Range, Variance, And Standard Deviation - Worksheet - Free Printable

Problem Explanation and Solution

The worksheet asks us to calculate the range, variance, and standard deviation for various data sets. Let's solve each problem step by step.

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#### 1. Find the range of the data set below.
Data set: \( 1, 22, 18, -8, -6, -6, -4 \)

- Range: The range is the difference between the maximum and minimum values in the data set.
- Maximum value: \( 22 \)
- Minimum value: \( -8 \)
- Range = Maximum - Minimum = \( 22 - (-8) = 22 + 8 = 30 \)

Answer: \( \boxed{30} \)

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#### 2. Find the range of the data set below.
Data set: \( 10, -11, 24, 25, 25, 10 \)

- Range:
- Maximum value: \( 25 \)
- Minimum value: \( -11 \)
- Range = Maximum - Minimum = \( 25 - (-11) = 25 + 11 = 36 \)

Answer: \( \boxed{36} \)

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#### 3. Find the range of the data set below.
Data set: \( 14, 18, 9, -7, -18, 20 \)

- Range:
- Maximum value: \( 20 \)
- Minimum value: \( -18 \)
- Range = Maximum - Minimum = \( 20 - (-18) = 20 + 18 = 38 \)

Answer: \( \boxed{38} \)

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#### 4. Find the variance of the data set below. Round your answer to the nearest hundredths place if applicable.
Data set: \( 3, 6, 4, 7 \)

- Variance: Variance (\( \sigma^2 \)) is calculated using the formula:
\[
\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}
\]
where \( \mu \) is the mean of the data set, \( x_i \) are the individual data points, and \( n \) is the number of data points.

1. Calculate the mean (\( \mu \)):
\[
\mu = \frac{3 + 6 + 4 + 7}{4} = \frac{20}{4} = 5
\]

2. Calculate the squared differences from the mean:
- For \( x_1 = 3 \): \( (3 - 5)^2 = (-2)^2 = 4 \)
- For \( x_2 = 6 \): \( (6 - 5)^2 = (1)^2 = 1 \)
- For \( x_3 = 4 \): \( (4 - 5)^2 = (-1)^2 = 1 \)
- For \( x_4 = 7 \): \( (7 - 5)^2 = (2)^2 = 4 \)

3. Sum the squared differences:
\[
\sum (x_i - \mu)^2 = 4 + 1 + 1 + 4 = 10
\]

4. Calculate the variance:
\[
\sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = \frac{10}{4} = 2.5
\]

Answer: \( \boxed{2.50} \)

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#### 5. Find the variance of the data set below. Round your answer to the nearest hundredths place if applicable.
Data set: \( 3, 3, 4, 14 \)

1. Calculate the mean (\( \mu \)):
\[
\mu = \frac{3 + 3 + 4 + 14}{4} = \frac{24}{4} = 6
\]

2. Calculate the squared differences from the mean:
- For \( x_1 = 3 \): \( (3 - 6)^2 = (-3)^2 = 9 \)
- For \( x_2 = 3 \): \( (3 - 6)^2 = (-3)^2 = 9 \)
- For \( x_3 = 4 \): \( (4 - 6)^2 = (-2)^2 = 4 \)
- For \( x_4 = 14 \): \( (14 - 6)^2 = (8)^2 = 64 \)

3. Sum the squared differences:
\[
\sum (x_i - \mu)^2 = 9 + 9 + 4 + 64 = 86
\]

4. Calculate the variance:
\[
\sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = \frac{86}{4} = 21.5
\]

Answer: \( \boxed{21.50} \)

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#### 6. Find the variance of the data set below. Round your answer to the nearest hundredths place if applicable.
Data set: \( 3, 6, 6 \)

1. Calculate the mean (\( \mu \)):
\[
\mu = \frac{3 + 6 + 6}{3} = \frac{15}{3} = 5
\]

2. Calculate the squared differences from the mean:
- For \( x_1 = 3 \): \( (3 - 5)^2 = (-2)^2 = 4 \)
- For \( x_2 = 6 \): \( (6 - 5)^2 = (1)^2 = 1 \)
- For \( x_3 = 6 \): \( (6 - 5)^2 = (1)^2 = 1 \)

3. Sum the squared differences:
\[
\sum (x_i - \mu)^2 = 4 + 1 + 1 = 6
\]

4. Calculate the variance:
\[
\sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = \frac{6}{3} = 2
\]

Answer: \( \boxed{2.00} \)

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Final Answers:

1. \( \boxed{30} \)
2. \( \boxed{36} \)
3. \( \boxed{38} \)
4. \( \boxed{2.50} \)
5. \( \boxed{21.50} \)
6. \( \boxed{2.00} \)