Math Problem Statement
Compute the (sample) variance and standard deviation of the data sample. (Round your answers to two decimal places.) 3.8, −3.5, 3.9, −0.2, −0.2 variance Incorrect: Your answer is incorrect. standard deviation
Solution
To compute the sample variance and standard deviation, let's go through the steps systematically:
Step 1: Calculate the Mean
First, we find the mean of the data sample.
Given dataTo compute the sample variance and standard deviation, let's go through the steps systematically:
Step 1: Calculate the Mean
First, we find the mean of the data sample.
Given
Step 2: Calculate the Squared Deviations
Now, we calculate each value's deviation from the mean, square those deviations, and sum them up.
Sum of squared deviations:
Step 3: Calculate the Sample Variance
To find the sample variance , divide the sum of squared deviations by , where is the sample size (in this case, 5):
Rounding to two decimal places, the variance is:
Step 4: Calculate the Sample Standard Deviation
The sample standard deviation is the square root of the variance:
Rounding to two decimal places, the standard deviation is:
Final Answers:
- Variance:
- Standard Deviation:
Would you like more details or have any questions?
Here are five related questions:
- How is sample variance different from population variance?
- Why do we divide by in sample variance calculations?
- What does a high standard deviation indicate about data dispersion?
- How does rounding affect the accuracy of variance and standard deviation?
- Can we calculate the variance and standard deviation without knowing the mean?
Tip: For a larger dataset, consider using statistical software or a calculator to avoid manual calculation errors.
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Math Problem Analysis
Mathematical Concepts
Statistics
Variance
Standard Deviation
Data Analysis
Formulas
Mean: \( \bar{x} = \frac{\sum x}{n} \)
Sample Variance: \( s^2 = \frac{\sum (x - \bar{x})^2}{n - 1} \)
Sample Standard Deviation: \( s = \sqrt{s^2} \)
Theorems
The formula for sample variance and standard deviation in statistics
Suitable Grade Level
Grades 10-12
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