To calculate the range, variance, and standard deviation of the given data set:

Data set:
29, 17, 15, 34, 24, 299, 23, 17, 20

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1. Range

The range is the difference between the maximum and minimum values.

$\text{Range} = \text{Maximum value} - \text{Minimum value}$ Maximum value = 299
Minimum value = 15

$\text{Range} = 299 - 15 = 284$

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2. Variance

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

Data set:
29, 17, 15, 34, 24, 299, 23, 17, 20

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1. Range

The range is the difference between the maximum and minimum values.

$\text{Range} = \text{Maximum value} - \text{Minimum value}$ Maximum value = 299
Minimum value = 15

$\text{Range} = 299 - 15 = 284$

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2. Variance

To calculate variance, we first need to find the mean (average) of the

$\text{Mean} = \frac{\sum \text{Data values}}{N}$ $\text{Mean} = \frac{29 + 17 + 15 + 34 + 24 + 299 + 23 + 17 + 20}{9} = \frac{478}{9} \approx 53.11$

Now, to calculate the variance, we use the formula:

$\text{Variance} (\sigma^2) = \frac{\sum (\text{Value} - \text{Mean})^2}{N}$

Let’s compute the squared differences from the mean for each data point:

$(29 - 53.11)^2 = (-24.11)^2 = 581.30$ $(17 - 53.11)^2 = (-36.11)^2 = 1304.00$ $(15 - 53.11)^2 = (-38.11)^2 = 1451.98$ $(34 - 53.11)^2 = (-19.11)^2 = 365.19$ $(24 - 53.11)^2 = (-29.11)^2 = 847.47$ $(299 - 53.11)^2 = (245.89)^2 = 60464.32$ $(23 - 53.11)^2 = (-30.11)^2 = 906.61$ $(17 - 53.11)^2 = (-36.11)^2 = 1304.00$ $(20 - 53.11)^2 = (-33.11)^2 = 1096.26$

Now, sum up all the squared differences:

$\sum = 581.30 + 1304.00 + 1451.98 + 365.19 + 847.47 + 60464.32 + 906.61 + 1304.00 + 1096.26 = 68720.13$

Finally, divide by the number of data points (9):

$\text{Variance} = \frac{68720.13}{9} \approx 7635.57$

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3. Standard Deviation

The standard deviation is the square root of the variance:

$\text{Standard deviation} (\sigma) = \sqrt{7635.57} \approx 87.39$

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Summary:

$\text{Range} = 284$ $\text{Variance} \approx 7635.57$ $\text{Standard deviation} \approx 87.39$

Would you like further clarification on any of these steps?

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Here are some related questions you might want to explore:

1. How does removing the outlier (299) affect the variance and standard deviation?
2. What is the difference between population variance and sample variance?
3. How would the range change if you added more data points?
4. Can you explain why the standard deviation is more robust than the range?
5. What is the significance of a high standard deviation in this data set?

Tip: Variance gives a rough idea of how data spreads around the mean, but standard deviation provides a more intuitive sense of this spread because it has the same units as the data.