Introduction to Standard Deviation Worksheet - Teaching and Practice - Free Printable

It looks like you've uploaded a worksheet about calculating the standard deviation of a data set. Since I can't view images directly, I’ll solve the problem based on the text you've provided. Let's go step by step and solve each part.

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Given Data Set:

$$
25, 23, 13, 31, 20, 28, 13, 19, 27, 33
$$

We are to compute the standard deviation using the table method.

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Step 1: Find the Mean

Add all the numbers:

$$
25 + 23 + 13 + 31 + 20 + 28 + 13 + 19 + 27 + 33 = 230
$$

Number of data points: $ n = 10 $

$$
\text{Mean} = \frac{230}{10} = 23
$$

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Step 2: Fill in the Table

We need a table with:
- Data (x)
- Deviation from mean (x - μ)
- Squared deviation ((x - μ)²)

Let’s compute each row.

| Data (x) | Deviation (x - 23) | Squared Deviation |
|----------|---------------------|-------------------|
| 25 | 25 - 23 = 2 | $2^2 = 4$ |
| 23 | 23 - 23 = 0 | $0^2 = 0$ |
| 13 | 13 - 23 = -10 | $(-10)^2 = 100$ |
| 31 | 31 - 23 = 8 | $8^2 = 64$ |
| 20 | 20 - 23 = -3 | $(-3)^2 = 9$ |
| 28 | 28 - 23 = 5 | $5^2 = 25$ |
| 13 | 13 - 23 = -10 | $(-10)^2 = 100$ |
| 19 | 19 - 23 = -4 | $(-4)^2 = 16$ |
| 27 | 27 - 23 = 4 | $4^2 = 16$ |
| 33 | 33 - 23 = 10 | $10^2 = 100$ |

Now sum up the squared deviations:

$$
4 + 0 + 100 + 64 + 9 + 25 + 100 + 16 + 16 + 100 = 434
$$

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Step 3: Compute Variance

$$
\text{Variance} = \frac{\sum (x - \mu)^2}{n} = \frac{434}{10} = 43.4
$$

This is called the population variance.

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Step 4: Compute Standard Deviation

$$
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{43.4} \approx 6.59
$$

So, the standard deviation is approximately 6.59.

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Answering the Questions:

#### 1. Mean:
$$
\boxed{23}
$$

#### 2. Complete the table:
See the table above.

#### 3. Sum of squared deviations:
$$
\boxed{434}
$$

#### 4. Variance:
$$
\boxed{43.4}
$$

#### 5. Standard Deviation:
$$
\boxed{\sqrt{43.4} \approx 6.59}
$$

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Part 6: What does the standard deviation mean?

> The standard deviation measures how spread out the data values are from the mean. A higher standard deviation means the data points are more spread out; a lower one means they're closer to the mean.

In this case, a standard deviation of ~6.59 means that, on average, the data points are about 6.59 units away from the mean (23).

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Bonus: Part 7 – Interpreting the Table (Optional)

You mentioned a second table for a different data set:
16, 18, 20, 22, 24, 26, 28, 30, 32

But since it's not fully visible, let me assume you want to find its standard deviation similarly.

Let’s do it quickly.

#### Step 1: Mean
$$
\text{Mean} = \frac{16+18+20+22+24+26+28+30+32}{9} = \frac{216}{9} = 24
$$

#### Step 2: Deviations and Squares

| x | x - 24 | (x - 24)² |
|---|--------|-----------|
| 16 | -8 | 64 |
| 18 | -6 | 36 |
| 20 | -4 | 16 |
| 22 | -2 | 4 |
| 24 | 0 | 0 |
| 26 | 2 | 4 |
| 28 | 4 | 16 |
| 30 | 6 | 36 |
| 32 | 8 | 64 |

Sum of squares:
$$
64 + 36 + 16 + 4 + 0 + 4 + 16 + 36 + 64 = 240
$$

Variance: $ \frac{240}{9} \approx 26.67 $

Standard Deviation: $ \sqrt{26.67} \approx 5.16 $

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Final Answer Summary:

- Mean: 23
- Variance: 43.4
- Standard Deviation: ≈ 6.59

✔ This completes the task.

If you have any additional questions or need clarification, feel free to ask!