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}

Question

simple

Answer 1

To calculate the variance for the data set $10,19,21,28,12,20,16$, follow these steps:

Step 1: Find the mean

First, we need to calculate the mean (average) of the data set.

$$\text{Mean}=\frac{\text{Sum of all data points}}{\text{Number of data points}}$$

Calculating the sum:

$$10+19+21+28+12+20+16=126$$

Counting the number of data points, we have 7 data points.

Now, calculate the mean:

$$\text{Mean}=\frac{126}{7}=18$$

Step 2: Calculate each data point's deviation from the mean

Next, we will calculate the deviation of each data point from the mean and then square it:

• $(10-18{)}^2=(-8{)}^2=64$
• $(19-18{)}^2=(1{)}^2=1$
• $(21-18{)}^2=(3{)}^2=9$
• $(28-18{)}^2=(10{)}^2=100$
• $(12-18{)}^2=(-6{)}^2=36$
• $(20-18{)}^2=(2{)}^2=4$
• $(16-18{)}^2=(-2{)}^2=4$

Step 3: Sum of squared deviations

Now, sum these squared deviations:

$$64+1+9+100+36+4+4=218$$

Step 4: Calculate the variance

Finally, the variance for a sample is calculated using the formula:

$$\text{Variance}=\frac{\text{Sum of squared deviations}}{N}$$

Where $N$ is the number of data points. In this case, $N=7$.

$$\text{Variance}=\frac{218}{7}\approx 31.14$$

Final Answer

Rounding this to the nearest hundredth, the variance of the data set $10,19,21,28,12,20,16$ is:

$$\text{Variance}\approx 31.14$$