Math Problem Statement
For the following distribution, use the computational variance formula to find the variance and standard deviation: 10, 15, 15, 17, 17, 17, 18, 18, 19, 22, 25, 27, 33, 35, 39, 50, 75, 76 show work
Solution
To find the variance and standard deviation of the given distribution using the computational variance formula, follow these steps:
Given Data:
The data is:
Step 1: Calculate the Mean ()
The formula for the mean is: where are the individual values, and is the number of data points.
First, find the sum of all the data points: The number of data points, .
Now calculate the mean:
Step 2: Use the Computational Formula for Variance
The computational variance formula is: where is the sum of the squares of the individual data points.
Let's calculate :
Step 3: Calculate the Variance
Now, apply the values into the variance formula: First, calculate the square of the mean: Now calculate the variance:
Step 4: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
Final Results:
- Variance:
- Standard Deviation:
Would you like more details or further clarifications?
Here are 5 related questions:
- How is the computational formula for variance derived from the basic definition?
- What is the difference between population and sample variance?
- Can you explain how the variance changes if outliers are present in the dataset?
- How do you interpret the standard deviation in the context of data distribution?
- How would you calculate the variance for a grouped frequency distribution?
Tip: Variance gives a measure of how spread out your data is, while standard deviation, being in the same units as the data, is easier to interpret.
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Math Problem Analysis
Mathematical Concepts
Statistics
Variance
Standard Deviation
Formulas
Mean: μ = Σx_i / n
Computational Variance Formula: σ^2 = (Σx_i^2 / n) - μ^2
Standard Deviation: σ = √(σ^2)
Theorems
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Suitable Grade Level
Grades 9-12
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