To find the five-number summary and the standard deviation of the heights of the basketball players, we first need to organize the data and perform the calculations.
Data:
Heights (in inches): 66, 70, 63, 75, 66
Step 1: Sort the Data
First, we will sort the data in ascending order: 63, 66, 66, 70, 75
Step 2: Five-Number Summary
The five-number summary includes:
- Minimum (min): The smallest value in the data set.
- First quartile (Q1): The median of the first half of the data.
- Median (Q2): The median of the entire data set.
- Third quartile (Q3): The median of the second half of the data.
- Maximum (max): The largest value in the data set.
Calculations:
- Minimum: 63
- Maximum: 75
- Median (Q2): The median of [63, 66, 66, 70, 75] is the middle value, which is 66.
- First Quartile (Q1): The first half is [63, 66], and the median of that is \( \frac{63 + 66}{2} = 64.5 \).
- Third Quartile (Q3): The second half is [66, 70, 75], and the median is 70.
Five-Number Summary:
- Minimum: 63
- Q1: 64.5
- Median (Q2): 66
- Q3: 70
- Maximum: 75
Step 3: Standard Deviation
To calculate the standard deviation, we'll follow these steps:
- Calculate the mean (average).
- Calculate the variance (average of the squared differences from the Mean).
- Take the square root of the variance to find the standard deviation.
Calculations:
-
Mean: \[ \text{Mean} = \frac{63 + 66 + 66 + 70 + 75}{5} = \frac{340}{5} = 68 \]
-
Variance: \[ \text{Variance} = \frac{(63 - 68)^2 + (66 - 68)^2 + (66 - 68)^2 + (70 - 68)^2 + (75 - 68)^2}{5} \] \[ = \frac{(-5)^2 + (-2)^2 + (-2)^2 + (2)^2 + (7)^2}{5} \] \[ = \frac{25 + 4 + 4 + 4 + 49}{5} = \frac{86}{5} = 17.2 \]
-
Standard Deviation: \[ \text{Standard Deviation} = \sqrt{17.2} \approx 4.14 \]
Final Results
- Five-Number Summary: Minimum = 63, Q1 = 64.5, Median (Q2) = 66, Q3 = 70, Maximum = 75
- Standard Deviation: Approximately 4.14