To calculate the variance of the data set \(10, 19, 21, 28, 12, 20, 16\), you can follow these steps:
Step 1: Calculate the Mean
First, find the mean (average) of the data set.
\[ \text{Mean} = \frac{\sum \text{data points}}{N} \] where \(N\) is the number of data points.
\[ \text{Mean} = \frac{10 + 19 + 21 + 28 + 12 + 20 + 16}{7} = \frac{126}{7} = 18 \]
Step 2: Calculate the Squared Differences from the Mean
Next, find the squared differences from the mean for each data point:
- \( (10 - 18)^2 = (-8)^2 = 64 \)
- \( (19 - 18)^2 = (1)^2 = 1 \)
- \( (21 - 18)^2 = (3)^2 = 9 \)
- \( (28 - 18)^2 = (10)^2 = 100 \)
- \( (12 - 18)^2 = (-6)^2 = 36 \)
- \( (20 - 18)^2 = (2)^2 = 4 \)
- \( (16 - 18)^2 = (-2)^2 = 4 \)
Step 3: Calculate the Variance
Now, find the variance using the formula:
\[ \text{Variance} = \frac{\sum \text{squared differences}}{N} \]
Calculate the sum of squared differences:
\[ \sum \text{squared differences} = 64 + 1 + 9 + 100 + 36 + 4 + 4 = 218 \]
Finally, compute the variance:
\[ \text{Variance} = \frac{218}{7} \approx 31.14 \]
Conclusion
The variance of the data set \(10, 19, 21, 28, 12, 20, 16\) is approximately \(31.14\).
Graphical Representation
Since I can't create a graph directly in this environment, I can describe how you can visualize this:
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Bar Graph: Create a bar graph with the data points \(10, 19, 21, 28, 12, 20, 16\) on the x-axis and their values on the y-axis. This gives a visual comparison of the data values.
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Scatter Plot: You can plot each data point on a scatter plot to see how they are distributed.
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Box Plot: A box plot can show the quartiles and give an idea of the spread and outliers.
To graph it yourself, use graphing software, Excel, or graph paper for manual plotting.