Consider the following data: 1,6,7,3,5,2 Step 1 of 3: Calculate the value of the sample variance. Round your answer to one decimal place. Step 2 of 3: Calculate the value of the sample standard deviation. Round your answer to one decimal place. Step 3 of 3: Calculate the value of the range
The Correct Answer and Explanation is:
Step 1: Calculate the Sample Variance
To calculate the sample variance, we follow these steps:
- Find the mean of the data.
- Subtract the mean from each data point and square the result.
- Sum up all the squared differences.
- Divide by (n – 1), where n is the number of data points.
Given data: 1, 6, 7, 3, 5, 2
Step 1.1: Find the mean
Mean=1+6+7+3+5+26=246=4\text{Mean} = \frac{1 + 6 + 7 + 3 + 5 + 2}{6} = \frac{24}{6} = 4Mean=61+6+7+3+5+2=624=4
Step 1.2: Calculate the squared differences from the mean
- (1 – 4)² = (-3)² = 9
- (6 – 4)² = (2)² = 4
- (7 – 4)² = (3)² = 9
- (3 – 4)² = (-1)² = 1
- (5 – 4)² = (1)² = 1
- (2 – 4)² = (-2)² = 4
Step 1.3: Sum of the squared differences
9+4+9+1+1+4=289 + 4 + 9 + 1 + 1 + 4 = 289+4+9+1+1+4=28
Step 1.4: Calculate the sample variance
Sample Variance=286−1=285=5.6\text{Sample Variance} = \frac{28}{6 – 1} = \frac{28}{5} = 5.6Sample Variance=6−128=528=5.6
So, the sample variance is 5.6 (rounded to one decimal place).
Step 2: Calculate the Sample Standard Deviation
The sample standard deviation is the square root of the sample variance: Sample Standard Deviation=5.6≈2.37\text{Sample Standard Deviation} = \sqrt{5.6} \approx 2.37Sample Standard Deviation=5.6≈2.37
Rounding to one decimal place, the sample standard deviation is 2.4.
Step 3: Calculate the Range
The range is simply the difference between the maximum and minimum values in the dataset:
- Maximum value = 7
- Minimum value = 1
Range=7−1=6\text{Range} = 7 – 1 = 6Range=7−1=6
So, the range is 6.
Summary:
- Sample Variance = 5.6
- Sample Standard Deviation = 2.4
- Range = 6
These values provide insights into the spread and variability of the dataset. The sample variance quantifies how spread out the data points are from the mean, the standard deviation gives a more intuitive measure of spread in the same units as the original data, and the range offers a simple but effective way of describing the extent between the smallest and largest data points.
