Question

kaylas family held a garage sale last weekend. they sold 5 items for: $5.00 $7.00 $4.00 $5.00 $6.00 what was the standard deviation of the prices of the items? if the answer is a decimal, round it to the nearest ten cents. standard deviation (σ): $

Explanation:

Step1: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 5\), \(x_1=5\), \(x_2 = 7\), \(x_3=4\), \(x_4 = 5\), \(x_5=6\).
\(\bar{x}=\frac{5 + 7+4 + 5+6}{5}=\frac{27}{5}=5.4\)

Step2: Calculate the squared differences from the mean

\((x_1-\bar{x})^2=(5 - 5.4)^2=(- 0.4)^2 = 0.16\)
\((x_2-\bar{x})^2=(7 - 5.4)^2=(1.6)^2 = 2.56\)
\((x_3-\bar{x})^2=(4 - 5.4)^2=(-1.4)^2 = 1.96\)
\((x_4-\bar{x})^2=(5 - 5.4)^2=(-0.4)^2 = 0.16\)
\((x_5-\bar{x})^2=(6 - 5.4)^2=(0.6)^2 = 0.36\)

Step3: Calculate the variance

The formula for the variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\)
\(\sigma^{2}=\frac{0.16+2.56 + 1.96+0.16+0.36}{5}=\frac{5.2}{5}=1.04\)

Step4: Calculate the standard deviation

The formula for the standard deviation \(\sigma=\sqrt{\sigma^{2}}\)
\(\sigma=\sqrt{1.04}\approx1.0\)

Answer:

\(1.0\)