To calculate the five-number summary and the standard deviation of the given heights (66, 70, 63, 75, 66), let's follow these steps:
Five-Number Summary
The five-number summary consists of:
- Minimum
- First Quartile (Q1)
- Median (Q2)
- Third Quartile (Q3)
- Maximum
First, we need to sort the data:
- Sorted data: 63, 66, 66, 70, 75
Now, we can calculate the five-number summary.
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Minimum:
- Minimum = 63
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First Quartile (Q1):
- Q1 is the median of the first half of the data (63, 66).
- Q1 = (63 + 66) / 2 = 64.5
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Median (Q2):
- Median = Middle value = 66 (the third number in the sorted list).
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Third Quartile (Q3):
- Q3 is the median of the second half of the data (66, 70, 75).
- Q3 = 70 (the second number in the second half).
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Maximum:
- Maximum = 75
Five-Number Summary:
- Minimum: 63
- Q1: 64.5
- Median (Q2): 66
- Q3: 70
- Maximum: 75
Standard Deviation
To calculate the standard deviation, we can use the following formula:
\[ \text{Standard Deviation} (s) = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]
Where \( x_i \) is each data point, \( \bar{x} \) is the mean of the data, and \( n \) is the number of data points.
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Calculate the mean (\( \bar{x} \)): \[ \bar{x} = \frac{63 + 66 + 66 + 70 + 75}{5} = \frac{340}{5} = 68 \]
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Calculate each deviation from the mean, square each deviation, and sum them: \[ (63 - 68)^2 = (-5)^2 = 25 \] \[ (66 - 68)^2 = (-2)^2 = 4 \] \[ (66 - 68)^2 = (-2)^2 = 4 \] \[ (70 - 68)^2 = (2)^2 = 4 \] \[ (75 - 68)^2 = (7)^2 = 49 \]
Now summing these squares: \[ 25 + 4 + 4 + 4 + 49 = 86 \]
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Calculate the variance: \[ \text{Variance} (s^2) = \frac{86}{5-1} = \frac{86}{4} = 21.5 \]
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Calculate the standard deviation: \[ s = \sqrt{21.5} \approx 4.64 \]
Summary of Results
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Five-number summary:
- Minimum: 63
- Q1: 64.5
- Median: 66
- Q3: 70
- Maximum: 75
-
Standard Deviation: Approximately \( 4.64 \)