To solve for the range and the sample standard deviation, we proceed as follows:

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1. Range

The range is calculated as:

$\text{Range} = \text{Maximum value} - \text{Minimum value}$

From the data, the maximum value is $10$ and the minimum value is $1$.

$\text{Range} = 10 - 1 = 9$

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2. Sample Standard Deviation

The formula for the sample standard deviation is:

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$

Where:

• $x_i$ are the data points,
• $\bar{x}$ is the sample mean,
• $n$ is the number of data points.

Step 1: Find the Mean ($\bar{x}$)

Sum all the values:

$\text{Sum} = 4 + 10 + 7 + 6 + 2 + 3 + 4 + 10 + 7 + 4 + 1 + 1 + 7 + 3 + 3 + 1 + 9 + 7 + 4 + 7 + 1 + 9 + 8 + 4$

$\text{Sum} = 124$

Number of values ($n$) is $24$, so the mean is:

$\bar{x} = \frac{\text{Sum}}{n} = \frac{124}{24} \approx 5.1667$

Step 2: Compute $(x_i - \bar{x})^2$

For each data point $x_i$, calculate the squared deviation from the mean. The values are:

$x_i: \{4, 10, 7, 6, 2, 3, 4, 10, 7, 4, 1, 1, 7, 3, 3, 1, 9, 7, 4, 7, 1, 9, 8, 4\}$

Let me calculate this systematically.import numpy as np

Data points

data = [4, 10, 7, 6, 2, 3, 4, 10, 7, 4, 1, 1, 7, 3, 3, 1, 9, 7, 4, 7, 1, 9, 8, 4]

Number of data points (n)

n = len(data)

Mean

mean = np.mean(data)

Deviations squared

squared_deviations = [(x - mean) ** 2 for x in data]

Sample variance

sample_variance = sum(squared_deviations) / (n - 1)

Sample standard deviation

sample_std_dev = np.sqrt(sample_variance)

mean, sample_std_dev(5.083333333333333, 2.962323803847341)(5.083333333333333, 2.962323803847341)successThe mean ($\bar{x}$) of the data is approximately $5.0833$.

The sample standard deviation is:

$s \approx 2.9623$

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Final Results

1. Range = $9$
2. Sample Standard Deviation = $2.9623$ (rounded to 4 decimal places)

Let me know if you'd like the step-by-step squared deviation calculations or have further questions!

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Related Questions:

1. How is the range influenced by outliers in a dataset?
2. What does the standard deviation tell us about the variability of the data?
3. How does the sample standard deviation differ from the population standard deviation?
4. Can you calculate the variance and interpret its significance here?
5. What happens to the mean and standard deviation if an extreme outlier is added?

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Tip: Always verify data entry carefully when calculating statistics to avoid errors in large datasets!