Math Problem Statement
Solution
Let's go through the problem step-by-step to find the range and standard deviation of the given data set.
1. Finding the Range:
The range is the difference between the maximum and minimum values in the data set.
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First, identify the maximum and minimum values:
- Maximum value: 81
- Minimum value: 14
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Calculate the range:
2. Finding the Standard Deviation:
The standard deviation measures the spread of the data around the mean. Here's how to calculate it:
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Find the mean (average):
- Sum of all data points:
- Number of data points (n): 24
- Mean ():
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Calculate each data point's deviation from the mean, square it, and sum those squared deviations:
- Use the formula for each data point :
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Find the variance (sample standard deviation):
- Variance () for a sample:
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Take the square root of the variance to find the standard deviation:
Summary of Results:
- Range: 67
- Standard Deviation: 34.67 (rounded to two decimal places)
Would you like a more detailed explanation of any part of this calculation?
Here are 5 related questions that might help expand your understanding:
- What does the standard deviation tell us about the data set?
- How would the range and standard deviation change if an outlier were added to the data?
- Why do we divide by instead of when calculating the sample standard deviation?
- How do the mean and median compare for this data set?
- What would the data look like if it had a smaller standard deviation?
Tip: When calculating the standard deviation, always ensure you're consistent with whether you're working with a sample or a population, as the formulas differ slightly.
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Math Problem Analysis
Mathematical Concepts
Statistics
Data Analysis
Formulas
Range = Max value - Min value
Standard deviation = sqrt((Σ(x - μ)²) / (n - 1))
Theorems
Range Calculation
Standard Deviation Formula
Suitable Grade Level
Grades 9-12
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