Math Problem Statement
Find the variance 20.2 33.1 22 58.5 23.4 110.7 30.4 24.9 75 60.1 40 45.9 65.4 76.4
Solution
To calculate the variance, we follow these steps:
- Find the mean (average) of the data.
- Subtract the mean from each data point and square the result (this is the squared deviation).
- Find the average of these squared deviations (this is the variance).
Here are the given numbers:
20.2, 33.1, 22, 58.5, 23.4, 110.7, 30.4, 24.9, 75, 60.1, 40, 45.9, 65.4, 76.4
Step 1: Find the mean
Step 2: Calculate the squared deviations
For each number , we compute .
Data point () | Deviation () | Squared deviation () |
---|---|---|
20.2 | -28.8 | 829.44 |
33.1 | -15.9 | 252.81 |
22 | -27 | 729 |
58.5 | 9.5 | 90.25 |
23.4 | -25.6 | 655.36 |
110.7 | 61.7 | 3806.89 |
30.4 | -18.6 | 345.96 |
24.9 | -24.1 | 580.81 |
75 | 26 | 676 |
60.1 | 11.1 | 123.21 |
40 | -9 | 81 |
45.9 | -3.1 | 9.61 |
65.4 | 16.4 | 268.96 |
76.4 | 27.4 | 750.76 |
Step 3: Find the variance
Now, sum the squared deviations and divide by the number of data points.
Thus, the variance is approximately 585.65.
Would you like a more detailed explanation or have any questions?
Related questions:
- What is the difference between variance and standard deviation?
- How is variance used in statistical analysis?
- What happens to the variance if an outlier is added?
- Can the variance ever be negative?
- How does the number of data points affect the variance?
Tip:
The standard deviation is simply the square root of the variance, which provides a measure of how spread out the data is in the same units as the original data.
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Math Problem Analysis
Mathematical Concepts
Statistics
Variance
Mean
Deviation
Formulas
Mean = (sum of data points) / number of data points
Variance = (sum of squared deviations from the mean) / number of data points
Theorems
-
Suitable Grade Level
Grades 9-12
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