To find the five-number summary and the standard deviation of the given data, we will follow these steps:
1. Organize the Data
First, let's arrange the heights in ascending order:
- 63, 66, 66, 70, 75
2. Five-Number Summary
The five-number summary consists of the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.
- Minimum (Min): The smallest value in the data set.
- Min = 63
- Maximum (Max): The largest value in the data set.
- Max = 75
- Median (Q2): The middle value.
- For an odd number of observations, the median is the middle number. Here, the middle value is the third number.
- Median = 66
- First Quartile (Q1): The median of the first half of the data (lower half).
- The first half is 63, 66. The median of these values is (63 + 66)/2 = 64.5.
- Third Quartile (Q3): The median of the second half of the data (upper half).
- The second half is 66, 70, 75. The median is 70.
So, the five-number summary is:
- Minimum: 63
- Q1: 64.5
- Median (Q2): 66
- Q3: 70
- Maximum: 75
Five-Number Summary:
\[ 63, 64.5, 66, 70, 75 \]
3. Standard Deviation
To compute the standard deviation, we will follow these steps:
Step 1: Calculate the mean
\[ \text{Mean} = \frac{63 + 66 + 66 + 70 + 75}{5} = \frac{340}{5} = 68 \]
Step 2: Calculate the variance
-
Find the squared differences from the mean:
- \((63 - 68)^2 = (-5)^2 = 25\)
- \((66 - 68)^2 = (-2)^2 = 4\)
- \((66 - 68)^2 = (-2)^2 = 4\)
- \((70 - 68)^2 = (2)^2 = 4\)
- \((75 - 68)^2 = (7)^2 = 49\)
-
Calculate the variance: \[ \text{Variance} = \frac{(25 + 4 + 4 + 4 + 49)}{n} = \frac{86}{5} = 17.2 \]
Step 3: Calculate the standard deviation
\[ \text{Standard Deviation} = \sqrt{Variance} = \sqrt{17.2} \approx 4.14 \]
Final Results:
- Five-Number Summary: 63, 64.5, 66, 70, 75
- Standard Deviation: Approximately 4.14