Math Problem Statement
Solution
Let's walk through the solution based on the provided steps.
Step 1: Calculate the sample mean ()
The data set is: .
The formula for the sample mean is: Where (since there are 3 values) and is the sum of the data points.
So, the sample mean is 64.
Step 2: Calculate the deviations and squares of the deviations
For each data point, we calculate the deviation from the mean () and the square of the deviation:
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Deviation of 58: Square of deviation:
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Deviation of 60: Square of deviation:
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Deviation of 74: Square of deviation:
Step 3: Calculate the sample variance and sample standard deviation
The formula for the sample variance () is: Where and is the sum of the squared deviations.
So, the sample variance is 76.
Now, the formula for the sample standard deviation () is:
Thus, the sample standard deviation is approximately 8.7.
Do you have any questions or need further details? Here are 5 related questions to expand your understanding:
- How would the sample mean change if another data point, 68, was added to the set?
- What is the difference between sample standard deviation and population standard deviation?
- How would the sample variance be affected if the data set consisted of larger numbers?
- What is the significance of squaring the deviations when calculating standard deviation?
- How can standard deviation help interpret data spread in real-world applications?
Tip: Squaring deviations ensures that both positive and negative deviations contribute equally to the variance.
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Math Problem Analysis
Mathematical Concepts
Statistics
Sample Standard Deviation
Variance
Formulas
Sample mean formula: \(\bar{x} = \frac{\sum{x}}{n}\)
Sample variance formula: \(s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}\)
Sample standard deviation formula: \(s = \sqrt{s^2}\)
Theorems
Standard deviation is a measure of data spread around the mean.
Suitable Grade Level
High School (Grades 9-12)
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