Suppose the mean and the standard deviation of the data $\left\{x_1, x_2, \cdots, x_9\right\}$ are $...

To simplify and clarify the explanation on updating the standard deviation when a new data point is added, consider the following:Mean Calculation:Original Data Mean:The mean of the original data set $\left\{x_1, x_2, \cdots, x_9\right\}$ is given by:$ \mu = \frac{1}{9} \sum_{i=1}^9 x_i \implies \sum_{i=1}^9 x_i = 9\mu $New Data Set Mean:When a new data point $x_{10}$ is added, maintaining the same mean $\mu$:$ \mu = \frac{1}{10} \sum_{i=1}^{10} x_i = \frac{1}{10}\left(9\mu + x_{10}\right) $Solving for $x_{10}$:$ 10\mu = 9\mu + x_{10} \implies x_{10} = \mu $Standard Deviation Calculation:Original Standard Deviation:The standard deviation of the original data set is:$ \sigma = \sqrt{\frac{1}{9} \sum_{i=1}^9 (x_i - \mu)^2} $Calculate Variance of the New Data Set:The variance for the new data set is:$ \text{Var} = \frac{1}{10} \sum_{i=1}^{10} (x_i - \mu)^2 $Breaking it down:$ \sum_{i=1}^{10} (x_i - \mu)^2 = \sum_{i=1}^9 (x_i - \mu)^2 + (x_{10} - \mu)^2 $Since $x_{10} = \mu$, $(x_{10} - \mu)^2 = 0$. Thus:$ \sum_{i=1}^{10} (x_i - \mu)^2 = \sum_{i=1}^9 (x_i - \mu)^2 $Equating Variance Values:Since $\sigma^2 = \frac{1}{9} \sum_{i=1}^9 (x_i - \mu)^2$, we have:$ \sum_{i=1}^9 (x_i - \mu)^2 = 9\sigma^2 $Substitute in new variance formula:$ \text{Var} = \frac{1}{10} \cdot 9\sigma^2 = \frac{9}{10}\sigma^2 $New Standard Deviation:The new standard deviation is:$ \sqrt{\text{Var}} = \sqrt{\frac{9}{10}\sigma^2} = \frac{3}{\sqrt{10}}\sigma = \frac{3\sigma}{\sqrt{10}} $Thus, the standard deviation of the updated data set $\left\{x_1, x_2, \cdots, x_9, x_{10}=\mu\right\}$ is $\frac{3\sigma}{\sqrt{10}}$.

Statistics - IAT-IISER Maths PYQs

To simplify and clarify the explanation on updating the standard deviation when a new data point is added, consider the following:Mean Calculation:Original Data Mean:The mean of the original data set

%5Cleft%5C%7Bx_1%2C%20x_2%2C%20%5Ccdots%2C%20x_9%5Cright%5C%7D

is given by:

%5Cmu%20%3D%20%5Cfrac%7B1%7D%7B9%7D%20%5Csum_%7Bi%3D1%7D%5E9%20x_i%20%5Cimplies%20%5Csum_%7Bi%3D1%7D%5E9%20x_i%20%3D%209%5Cmu

New Data Set Mean:When a new data point

x_%7B10%7D

is added, maintaining the same mean

%5Cmu

%5Cmu%20%3D%20%5Cfrac%7B1%7D%7B10%7D%20%5Csum_%7Bi%3D1%7D%5E%7B10%7D%20x_i%20%3D%20%5Cfrac%7B1%7D%7B10%7D%5Cleft(9%5Cmu%20%2B%20x_%7B10%7D%5Cright)

Solving for

x_%7B10%7D

10%5Cmu%20%3D%209%5Cmu%20%2B%20x_%7B10%7D%20%5Cimplies%20x_%7B10%7D%20%3D%20%5Cmu

Standard Deviation Calculation:Original Standard Deviation:The standard deviation of the original data set is:

%5Csigma%20%3D%20%5Csqrt%7B%5Cfrac%7B1%7D%7B9%7D%20%5Csum_%7Bi%3D1%7D%5E9%20(x_i%20-%20%5Cmu)%5E2%7D

Calculate Variance of the New Data Set:The variance for the new data set is:

%5Ctext%7BVar%7D%20%3D%20%5Cfrac%7B1%7D%7B10%7D%20%5Csum_%7Bi%3D1%7D%5E%7B10%7D%20(x_i%20-%20%5Cmu)%5E2

Breaking it down:

%5Csum_%7Bi%3D1%7D%5E%7B10%7D%20(x_i%20-%20%5Cmu)%5E2%20%3D%20%5Csum_%7Bi%3D1%7D%5E9%20(x_i%20-%20%5Cmu)%5E2%20%2B%20(x_%7B10%7D%20-%20%5Cmu)%5E2

Since

x_%7B10%7D%20%3D%20%5Cmu

(x_%7B10%7D%20-%20%5Cmu)%5E2%20%3D%200

. Thus:

%5Csum_%7Bi%3D1%7D%5E%7B10%7D%20(x_i%20-%20%5Cmu)%5E2%20%3D%20%5Csum_%7Bi%3D1%7D%5E9%20(x_i%20-%20%5Cmu)%5E2

Equating Variance Values:Since

%5Csigma%5E2%20%3D%20%5Cfrac%7B1%7D%7B9%7D%20%5Csum_%7Bi%3D1%7D%5E9%20(x_i%20-%20%5Cmu)%5E2

, we have:

%5Csum_%7Bi%3D1%7D%5E9%20(x_i%20-%20%5Cmu)%5E2%20%3D%209%5Csigma%5E2

Substitute in new variance formula:

%5Ctext%7BVar%7D%20%3D%20%5Cfrac%7B1%7D%7B10%7D%20%5Ccdot%209%5Csigma%5E2%20%3D%20%5Cfrac%7B9%7D%7B10%7D%5Csigma%5E2

New Standard Deviation:The new standard deviation is:

%5Csqrt%7B%5Ctext%7BVar%7D%7D%20%3D%20%5Csqrt%7B%5Cfrac%7B9%7D%7B10%7D%5Csigma%5E2%7D%20%3D%20%5Cfrac%7B3%7D%7B%5Csqrt%7B10%7D%7D%5Csigma%20%3D%20%5Cfrac%7B3%5Csigma%7D%7B%5Csqrt%7B10%7D%7D

Thus, the standard deviation of the updated data set

%5Cleft%5C%7Bx_1%2C%20x_2%2C%20%5Ccdots%2C%20x_9%2C%20x_%7B10%7D%3D%5Cmu%5Cright%5C%7D

%5Cfrac%7B3%5Csigma%7D%7B%5Csqrt%7B10%7D%7D

Statistics

High-Yield Trend

Questions
1 MCQs

Official Solution

Statistics

High-Yield Trend

Questions 1 MCQs

Official Solution

Questions
1 MCQs