Calculate the variance for the data set. Round your answer to the nearest hundredth. Show all of your steps. {10, 19, 21, 28, 12, 20, 16}

1 answer

To calculate the variance for the given data set \({10, 19, 21, 28, 12, 20, 16}\), we will follow these steps:

  1. Calculate the mean (average) of the data set.
  2. Find the squared differences from the mean for each data point.
  3. Calculate the average of those squared differences to find the variance.

Step 1: Calculate the Mean

First, we find the mean of the data set.

\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{10 + 19 + 21 + 28 + 12 + 20 + 16}{7} \]

Calculating the sum:

\[ 10 + 19 + 21 + 28 + 12 + 20 + 16 = 126 \]

Now, we divide by the number of data points (which is 7):

\[ \text{Mean} = \frac{126}{7} = 18 \]

Step 2: Find the Squared Differences

Next, we calculate the squared differences from the mean for each data point:

  • For \(10\): \[ (10 - 18)^2 = (-8)^2 = 64 \]
  • For \(19\): \[ (19 - 18)^2 = (1)^2 = 1 \]
  • For \(21\): \[ (21 - 18)^2 = (3)^2 = 9 \]
  • For \(28\): \[ (28 - 18)^2 = (10)^2 = 100 \]
  • For \(12\): \[ (12 - 18)^2 = (-6)^2 = 36 \]
  • For \(20\): \[ (20 - 18)^2 = (2)^2 = 4 \]
  • For \(16\): \[ (16 - 18)^2 = (-2)^2 = 4 \]

Now, we list the squared differences:

\[ {64, 1, 9, 100, 36, 4, 4} \]

Step 3: Calculate the Variance

Now, we find the average of the squared differences (this gives us the variance).

First, compute the sum of the squared differences:

\[ 64 + 1 + 9 + 100 + 36 + 4 + 4 = 218 \]

Now, we divide by the number of data points (7) to find the variance:

\[ \text{Variance} = \frac{218}{7} \approx 31.14 \]

Final Result

Thus, the variance of the data set \({10, 19, 21, 28, 12, 20, 16}\) rounded to the nearest hundredth is:

\[ \boxed{31.14} \]