Math Problem Statement
Find the range, the standard deviation, and the variance for the given samples. Round non-integer results to the nearest tenth. 94, 59, 62, 58, 56, 46, 86, 52, 48
Solution
Let's calculate the range, standard deviation, and variance for the given sample:
Given Data:
94, 59, 62, 58, 56, 46, 86, 52, 48
1. Range:
The range is the difference between the maximum and minimum values in the data set.
- Maximum value = 94
- Minimum value = 46
2. Variance and Standard Deviation:
First, let's find the variance and then the standard deviation (which is the square root of the variance).
Step 1: Find the Mean ()
Step 2: Find the Squared Differences from the Mean
For each number, we subtract the mean and square the result.
Step 3: Find the Variance ()
Variance is the average of these squared differences:
Step 4: Find the Standard Deviation ()
The standard deviation is the square root of the variance:
Final Results:
- Range = 48
- Variance = 245.7
- Standard Deviation = 15.7
Would you like further details on any of the calculations or have any questions? Here are five related questions to practice:
- How would the range change if the sample included an additional value of 100?
- What effect does an outlier (a value much higher or lower than the others) have on the standard deviation?
- Calculate the mean for a new data set: 34, 76, 54, 89, 67, 23, 90.
- If you added a value of 80 to this sample, how would it affect the mean and standard deviation?
- Explain the difference between population variance and sample variance.
Tip: Standard deviation is sensitive to extreme values; a few high or low numbers can significantly affect it.
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Math Problem Analysis
Mathematical Concepts
Statistics
Descriptive Statistics
Measures of Dispersion
Formulas
Range = Max value - Min value
Variance (σ^2) = Σ((x - μ)^2) / N
Standard Deviation (σ) = √Variance
Mean (μ) = Σx / N
Theorems
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Suitable Grade Level
Grades 9-12
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