If the mean of the data: 7,8,9,7,8,7,$\lambda$,8 is 8, then the variance of this data is :-
Finding Lambda using Mean
The dataset is given as: 7, 8, 9, 7, 8, 7, $\lambda$, 8. There are 8 observations (n=8).
The mean ($\mu$) is given as 8.
The formula for the mean is: $\mu = \frac{\sum x_i}{n}$
Substituting the values:
$8 = \frac{7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8}{8}$
$8 = \frac{54 + \lambda}{8}$
Multiply both sides by 8:
$64 = 54 + \lambda$
Solve for $\lambda$:
$\lambda = 64 - 54 = 10$
Calculating Variance
Now the complete dataset is: 7, 8, 9, 7, 8, 7, 10, 8.
The mean ($\mu$) is 8.
The formula for variance ($\sigma^2$) is: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$
Calculate the squared differences from the mean:
- $(7 - 8)^2 = (-1)^2 = 1$
- $(8 - 8)^2 = (0)^2 = 0$
- $(9 - 8)^2 = (1)^2 = 1$
- $(7 - 8)^2 = (-1)^2 = 1$
- $(8 - 8)^2 = (0)^2 = 0$
- $(7 - 8)^2 = (-1)^2 = 1$
- $(10 - 8)^2 = (2)^2 = 4$
- $(8 - 8)^2 = (0)^2 = 0$
Sum of the squared differences:
$\sum (x_i - \mu)^2 = 1 + 0 + 1 + 1 + 0 + 1 + 4 + 0 = 8$
Calculate the variance:
$\sigma^2 = \frac{8}{8} = 1$
The variance of the data is 1.
Similar Questions
- Let $X = \{x \in \mathbb{N} : 1 \leq x \leq 19\}$ and for some $a, b \in \mathbb{R}, Y = \{ax + b : x \in X\}$. If the mean and variance of the elements of $Y$ are 30 and 750, respectively, then the sum of all possible values of $b$ is
- A random variable $X$ takes values $0, 1, 2, 3$ with probabilities $\frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30}, b$ respectively, where $a, b \in \mathbf{R}$. Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that $\sigma^2 + \mu^2 = 2$. Then $\frac{a}{b}$ is equal to :
If the mean and median of the data
x 0-10 10-20 20-30 30-40 40-50 f 3 6 2 x y $\Sigma f = 20$ are equal, then $xy^2$ is equal to
Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2\times2$ matrices. The probability that such formed matrices have all different entries and are non-singular, is :
Consider the following frequency distribution :
Class 0 - 6 6 - 12 12 - 18 18 - 24 24 - 30 Frequency a b 12 9 5 If mean = $\frac{309}{22}$ and median=$14$, then the value $(a - b)^2$ is equal to ________.
- For 10 observations $x_1, x_2, \dots, x_{10}$, if $\sum_{i=1}^{10} (x_i + 2)^2 = 180$ and $\sum_{i=1}^{10} (x_i - 1)^2 = 90$, then their standard deviation is:
- The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation $\alpha$ in this data is replaced by $\beta$, then the mean and variance become 10.1 and 1.99, respectively. Then $\alpha + \beta$ equals
- The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistakeone observation is taken as 50 instead of 40. If the correct mean and the correct standarddeviation are $\mu$ and $\sigma$ respectively, then $10(\mu +\sigma)$ is equal to
- The probability distribution of a random variable X is given below :
X $4\text{k}$ $\frac{30}{7}\text{k}$ $\frac{32}{7}\text{k}$ $\frac{34}{7}\text{k}$ $\frac{36}{7}\text{k}$ $\frac{38}{7}\text{k}$ $\frac{40}{7}\text{k}$ $6\text{k}$ P(X) $\frac{2}{15}$ $\frac{1}{15}$ $\frac{2}{15}$ $\frac{1}{5}$ $\frac{1}{15}$ $\frac{2}{15}$ $\frac{1}{5}$ $\frac{1}{15}$
If $\text{E}(\text{X}) = \frac{263}{15}$, then $\text{P}(\text{X} < 20)$ is equal to : Let the mean and variance of 8 numbers $-10, -7, -1, x, y, 9, 2, 16$ be $\frac{7}{2}$ and $\frac{293}{4}$, respectively.
Then the mean of 4 numbers $x, y, x + y + 1, |x - y|$ is :
Important Questions from Measures of Dispersion and Probability
- Let $X = \{x \in \mathbb{N} : 1 \leq x \leq 19\}$ and for some $a, b \in \mathbb{R}, Y = \{ax + b : x \in X\}$. If the mean and variance of the elements of $Y$ are 30 and 750, respectively, then the sum of all possible values of $b$ is
- A random variable $X$ takes values $0, 1, 2, 3$ with probabilities $\frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30}, b$ respectively, where $a, b \in \mathbf{R}$. Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that $\sigma^2 + \mu^2 = 2$. Then $\frac{a}{b}$ is equal to :
If the mean and median of the data
x 0-10 10-20 20-30 30-40 40-50 f 3 6 2 x y $\Sigma f = 20$ are equal, then $xy^2$ is equal to
Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2\times2$ matrices. The probability that such formed matrices have all different entries and are non-singular, is :
Consider the following frequency distribution :
Class 0 - 6 6 - 12 12 - 18 18 - 24 24 - 30 Frequency a b 12 9 5 If mean = $\frac{309}{22}$ and median=$14$, then the value $(a - b)^2$ is equal to ________.
