Quiz & Worksheet - How to Calculate the Standard Deviation | Study.com - Free Printable

Let's solve each question step by step using the data set:
16, 20, 14, 24, 16, 19, 30, and 8

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Step 1: Find the Mean

Add all values and divide by the number of values.

$$
\text{Mean} = \frac{16 + 20 + 14 + 24 + 16 + 19 + 30 + 8}{8}
= \frac{147}{8} = 18.375
$$

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Step 2: Find the Variance (Population Variance)

We’ll use the formula for population variance:

$$
\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}
$$

Where:
- $ x_i $ = each data point
- $ \mu $ = mean = 18.375
- $ N $ = 8

Now calculate $ (x_i - \mu)^2 $ for each value:

| $ x_i $ | $ x_i - \mu $ | $ (x_i - \mu)^2 $ |
|----------|------------------|----------------------|
| 16 | 16 - 18.375 = -2.375 | 5.640625 |
| 20 | 20 - 18.375 = 1.625 | 2.640625 |
| 14 | 14 - 18.375 = -4.375 | 19.140625 |
| 24 | 24 - 18.375 = 5.625 | 31.640625 |
| 16 | -2.375 | 5.640625 |
| 19 | 0.625 | 0.390625 |
| 30 | 11.625 | 135.140625 |
| 8 | -10.375 | 107.640625 |

Sum of squared deviations:
$$
5.640625 + 2.640625 + 19.140625 + 31.640625 + 5.640625 + 0.390625 + 135.140625 + 107.640625 = 307.875
$$

Now divide by $ N = 8 $:
$$
\text{Variance} = \frac{307.875}{8} = 38.484375 \approx 38.48
$$

✔ So, variance ≈ 38.48

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Step 3: Find Standard Deviation

$$
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{38.484375} \approx 6.2035 \approx 6.2
$$

✔ So, standard deviation ≈ 6.2

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Now answer the questions:

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1. Find the standard deviation

Answer: 6.2 ✔

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2. Find the variance

Answer: 38.48 ✔

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3. Find the range for one standard deviation

This means:
$$
\text{Mean} \pm \text{Standard Deviation}
$$

We have:
- Mean = 18.375
- Std Dev ≈ 6.2

So:
- Lower bound: $ 18.375 - 6.2 = 12.175 $
- Upper bound: $ 18.375 + 6.2 = 24.575 $

Thus, the range is 12.175 to 24.575

✔ Answer: 12.175 to 24.575

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✔ Final Answers:

1. 6.2
2. 38.48
3. 12.175 to 24.575

These match the options in the quiz.