I see you've uploaded an image with four data sets, and you want to find the range and standard deviation for each. Let me start by calculating these for each of the sets.

Steps:

1. Range: The difference between the maximum and minimum values of the data set.
2. Standard Deviation: $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$ where $\mu$ is the mean, $x_i$ are the data points, and $N$ is the number of data points.

Let's go through the calculations.

Data Set 1: $22, 18, 19, 25, 27, 21, 24$

1. Range:
   Max = 27, Min = 18
   Range = $27 - 18 = 9$

2. Mean:
   $\mu = \frac{22 + 18 + 19 + 25 + 27 + 21 + 24}{7} = \frac{156}{7} \approx 22.29$

3. Standard Deviation: $\sigma \approx \sqrt{\frac{(22-22.29)^2 + (18-22.29)^2 + (19-22.29)^2 + (25-22.29)^2 + (27-22.29)^2 + (21-22.29)^2 + (24-22.29)^2}{7}}$

I'll compute this for you.

Data Set 2: $38, 46, 37, 42, 39, 40, 48, 42$

1. Range:
   Max = 48, Min = 37
   Range = $48 - 37 = 11$

2. Mean:
   $\mu = \frac{38 + 46 + 37 + 42 + 39 + 40 + 48 + 42}{8} = \frac{332}{8} = 41.5$

3. Standard Deviation:
   $\sigma \approx \sqrt{\frac{(38-41.5)^2 + (46-41.5)^2 + \dots + (42-41.5)^2}{8}}$

I'll compute this as well.

Data Set 3: $8.4, 7.7, 8.6, 7.5, 8.9, 7.8, 8.6, 9.1, 8.0$

1. Range:
   Max = 9.1, Min = 7.5
   Range = $9.1 - 7.5 = 1.6$

2. Mean:
   $\mu = \frac{8.4 + 7.7 + 8.6 + 7.5 + 8.9 + 7.8 + 8.6 + 9.1 + 8.0}{9} = \frac{74.6}{9} \approx 8.29$

3. Standard Deviation:
   $\sigma \approx \sqrt{\frac{(8.4 - 8.29)^2 + (7.7 - 8.29)^2 + \dots + (8.0 - 8.29)^2}{9}}$

I'll calculate this as well.

Data Set 4: $1.25, 3.69, 5.67, 4.89, 0.12, 4.35, 2.78$

1. Range:
   Max = 5.67, Min = 0.12
   Range = $5.67 - 0.12 = 5.55$

2. Mean:
   $\mu = \frac{1.25 + 3.69 + 5.67 + 4.89 + 0.12 + 4.35 + 2.78}{7} = \frac{22.75}{7} \approx 3.25$

3. Standard Deviation: $\sigma \approx \sqrt{\frac{(1.25 - 3.25)^2 + (3.69 - 3.25)^2 + \dots + (2.78 - 3.25)^2}{7}}$

Let me compute the standard deviations for all four sets and provide the final answers.Here are the results for the range and standard deviation of each data set:

Data Set 1: $22, 18, 19, 25, 27, 21, 24$

• Range: 9
• Standard Deviation: 3.01

Data Set 2: $38, 46, 37, 42, 39, 40, 48, 42$

• Range: 11
• Standard Deviation: 3.61

Data Set 3: $8.4, 7.7, 8.6, 7.5, 8.9, 7.8, 8.6, 9.1, 8.0$

• Range: 1.6
• Standard Deviation: 0.53

Data Set 4: $1.25, 3.69, 5.67, 4.89, 0.12, 4.35, 2.78$

• Range: 5.55
• Standard Deviation: 1.85

Do you want further details on any of these? Feel free to ask any questions!

Additional Questions:

1. How is the range of a data set useful in understanding its distribution?
2. What does a higher standard deviation indicate about the spread of data?
3. How can we interpret the standard deviation for each data set here?
4. Would the results change significantly if the data sets had more values?
5. Can you explain the difference between variance and standard deviation?

Tip:

When working with large data sets, standard deviation helps you understand how spread out the data is from the mean, with smaller values indicating closer clustering around the mean.