To solve this problem, we need to find two things based on the list of temperatures provided:

1. Range of the temperatures: The range is calculated as the difference between the highest and lowest values in the dataset.

   • Given temperatures: $36, 52, 63, 78, 76, 80, 82, 93, 96, 90, 75, 52$.
   • Highest temperature: $96$
   • Lowest temperature: $36$
   • Range = $96 - 36 = 60$

   So, the range is 60 degrees.

2. Sample Standard Deviation: This requires calculating the standard deviation for a sample. Here’s the formula:

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

   where:

   • $x_i$ represents each temperature,
   • $\bar{x}$ is the mean of the temperatures,
   • $n$ is the number of temperatures.

   Let's calculate it step-by-step:

   1. Calculate the mean ($\bar{x}$): $\bar{x} = \frac{36 + 52 + 63 + 78 + 76 + 80 + 82 + 93 + 96 + 90 + 75 + 52}{12}$ $\bar{x} = \frac{873}{12} \approx 72.75$

   2. Find the squared differences from the mean and sum them.

   3. Divide by $n - 1$ (which is 11 here) and take the square root.

I’ll perform these calculations for the standard deviation.import numpy as np

Given temperatures

temperatures = [36, 52, 63, 78, 76, 80, 82, 93, 96, 90, 75, 52]

Mean calculation

mean_temp = np.mean(temperatures)

Sample standard deviation calculation

std_dev = np.std(temperatures, ddof=1) # ddof=1 for sample standard deviation

mean_temp, std_dev(72.75, 18.429842400549468)(72.75, 18.429842400549468)successThe calculations are as follows:

• Mean temperature ($\bar{x}$) ≈ 72.75
• Sample Standard Deviation ≈ 18.43

So, the range is 60 degrees, and the sample standard deviation is approximately 18.43 degrees.

Would you like further details on these calculations, or do you have any other questions?

Here are some related questions you might find helpful:

1. How does the standard deviation differ from the range in measuring data spread?
2. What are the steps involved in calculating the sample variance?
3. Why is the sample standard deviation divided by $n - 1$ instead of $n$?
4. How would the values change if these were population data rather than a sample?
5. What other measures of variability could be useful for this data set?

Tip: Remember that standard deviation gives you insight into how spread out the data points are around the mean. A higher standard deviation indicates more variability.