Standard-deviation-practice-worksheet (Auto Recovered) - Name Date ... - Free Printable

You're working on Practice Problem #1 from the worksheet “9.4 CALCULATING STANDARD DEVIATION”. Let’s walk through the correct calculations step-by-step and fix the errors in the provided table.

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## ✔ Step 1: Find the Mean (x̄)

Test Scores:
22, 99, 102, 33, 57, 75, 100, 81, 62, 29

There are 10 scores, so n = 10.

Sum of scores:
22 + 99 + 102 + 33 + 57 + 75 + 100 + 81 + 62 + 29 =
Let’s add them:

- 22 + 99 = 121
- 121 + 102 = 223
- 223 + 33 = 256
- 256 + 57 = 313
- 313 + 75 = 388
- 388 + 100 = 488
- 488 + 81 = 569
- 569 + 62 = 631
- 631 + 29 = 660

Mean (x̄) = Sum / n = 660 / 10 = 66

✔ So, Mean = 66, n = 10

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## ✔ Step 2: Fill in the Table Correctly

The table has a critical error: it incorrectly calculates (Difference from the mean)² as difference × difference — which is actually correct — BUT some of the multiplication results are wrong!

Let’s recalculate each row correctly.

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

> ❗ Note: The original table had errors like:
> - "-44 x -44 = -88" → Wrong! Should be +1936
> - "-37 x -37 = -74" → Wrong! Should be +1369
> - "-33 x -33 = -66" → Wrong! Should be +1089
> - "-9 x -9 = 18" → Wrong! Should be +81
> - "9 x 9 = 18" → Wrong! Should be +81
> - etc.

✔ So, sum of (x - x̄)² = 1936 + 1369 + 1089 + 81 + 16 + 81 + 225 + 1089 + 1156 + 1296

Let’s add them:

Group for easier addition:

- 1936 + 1369 = 3305
- 1089 + 81 = 1170
- 16 + 81 = 97
- 225 + 1089 = 1314
- 1156 + 1296 = 2452

Now add those:

- 3305 + 1170 = 4475
- 4475 + 97 = 4572
- 4572 + 1314 = 5886
- 5886 + 2452 = 8338

✔ So, Σ(x - x̄)² = 8338

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## ✔ Step 3: Calculate Variance

Variance = Σ(x - x̄)² / (n - 1) = 8338 / (10 - 1) = 8338 / 9 ≈ 926.444...

We’ll keep it as a fraction or decimal for now.

✔ Variance ≈ 926.44

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## ✔ Step 4: Calculate Standard Deviation

Standard deviation = √(variance) = √(8338 / 9) = √(926.444...) ≈ 30.44

Let’s compute more precisely:

√(8338 ÷ 9) = √926.444... ≈ 30.44

(You can use calculator: √926.444 ≈ 30.44)

✔ Standard deviation ≈ 30.44

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## 📝 Final Answers to Fill In:

- Mean: 66
- n: 10
- Sum of (x - x̄)²: 8338
- Variance (Σ(x - x̄)² / (n - 1)): 8338 / 9 ≈ 926.44
- Standard deviation = √(variance) = √(8338/9) ≈ 30.44

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## 💡 Why This Matters

The standard deviation tells us how spread out the test scores are from the mean (66). A value of ~30.44 means scores vary quite a bit — some are much lower (like 22), some much higher (like 102).

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## ✍️ Corrections to Original Worksheet Table:

The worksheet table contains multiple arithmetic errors. Here's the corrected version of the last column:

| (Difference from the mean)² |
|-----------------------------|
| 1936 |
| 1369 |
| 1089 |
| 81 |
| 16 |
| 81 |
| 225 |
| 1089 |
| 1156 |
| 1296 |

Sum = 8338 ✔️

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✔ Final Answer: Standard Deviation ≈ 30.44

You can round to two decimal places unless instructed otherwise.

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Let me know if you want to see this done with a calculator or using Excel formulas!