$$ \begin{array}{lllllllllll}5.32 & 5.27 & 2.63 & 3.16 & 3.42 & 2.31 & 6.76 & 5.13 & 4.78 & 4.37 & 2.77 \ 1.87\end{array} $$ The range is 4.89 dollars. (Type an integer or decimal rounded to two decimal places as needed.) The variance is $$ \square $$ (Type an integer or decimal rounded to two decimal places as needed.)

Question

UpStudy AI · Accepted Answer

To calculate the variance of the given dataset, follow these steps:

### **Dataset:**
$$ 5.32, \, 5.27, \, 2.63, \, 3.16, \, 3.42, \, 2.31, \, 6.76, \, 5.13, \, 4.78, \, 4.37, \, 2.77, \, 1.87 $$

### **Step 1: Calculate the Mean**
$$
\text{Mean} = \frac{\sum x_i}{n} = \frac{5.32 + 5.27 + 2.63 + 3.16 + 3.42 + 2.31 + 6.76 + 5.13 + 4.78 + 4.37 + 2.77 + 1.87}{12} = \frac{47.79}{12} = 3.98
$$

### **Step 2: Calculate Each Squared Deviation from the Mean**
$$
\begin{align*}
(5.32 - 3.98)^2 &= 1.34^2 = 1.79 \
(5.27 - 3.98)^2 &= 1.29^2 = 1.66 \
(2.63 - 3.98)^2 &= (-1.35)^2 = 1.83 \
(3.16 - 3.98)^2 &= (-0.82)^2 = 0.68 \
(3.42 - 3.98)^2 &= (-0.56)^2 = 0.32 \
(2.31 - 3.98)^2 &= (-1.67)^2 = 2.80 \
(6.76 - 3.98)^2 &= 2.78^2 = 7.71 \
(5.13 - 3.98)^2 &= 1.15^2 = 1.32 \
(4.78 - 3.98)^2 &= 0.80^2 = 0.64 \
(4.37 - 3.98)^2 &= 0.39^2 = 0.15 \
(2.77 - 3.98)^2 &= (-1.21)^2 = 1.47 \
(1.87 - 3.98)^2 &= (-2.11)^2 = 4.46 \
\end{align*}
$$

### **Step 3: Sum All Squared Deviations**
$$
\sum (x_i - \text{Mean})^2 = 1.79 + 1.66 + 1.83 + 0.68 + 0.32 + 2.80 + 7.71 + 1.32 + 0.64 + 0.15 + 1.47 + 4.46 = 24.82
$$

### **Step 4: Calculate the Variance**
$$
\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n} = \frac{24.82}{12} = 2.07
$$

### **Final Answer:**
$$
\boxed{2.07}
$$
The variance is 2.07.

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