Let's calculate the mean, range, and standard deviation for each of the three samples.

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Formulas Used:

1. Mean:
   $\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}$

2. Range:
   $\text{Range} = \text{Maximum value} - \text{Minimum value}$

3. Standard Deviation:
   $\sigma = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{N}}$
   Where:

• $x_i$: Each data point
• $\bar{x}$: Mean of the data
• $N$: Number of data points

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Sample A: $25, 33, 41, 49, 57, 65, 73$

1. Mean:
   $\text{Mean} = \frac{25 + 33 + 41 + 49 + 57 + 65 + 73}{7} = \frac{343}{7} = 49$

2. Range:
   $\text{Range} = 73 - 25 = 48$

3. Standard Deviation:

(x_i - \bar{x})^2 &= (25-49)^2 + (33-49)^2 + (41-49)^2 + (49-49)^2 + (57-49)^2 + (65-49)^2 + (73-49)^2 \\ &= 576 + 256 + 64 + 0 + 64 + 256 + 576 = 1792 \\ \text{Variance} &= \frac{1792}{7} = 256 \\ \text{Standard Deviation} &= \sqrt{256} = 16 \end{aligned}$$ --- ### **Sample B**: $$25, 29, 33, 49, 65, 69, 73$$ 1. **Mean**: $$\text{Mean} = \frac{25 + 29 + 33 + 49 + 65 + 69 + 73}{7} = \frac{343}{7} = 49$$ 2. **Range**: $$\text{Range} = 73 - 25 = 48$$ 3. **Standard Deviation**: $$\begin{aligned} (x_i - \bar{x})^2 &= (25-49)^2 + (29-49)^2 + (33-49)^2 + (49-49)^2 + (65-49)^2 + (69-49)^2 + (73-49)^2 \\ &= 576 + 400 + 256 + 0 + 256 + 400 + 576 = 2464 \\ \text{Variance} &= \frac{2464}{7} = 352 \\ \text{Standard Deviation} &= \sqrt{352} \approx 18.76 \end{aligned}$$ --- ### **Sample C**: $$25, 25, 25, 49, 73, 73, 73$$ 1. **Mean**: $$\text{Mean} = \frac{25 + 25 + 25 + 49 + 73 + 73 + 73}{7} = \frac{343}{7} = 49$$ 2. **Range**: $$\text{Range} = 73 - 25 = 48$$ 3. **Standard Deviation**: $$\begin{aligned} (x_i - \bar{x})^2 &= (25-49)^2 + (25-49)^2 + (25-49)^2 + (49-49)^2 + (73-49)^2 + (73-49)^2 + (73-49)^2 \\ &= 576 + 576 + 576 + 0 + 576 + 576 + 576 = 3456 \\ \text{Variance} &= \frac{3456}{7} = 493.71 \\ \text{Standard Deviation} &= \sqrt{493.71} \approx 22.22 \end{aligned}$$ --- ### **Summary**: | Sample | Mean | Range | Standard Deviation | |--------|------|-------|---------------------| | A | 49 | 48 | 16 | | B | 49 | 48 | 18.76 | | C | 49 | 48 | 22.22 | --- Would you like any of these steps explained in further detail or assistance with a related question? #### Questions: 1. Why do Sample B and C have a higher standard deviation despite having the same mean and range? 2. How does the presence of repeated values (as in Sample C) affect the variance? 3. What are practical applications of understanding standard deviation in real-world scenarios? 4. Could this data be represented graphically, and if so, what graph type would work best? 5. How would these calculations differ for a population vs. a sample dataset? #### Tip: When calculating standard deviation, always ensure you differentiate between a sample and population formula ($$N-1$$ for samples).