Understanding Variance: A Practical Example
In our recent Data Science and AI lecture, we explored the concept of variance—a crucial measure in statistics that indicates how spread out a set of data points is around their mean. Today, we’ll dive into a practical example involving a batsman’s scores over ten matches.
The Problem
Let’s consider the following scores:
38, 70, 48, 34, 42, 55, 63, 46, 54, 44
We will calculate the variance and standard deviation of these scores.
Step 1: Calculate the Mean
First, we find the mean (\( \bar{x} \)) of the scores:
\[
\bar{x} = \frac{38 + 70 + 48 + 34 + 42 + 55 + 63 + 46 + 54 + 44}{10} = 48
\]
Step 2: Find the Deviations
Next, we compute the deviations (\( d_i = x_i – \bar{x} \)) from the mean:
| (\( x_i \)) | (\( d_i = x_i – 48 \)) | |
|---|---|---|
| 38 | -10 | 100 |
| 70 | 22 | 484 |
| 48 | 0 | 0 |
| 34 | -14 | 196 |
| 42 | -6 | 36 |
| 55 | 7 | 49 |
| 63 | 15 | 225 |
| 46 | -2 | 4 |
| 54 | 6 | 36 |
| 44 | -4 | 16 |
Step 3: Calculate Variance
Now, we sum the squared deviations:
\[
\sum d_i^2 = 100 + 484 + 0 + 196 + 36 + 49 + 225 + 4 + 36 + 16 = 1126
\]
Using the formula for variance (\( Var(X) \)):
\[
Var(X) = \frac{\sum d_i^2}{n} = \frac{1126}{10} = 112.6
\]
Step 4: Calculate Standard Deviation
Finally, the standard deviation (\( S.D. \)) is the square root of the variance:
\[
S.D. = \sqrt{Var(X)} = \sqrt{112.6} \approx 10.61
\]
Presentation
variance ex 8Video
Conclusion
Understanding variance and standard deviation provides valuable insights into data variability. This example illustrates how these concepts can be practically applied to analyze performance metrics in sports.
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