Question

Zachary is a party planner who bought 5 decorations for a birthday party. The prices of the decorations were:
$1.00 $3.00 $2.00 $6.00 $9.00
What was the standard deviation of the prices of the decorations?

If the answer is a decimal, round it to the nearest ten cents.

standard deviation (σ): $

Answer 1

To calculate the standard deviation of the prices of the decorations, follow these steps:

1. **Find the mean (average) of the data.**
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{1.00 + 3.00 + 2.00 + 6.00 + 9.00}{5}
\]
\[
\text{Mean} = \frac{21.00}{5} = 4.20
\]

2. **Calculate the deviations from the mean and square them.**
\[
(1.00 - 4.20)^2 = (-3.20)^2 = 10.24
\]
\[
(3.00 - 4.20)^2 = (-1.20)^2 = 1.44
\]
\[
(2.00 - 4.20)^2 = (-2.20)^2 = 4.84
\]
\[
(6.00 - 4.20)^2 = (1.80)^2 = 3.24
\]
\[
(9.00 - 4.20)^2 = (4.80)^2 = 23.04
\]

3. **Sum the squared deviations.**
\[
10.24 + 1.44 + 4.84 + 3.24 + 23.04 = 43.80
\]

4. **Calculate the variance.**
The formula for the variance \( \sigma^2 \) when dealing with a sample is:
\[
\text{Variance} = \frac{\text{Sum of squared deviations}}{n}
\]
where \( n \) is the number of data points. Here, \( n = 5 \).
\[
\text{Variance} = \frac{43.80}{5} = 8.76
\]

5. **Calculate the standard deviation.**
\[
\sigma = \sqrt{\text{Variance}} = \sqrt{8.76} \approx 2.96
\]

Finally, round the result to the nearest ten cents:
\[
\boxed{2.96}
\]