 = 4321
- 4321 + 1089 = 5410
- 5410 + 81 = 5491
- 5491 + 81 = 5572
- 5572 + 1156 = 6728
- 6728 + 225 = 6953
- 6953 + 16 = 6969
- 6969 + 1369 = 8338

So,
$$
\sum(x - \bar{x})^2 = 8338
$$

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Step 4: Calculate Variance

Variance = $ \frac{\sum(x - \bar{x})^2}{n - 1} $

Since this is a sample (not population), we use $ n - 1 $ (degrees of freedom).

$$
\text{Variance} = \frac{8338}{10 - 1} = \frac{8338}{9} \approx 926.44
$$

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Step 5: Standard Deviation

Standard Deviation = $ \sqrt{\text{Variance}} $

$$
\text{Standard Deviation} = \sqrt{926.44} \approx 30.44
$$

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Final Answers:

- Mean: 66
- n: 10
- Sum of (Difference from the Mean)²: 8338
- Variance: $ \frac{8338}{9} \approx 926.44 $
- Standard Deviation: $ \sqrt{926.44} \approx 30.44 $

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✔ Final Answer:

Standard deviation ≈ 30.44

This means, on average, each test score is about 30.44 points away from the mean.

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Summary of Filled-in Worksheet:

| Test Score (x) | Difference from the mean (x - x̄) | (Difference from the mean)² |
|----------------|-----------------------------------|-------------------------------|
| 22 | -44 | 1936 |
| 99 | 33 | 1089 |
| 102 | 36 | 1296 |
| 33 | -33 | 1089 |
| 57 | -9 | 81 |
| 75 | 9 | 81 |
| 100 | 34 | 1156 |
| 81 | 15 | 225 |
| 62 | -4 | 16 |
| 29 | -37 | 1369 |
| | | Sum = 8338 |

$$
\text{Variance} = \frac{8338}{9} = 926.44
$$
$$
\text{Standard deviation} = \sqrt{926.44} \approx \boxed{30.44}
$$