What is the value of the sample variance (S²) calculated from the data 4, 9, and 5?
Understand the Problem
The question is asking us to calculate the sample variance from the provided data set of numbers (4, 9, and 5). Sample variance is calculated using the formula S² = Σ(x - x̄)² / (n - 1), where x̄ is the sample mean, and n is the number of observations. We will first calculate the mean, then find the squared deviations from the mean, and finally calculate the variance using the formula.
Answer
$7$
Answer for screen readers
The sample variance of the data set (4, 9, 5) is $7$.
Steps to Solve
- Calculate the mean of the data set
To find the mean ($\bar{x}$), use the formula: $$ \bar{x} = \frac{Σx}{n} $$ where $Σx$ is the sum of all data points and $n$ is the number of observations. For the data set (4, 9, 5), we calculate: $$ Σx = 4 + 9 + 5 = 18 $$ Since there are 3 numbers in the data set, $n = 3$: $$ \bar{x} = \frac{18}{3} = 6 $$
- Find the squared deviations from the mean
Next, calculate the squared deviations $(x - \bar{x})^2$ for each data point:
- For 4: $(4 - 6)^2 = (-2)^2 = 4$
- For 9: $(9 - 6)^2 = (3)^2 = 9$
- For 5: $(5 - 6)^2 = (-1)^2 = 1$
Now sum these squared deviations: $$ Σ(x - \bar{x})^2 = 4 + 9 + 1 = 14 $$
- Calculate the sample variance
Finally, use the sample variance formula: $$ S² = \frac{Σ(x - \bar{x})^2}{n - 1} $$ Substituting the values we found: $$ S² = \frac{14}{3 - 1} = \frac{14}{2} = 7 $$
The sample variance of the data set (4, 9, 5) is $7$.
More Information
Sample variance provides a measure of how much the data points in a sample vary from the sample mean. It is commonly used in statistics to understand the spread of data.
Tips
- Forgetting to subtract 1 from the sample size when calculating the sample variance. Always use $n - 1$ instead of $n$ to ensure an unbiased estimate of the population variance.
- Miscalculating the mean or the squared deviations. It's important to carefully check calculations at each step.
