The unbiased sample variance for the set of numbers: $S = \{40,45,50,55,60\}$ is_____. (write answer with one decimal place)
Calculate Unbiased Sample Variance for Dataset S
The unbiased sample variance ($s^2$) measures how spread out the numbers in a sample are. It's calculated using the sample data points and the sample mean. The formula uses $n-1$ in the denominator, making it unbiased.
Step 1: Calculate the Sample Mean ($\bar{x}$)
Find the average of the numbers in the set $S = \{40, 45, 50, 55, 60\}$.
Sum of numbers = $40 + 45 + 50 + 55 + 60 = 250$
Number of observations ($n$) = 5
Mean ($\bar{x}$) = $\frac{\sum x_i}{n} = \frac{250}{5} = 50$
Step 2: Calculate Squared Differences from the Mean
Find the difference between each number and the mean ($\bar{x}=50$), and then square each difference.
| Data Point ($x_i$) | Difference ($x_i - \bar{x}$) | Squared Difference ($(x_i - \bar{x})^2$) |
|---|---|---|
| 40 | $40 - 50 = -10$ | $(-10)^2 = 100$ |
| 45 | $45 - 50 = -5$ | $(-5)^2 = 25$ |
| 50 | $50 - 50 = 0$ | $(0)^2 = 0$ |
| 55 | $55 - 50 = 5$ | $(5)^2 = 25$ |
| 60 | $60 - 50 = 10$ | $(10)^2 = 100$ |
Step 3: Sum the Squared Differences
Add up all the squared differences calculated.
Sum of squared differences = $\sum (x_i - \bar{x})^2 = 100 + 25 + 0 + 25 + 100 = 250$
Step 4: Calculate the Unbiased Sample Variance ($s^2$)
Apply the formula for unbiased sample variance:
$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$
Substitute the sum of squared differences and $n-1$:
$s^2 = \frac{250}{5 - 1} = \frac{250}{4}$
$s^2 = 62.5$
Final Result
The unbiased sample variance for the set $S$ is 62.5. This value is between 61 and 63.
Important Questions from Variance
Consider a distribution with the following probability density function $$f(x) = \begin{cases} 0.5, & 0 < x < 2 \\ 0.0, & Otherwise \end{cases}$$ Given that the mean of the above probability distribution is 1, the variance (rounded off to two decimal places) is _______________.
- Let $X$ and $Y$ be two independent random variables. $X$ follows $Bernoulli(p = 0.3)$ distribution and $Y$ follows $Normal(\mu = 0, \sigma^2 = 100)$ distribution.
Which of the following options is the variance of $(2X - 1)Y$? - For a given data set $\{x_1, x_2, \ldots, x_n\}$, where $n = 100$, it is known that
$$ \frac{1}{2000} \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2 = 99 $$
Let us denote $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$.
The value of $\frac{1}{99} \sum_{i=1}^{n} (x_i - \bar{x})^2$ is __________ . (Answer in integer) - A random variable $X$ has the sample space $\{0,1\}$. The probability $P(X = 0) = 1/4$ and $P(X = 1) = 3/4$.
What is the variance of the random variable?
Hint: $\text{Mean } (\mu) = \sum_{i=1}^{n} x_i p(x_i) ; \text{Variance } (\sigma^2) = \sum_{i=1}^{n} (x_i - \mu)^2 p(x_i)$ A set of 11 (x, y) data points is least-squares fitted to a quadratic polynomial. If the sum of squares of error is 2.4, the variance of error is ________ (round off to 1decimal place).
