To calculate the five-number summary and the standard deviation of the heights of the starting 5 players (66, 70, 63, 75, 66), we can follow these steps.
Five-Number Summary
The five-number summary consists of the following:
- Minimum
- First Quartile (Q1)
- Median (Q2)
- Third Quartile (Q3)
- Maximum
Step 1: Sort the Data
First, we need to arrange the data in ascending order: \[ 63, 66, 66, 70, 75 \]
Step 2: Find the Minimum and Maximum
- Minimum: 63
- Maximum: 75
Step 3: Find the Median (Q2)
To find the median, we can look for the middle value. Since we have 5 values, the median is the 3rd value in the ordered list:
- Median (Q2): 66
Step 4: Find Q1 and Q3
-
Q1: This is the median of the first half of the data (63, 66). So Q1 is the average of 63 and 66: \[ Q1 = \frac{63 + 66}{2} = \frac{129}{2} = 64.5 \]
-
Q3: This is the median of the second half of the data (66, 70, 75). Since there are three numbers, the Q3 is the second value:
- Q3: 70
Five-Number Summary:
- Minimum: 63
- Q1: 64.5
- Median (Q2): 66
- Q3: 70
- Maximum: 75
Standard Deviation
To find the standard deviation, we can use the following formula: \[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}} \]
Where:
- \(x_i\) = each data point
- \(\mu\) = mean of the data points
- \(n\) = number of data points
Step 1: Calculate the Mean
\[ \mu = \frac{63 + 66 + 66 + 70 + 75}{5} = \frac{340}{5} = 68 \]
Step 2: Calculate the Squared Differences from the Mean
\[ \begin{align*} (63 - 68)^2 & = (-5)^2 = 25 \ (66 - 68)^2 & = (-2)^2 = 4 \ (66 - 68)^2 & = (-2)^2 = 4 \ (70 - 68)^2 & = (2)^2 = 4 \ (75 - 68)^2 & = (7)^2 = 49 \ \end{align*} \]
Step 3: Sum the Squared Differences
\[ \sum (x_i - \mu)^2 = 25 + 4 + 4 + 4 + 49 = 86 \]
Step 4: Calculate the Variance
\[ \sigma^2 = \frac{86}{5} = 17.2 \]
Step 5: Calculate the Standard Deviation
\[ \sigma = \sqrt{17.2} \approx 4.14 \]
Summary of Results
- Five-Number Summary: \(63, 64.5, 66, 70, 75\)
- Standard Deviation: Approximately \(4.14\) (rounded to two decimal places)