To analyze the heights of the starting 5 players on the intramural basketball team (66, 70, 63, 75, 66), we will calculate both the five-number summary and the standard deviation.
Step 1: Five-Number Summary
The five-number summary consists of the following five statistics:
- Minimum
- First Quartile (Q1)
- Median (Q2)
- Third Quartile (Q3)
- Maximum
1. Sort the data:
First, we sort the data in ascending order: 63, 66, 66, 70, 75
2. Calculate the minimum and maximum:
- Minimum = 63
- Maximum = 75
3. Calculate the median (Q2):
The median is the middle value of the sorted data. Since we have 5 data points, the median is the 3rd value:
- Median (Q2) = 66
4. Calculate Q1 (First Quartile):
Q1 is the median of the lower half of the data: Lower half = 63, 66 The median of this set (the average of these two values) is:
- Q1 = (63 + 66) / 2 = 64.5
5. Calculate Q3 (Third Quartile):
Q3 is the median of the upper half of the data: Upper half = 66, 70, 75 The median of this set is:
- Q3 = 70
Summary of Five-Number Summary:
- Minimum = 63
- Q1 = 64.5
- Median (Q2) = 66
- Q3 = 70
- Maximum = 75
So, the five-number summary is: (63, 64.5, 66, 70, 75)
Step 2: Standard Deviation
To calculate the standard deviation, we can follow these steps:
1. Calculate the mean:
\[ \text{Mean} = \frac{(63 + 66 + 66 + 70 + 75)}{5} = \frac{340}{5} = 68 \]
2. Calculate the squared differences from the mean:
- For 63: \( (63 - 68)^2 = (-5)^2 = 25 \)
- For 66: \( (66 - 68)^2 = (-2)^2 = 4 \)
- For 66: \( (66 - 68)^2 = (-2)^2 = 4 \)
- For 70: \( (70 - 68)^2 = (2)^2 = 4 \)
- For 75: \( (75 - 68)^2 = (7)^2 = 49 \)
3. Compute the variance:
\[ \text{Variance} = \frac{(25 + 4 + 4 + 4 + 49)}{5} = \frac{86}{5} = 17.2 \]
4. Compute the standard deviation:
\[ \text{Standard Deviation} = \sqrt{17.2} \approx 4.14 \]
Conclusion
- Five-Number Summary: (63, 64.5, 66, 70, 75)
- Standard Deviation: approximately 4.14 (rounded to two decimal places).