Sure! Let's break it down into simple steps.
Five Number Summary
The five number summary includes:
- Minimum
- First Quartile (Q1)
- Median
- Third Quartile (Q3)
- Maximum
Step 1: Organize the Data
First, we sort the heights from smallest to largest:
- 63, 66, 66, 70, 75
Step 2: Find the Minimum and Maximum
- Minimum: The smallest number is 63.
- Maximum: The largest number is 75.
Step 3: Find the Median
The median is the middle number. Since there are 5 numbers:
- The median is the third number in the sorted list, which is 66.
Step 4: Find Q1 and Q3
- Q1 (first quartile) is the median of the first half (63, 66). The median of these two numbers is (63 + 66) / 2 = 64.5.
- Q3 (third quartile) is the median of the second half (66, 70, 75). Q3 is the second number in this set, which is 70.
Five Number Summary:
- Minimum: 63
- Q1: 64.5
- Median: 66
- Q3: 70
- Maximum: 75
So, the five number summary is 63, 64.5, 66, 70, 75.
Standard Deviation
To find the standard deviation, we follow these steps:
Step 1: Find the Mean
Add all the heights and divide by how many there are. (63 + 66 + 66 + 70 + 75) / 5 = 68.
Step 2: Calculate Each Deviation from the Mean
Now we subtract the mean from each height and square the result:
- (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
Step 3: Calculate the Average of Those Squared Deviations
Add them up: 25 + 4 + 4 + 4 + 49 = 86. Now, divide by the number of measurements (which is 5) for the variance: Variance = 86 / 5 = 17.2.
Step 4: Take the Square Root of the Variance
Now, we take the square root to find the standard deviation: Standard Deviation = √17.2 ≈ 4.14.
So, the standard deviation of the heights is about 4.14 inches!
Summary:
Five Number Summary: 63, 64.5, 66, 70, 75
Standard Deviation: 4.14 inches