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:

  1. Find the mean (average) of the data.
  2. Subtract the mean from each data point and square the result (this is the squared deviation).
  3. 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

Mean=xin=20.2+33.1+22+58.5+23.4+110.7+30.4+24.9+75+60.1+40+45.9+65.4+76.414\text{Mean} = \frac{\sum x_i}{n} = \frac{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}{14} =68614=49= \frac{686}{14} = 49

Step 2: Calculate the squared deviations

For each number xix_i, we compute (ximean)2(x_i - \text{mean})^2.

Data point (xix_i)Deviation (xi49x_i - 49)Squared deviation ((xi49)2(x_i - 49)^2)
20.2-28.8829.44
33.1-15.9252.81
22-27729
58.59.590.25
23.4-25.6655.36
110.761.73806.89
30.4-18.6345.96
24.9-24.1580.81
7526676
60.111.1123.21
40-981
45.9-3.19.61
65.416.4268.96
76.427.4750.76

Step 3: Find the variance

Now, sum the squared deviations and divide by the number of data points.

Variance=(xi49)2n=8199.114585.65\text{Variance} = \frac{\sum (x_i - 49)^2}{n} = \frac{8199.1}{14} \approx 585.65

Thus, the variance is approximately 585.65.

Would you like a more detailed explanation or have any questions?

Related questions:

  1. What is the difference between variance and standard deviation?
  2. How is variance used in statistical analysis?
  3. What happens to the variance if an outlier is added?
  4. Can the variance ever be negative?
  5. 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

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Suitable Grade Level

Grades 9-12