Problem: Calculate the standard deviation for the number of baskets scored by a junior high basketball team in ten games. The scores are: \( 8, 4, 6, 6, 7, 7, 9, 4, 8, 5 \).
Solution:
We will follow the steps provided in the problem to calculate the standard deviation.
####
Step 1: Arrange the scores in ascending order.
The given scores are: \( 8, 4, 6, 6, 7, 7, 9, 4, 8, 5 \).
Arranging them in ascending order:
\[ 4, 4, 5, 6, 6, 7, 7, 8, 8, 9 \]
####
Step 2: Find the mean of the data set.
The mean (\( \bar{x} \)) is calculated as:
\[
\bar{x} = \frac{\text{Sum of all scores}}{\text{Number of scores}}
\]
First, calculate the sum of the scores:
\[
4 + 4 + 5 + 6 + 6 + 7 + 7 + 8 + 8 + 9 = 64
\]
There are 10 scores, so:
\[
\bar{x} = \frac{64}{10} = 6.4
\]
Mean (\( \bar{x} \)): 6.4
####
Step 3: Find the difference from the mean for each score.
Subtract the mean (\( 6.4 \)) from each score and record the differences.
| Number | Difference from the Mean |
|--------|---------------------------|
| 4 | \( 4 - 6.4 = -2.4 \) |
| 4 | \( 4 - 6.4 = -2.4 \) |
| 5 | \( 5 - 6.4 = -1.4 \) |
| 6 | \( 6 - 6.4 = -0.4 \) |
| 6 | \( 6 - 6.4 = -0.4 \) |
| 7 | \( 7 - 6.4 = 0.6 \) |
| 7 | \( 7 - 6.4 = 0.6 \) |
| 8 | \( 8 - 6.4 = 1.6 \) |
| 8 | \( 8 - 6.4 = 1.6 \) |
| 9 | \( 9 - 6.4 = 2.6 \) |
####
Step 4: Square each difference.
Square the values in the "Difference from the Mean" column.
| Number | Difference from the Mean | Square of the Difference |
|--------|---------------------------|---------------------------|
| 4 | \(-2.4\) | \( (-2.4)^2 = 5.76 \) |
| 4 | \(-2.4\) | \( (-2.4)^2 = 5.76 \) |
| 5 | \(-1.4\) | \( (-1.4)^2 = 1.96 \) |
| 6 | \(-0.4\) | \( (-0.4)^2 = 0.16 \) |
| 6 | \(-0.4\) | \( (-0.4)^2 = 0.16 \) |
| 7 | \(0.6\) | \( (0.6)^2 = 0.36 \) |
| 7 | \(0.6\) | \( (0.6)^2 = 0.36 \) |
| 8 | \(1.6\) | \( (1.6)^2 = 2.56 \) |
| 8 | \(1.6\) | \( (1.6)^2 = 2.56 \) |
| 9 | \(2.6\) | \( (2.6)^2 = 6.76 \) |
####
Step 5: Count the number of items in the data set.
The number of items (\( n \)) is 10.
####
Step 6: Find the sum of the squared differences.
Add up all the values in the "Square of the Difference" column:
\[
5.76 + 5.76 + 1.96 + 0.16 + 0.16 + 0.36 + 0.36 + 2.56 + 2.56 + 6.76 = 26.4
\]
####
Step 7: Divide the sum of the squared differences by \( n \).
Divide the sum by the number of items (\( n = 10 \)):
\[
\frac{26.4}{10} = 2.64
\]
####
Step 8: Take the square root of the result from Step 7.
The standard deviation (\( \sigma \)) is the square root of the value obtained in Step 7:
\[
\sigma = \sqrt{2.64} \approx 1.62
\]
Final Answer:
\[
\boxed{1.62}
\]
Parent Tip: Review the logic above to help your child master the concept of calculating standard deviation worksheet answers.
