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Let's go through the problem step-by-step to find the range and standard deviation of the given data set.

### 1. **Finding the Range:**
The range is the difference between the maximum and minimum values in the data set.

- First, identify the maximum and minimum values:
  - **Maximum value:** 81
  - **Minimum value:** 14

- Calculate the range:
  \[
  \text{Range} = \text{Maximum} - \text{Minimum} = 81 - 14 = 67
  \]

### 2. **Finding the Standard Deviation:**
The standard deviation measures the spread of the data around the mean. Here's how to calculate it:

1. **Find the mean (average):**
   - Sum of all data points:
   \[
   37 + 51 + 86 + 44 + 59 + 85 + 81 + 60 + 30 + 35 + 46 + 26 + 62 + 50 + 56 + 66 + 61 + 65 + 82 + 41 + 88 + 76 + 92 + 14 = 1282
   \]
   - Number of data points (n): 24
   - Mean (\(\mu\)):
   \[
   \mu = \frac{\sum X}{n} = \frac{1282}{24} \approx 53.42
   \]

2. **Calculate each data point's deviation from the mean, square it, and sum those squared deviations:**
   - Use the formula \((x_i - \mu)^2\) for each data point \(x_i\):
   \[
   \sum (x_i - \mu)^2 \approx 27635.48
   \]

3. **Find the variance (sample standard deviation):**
   - Variance (\(s^2\)) for a sample:
   \[
   s^2 = \frac{\sum (x_i - \mu)^2}{n-1} = \frac{27635.48}{23} \approx 1201.54
   \]

4. **Take the square root of the variance to find the standard deviation:**
   \[
   s = \sqrt{1201.54} \approx 34.67
   \]

### Summary of Results:
- **Range:** 67
- **Standard Deviation:** 34.67 (rounded to two decimal places)

Would you like a more detailed explanation of any part of this calculation? 

Here are 5 related questions that might help expand your understanding:
1. What does the standard deviation tell us about the data set?
2. How would the range and standard deviation change if an outlier were added to the data?
3. Why do we divide by \(n-1\) instead of \(n\) when calculating the sample standard deviation?
4. How do the mean and median compare for this data set?
5. What would the data look like if it had a smaller standard deviation?

**Tip:** When calculating the standard deviation, always ensure you're consistent with whether you're working with a sample or a population, as the formulas differ slightly.

Using your calculator, find the range and standard deviation, rounded to two decimal places. The table below gives the number of hours spent watching TV last week by a sample of 24 children. Data: 37, 51, 86, 44, 59, 85, 81, 60, 30, 35, 46, 26, 62, 50, 56, 66, 61, 65, 82, 41, 88, 76, 92, 14.

Learn how to calculate the range and standard deviation for a data set of 24 children's TV watching hours. This problem covers basic statistics concepts and includes detailed calculations of the range and sample standard deviation.

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