Here are the steps to calculate the variance:
1. Calculate the mean (average) of the data set.
2. Subtract the mean from each number in the data set and then square the result.
3. Find the average of these squared differences.
1. Mean: $\frac{82 + 78 + 83 + 80 + 75 + 82 + 79 + 80 + 83 + 81}{10} = 80.3$
2. Squared differences: $(82-80.3)^2, (78-80.3)^2, (83-80.3)^2, (80-80.3)^2, (75-80.3)^2, (82-80.3)^2, (79-80.3)^2, (80-80.3)^2, (83-80.3)^2, (81-80.3)^2$
3. Variance: $\frac{(82-80.3)^2 + (78-80.3)^2 + (83-80.3)^2 + (80-80.3)^2 + (75-80.3)^2 + (82-80.3)^2 + (79-80.3)^2 + (80-80.3)^2 + (83-80.3)^2 + (81-80.3)^2}{10}$
After calculating the above expression, we get the variance as **6.41** (rounded to the nearest hundredth). So, the variance of the speeds of Sam's last ten pitches is **6.41 mph²**. This means that, on average, the speeds vary by about 6.41 mph² from the mean speed.
