To calculate the variance of the data set \(4, 16, 21, 32, 11, 12\), we will follow these steps:
- Calculate the mean (average) of the data set.
- Subtract the mean from each data point and square the result (this is called the squared deviation).
- Calculate the average of those squared deviations to find the variance.
Step 1: Calculate the Mean
The formula for the mean (\(\mu\)) is:
\[ \mu = \frac{\sum x_i}{n} \]
where \(x_i\) is each value in the data set and \(n\) is the number of values.
Calculating the sum of the data set:
\[ 4 + 16 + 21 + 32 + 11 + 12 = 96 \]
The number of values \(n\) is \(6\).
Calculating the mean:
\[ \mu = \frac{96}{6} = 16 \]
Step 2: Calculate the Squared Deviations
Now we calculate each squared deviation from the mean:
-
For \(4\): \[ (4 - 16)^2 = (-12)^2 = 144 \]
-
For \(16\): \[ (16 - 16)^2 = (0)^2 = 0 \]
-
For \(21\): \[ (21 - 16)^2 = (5)^2 = 25 \]
-
For \(32\): \[ (32 - 16)^2 = (16)^2 = 256 \]
-
For \(11\): \[ (11 - 16)^2 = (-5)^2 = 25 \]
-
For \(12\): \[ (12 - 16)^2 = (-4)^2 = 16 \]
Step 3: Calculate the Variance
Now we sum the squared deviations:
\[ 144 + 0 + 25 + 256 + 25 + 16 = 466 \]
To find the variance (\(\sigma^2\)), we divide the total squared deviations by the number of values \(n\) (if you are calculating the population variance) or by \(n-1\) (if you are calculating the sample variance).
Since we are not specified whether it's a sample or a population, I will calculate both:
-
Population Variance: \[ \sigma^2 = \frac{466}{6} \approx 77.67 \]
-
Sample Variance (using \(n - 1\)): \[ s^2 = \frac{466}{5} = 93.2 \]
Final Results
- The population variance is approximately 77.67.
- The sample variance is 93.2.
You can use either based on whether your data set represents a complete population or a sample.
1 answer