Let $X = \{11, 12, 13, \ldots, 40, 41\}$ and $Y = \{61, 62, 63, \ldots, 90, 91\}$ be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their respective means and $\sigma^2$ is the variance of all the observations in $X \cup Y$, then $|\bar{x} + \bar{y} - \sigma^2|$ is equal to ______.
Let \(x_1, x_2, x_3, x_4, x_5\) be the observations with mean \(m\) and standard deviation \(s\). The standard deviation of the observations \(kx_1, kx_2, kx_3, kx_4, kx_5\) is
For the frequency distribution (class: 0-10, 10-20, 20-30, 30-40, 40-50; frequency: 2,3,x,5,4) with mean 28, the variance is ________.
The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then \(\frac{y}{x}\) is equal to
Mean and variance of 8 numbers $x,y,10,12,6,12,4,8$ are 9 and 9.25. If $x>y$, then $3x-2y$ is equal to _______
If both the mean and the standard deviation of 50 observations \(x_1, x_2, \ldots, x_{50}\) are equal to 16, then the mean of \((x_1 - 4)^2, (x_2 - 4)^2, \ldots, (x_{50} - 4)^2\) is ______.
If the mean and the variance of the data (Class 4–8, 8–12, 12–16, 16–20; Frequency 3, $\lambda$, 4, 7) are $\mu$ and 19 respectively, then the value of $\lambda+\mu$ is
Let $\alpha,\beta\in\mathbb{R}$. Let the mean and the variance of 6 observations $-3,4,7,-6,\alpha,\beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:
For 50 observations \(a, 2a, 3a, \ldots, 50a\), the mean deviation about the median is minimized and \(\dfrac{1}{n}\sum|x_i - A|\) is minimized when \(A\) is the median. If \(625a = 2500\), find \(a\).
Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that $\displaystyle\sum_{i=1}^{10}(x_i - 2) = 30$, $\displaystyle\sum_{i=1}^{10}(x_i - \beta)^2 = 98$, $\beta > 2$, and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1 - 1) + 4\beta, 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta$, then $\frac{\beta\mu}{\sigma^2}$ is equal to:
For a statistical data $x_{1},x_{2},\dots,x_{10}$ of $10$ values, a student obtained the mean as $5.5$ and $\sum_{i=1}^{10}x_{i}^{2}=371$. He later found that he had noted two values in the data incorrectly as $4$ and $5$, instead of the correct values $6$ and $8$, respectively. The variance of the corrected data is:
If the arithmetic mean of the numbers \(x_1, x_2, x_3, \ldots, x_n\) is \(\bar{x}\), then the arithmetic mean of the numbers \(ax_1 + b,\ ax_2 + b,\ ax_3 + b,\ \ldots,\ ax_n + b\), where \(a, b\) are two constants, would be
Consider data on $X$ taking values $0, 2, 4, 8, \ldots, 2^n$ with frequencies ${}^nC_0, {}^nC_1, \ldots, {}^nC_n$ respectively. If the mean of this data is $\dfrac{728}{2^n}$, then $n$ is equal to
Frequency distribution with classes $[0,10),[10,20),\ldots,[50,60)$ and frequencies $3,c,d,11,5,5$ has mean $31$ and median $340/11$. Value of $\left[\tan^{-1}\!\dfrac{2cd}{d^2-c^2}\right]$ (GIF) is