Answer:

To find the range, variance, and standard deviation, we first need to calculate the mean (average) of the given set.

1. Calculate the mean:

\[ \text{Mean} = \frac{18 + 19 + 25 + 22 + 21 + 19 + 20 + 27}{8} = \frac{171}{8} = 21.375 \]

2. Calculate the squared differences from the mean for each number:

\[ (18 - 21.375)^2, (19 - 21.375)^2, (25 - 21.375)^2, (22 - 21.375)^2, (21 - 21.375)^2, (19 - 21.375)^2, (20 - 21.375)^2, (27 - 21.375)^2 \]

3. Calculate the variance:

\[ \text{Variance} = \frac{\text{Sum of squared differences}}{\text{Number of observations}} \]

\[ \text{Variance} = \frac{153.875}{8} = 19.234375 \]

4. Calculate the standard deviation:

\[ \text{Standard Deviation} = \sqrt{\text{Variance}} \]

\[ \text{Standard Deviation} \approx \sqrt{19.234375} \approx 4.38 \]

5. Calculate the range:

\[ \text{Range} = \text{Maximum value} - \text{Minimum value} \]

\[ \text{Range} = 27 - 18 = 9 \]

So, for the given set, the range is 9, the variance is approximately 19.23, and the standard deviation is approximately 4.38.

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