Standard Deviation Calculation Steps
We are given information about 10 observations ($n=10$) and two summation formulas:
- $\sum_{i=1}^{10} (x_i + 2)^2 = 180$
- $\sum_{i=1}^{10} (x_i - 1)^2 = 90$
We need to find the standard deviation ($\sigma$). First, let's expand the given equations.
Expanding the Summation Formulas
Let $S_1 = \sum_{i=1}^{10} x_i$ and $S_2 = \sum_{i=1}^{10} x_i^2$. Using the algebraic identity $(a \pm b)^2 = a^2 \pm 2ab + b^2$, we expand the terms:
- First Equation Expansion: $ \sum_{i=1}^{10} (x_i + 2)^2 = \sum_{i=1}^{10} (x_i^2 + 4x_i + 4) $ $ = \sum_{i=1}^{10} x_i^2 + 4 \sum_{i=1}^{10} x_i + \sum_{i=1}^{10} 4 $ $ = S_2 + 4S_1 + (10 \times 4) = 180 $ $ \implies S_2 + 4S_1 + 40 = 180 $ $ \implies S_2 + 4S_1 = 140 $ (Equation A)
- Second Equation Expansion: $ \sum_{i=1}^{10} (x_i - 1)^2 = \sum_{i=1}^{10} (x_i^2 - 2x_i + 1) $ $ = \sum_{i=1}^{10} x_i^2 - 2 \sum_{i=1}^{10} x_i + \sum_{i=1}^{10} 1 $ $ = S_2 - 2S_1 + (10 \times 1) = 90 $ $ \implies S_2 - 2S_1 + 10 = 90 $ $ \implies S_2 - 2S_1 = 80 $ (Equation B)
Solving for Sums ($S_1$ and $S_2$)
Now we have a system of two linear equations with two variables ($S_1$ and $S_2$):
- Equation A: $S_2 + 4S_1 = 140$
- Equation B: $S_2 - 2S_1 = 80$
Subtract Equation B from Equation A:
$ (S_2 + 4S_1) - (S_2 - 2S_1) = 140 - 80 $ $ 6S_1 = 60 $ $ S_1 = \frac{60}{6} = 10 $Substitute $S_1 = 10$ into Equation B:
$ S_2 - 2(10) = 80 $ $ S_2 - 20 = 80 $ $ S_2 = 100 $Calculating Mean and Variance
The mean ($\bar{x}$) of the observations is:
$ \bar{x} = \frac{S_1}{n} = \frac{10}{10} = 1 $The variance ($\sigma^2$) is calculated using the formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$:
$ \sigma^2 = \frac{S_2}{n} - (\bar{x})^2 $ $ \sigma^2 = \frac{100}{10} - (1)^2 $ $ \sigma^2 = 10 - 1 = 9 $Determining Standard Deviation
The standard deviation ($\sigma$) is the square root of the variance:
$ \sigma = \sqrt{\sigma^2} = \sqrt{9} $ $ \sigma = 3 $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
If the mean of the data: 7,8,9,7,8,7,$\lambda$,8 is 8, then the variance of this data is :-
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 ________.
- 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
If the mean of the data: 7,8,9,7,8,7,$\lambda$,8 is 8, then the variance of this data is :-
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 :
