To calculate the five-number summary and the standard deviation of the heights (66, 70, 63, 75, 66), follow these steps:
Five-number summary:
- Minimum: The smallest value in the dataset.
- Min = 63
- First Quartile (Q1): The median of the first half of the data.
- First half: 63, 66 (Median of Q1 = (63 + 66)/2 = 64.5)
- Median (Q2): The median of the dataset.
- Ordered data: 63, 66, 66, 70, 75 (Median = 66)
- Third Quartile (Q3): The median of the second half of the data.
- Second half: 66, 70, 75 (Median of Q3 = 70)
- Maximum: The largest value in the dataset.
- Max = 75
Five-number summary: Min = 63, Q1 = 64.5, Median = 66, Q3 = 70, Max = 75
Standard Deviation:
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Calculate the mean: \[ \text{Mean} = \frac{66 + 70 + 63 + 75 + 66}{5} = \frac{340}{5} = 68 \]
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Calculate the variance:
- Find the squared differences from the mean: \[ (66 - 68)^2 = 4, \quad (70 - 68)^2 = 4, \quad (63 - 68)^2 = 25, \quad (75 - 68)^2 = 49, \quad (66 - 68)^2 = 4 \]
- Calculate the average of these squared differences: \[ \text{Variance} = \frac{4 + 4 + 25 + 49 + 4}{5} = \frac{86}{5} = 17.2 \]
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Calculate the standard deviation: \[ \text{Standard Deviation} = \sqrt{17.2} \approx 4.14 \]
Summary of Calculations:
- Five-number summary: 63, 64.5, 66, 70, 75
- Standard deviation: Approximately 4.14
Answer in one simple short sentence: The five-number summary is 63, 64.5, 66, 70, 75, and the standard deviation is about 4.14.