Question

the standard deviation of a data is 2.8 ,if 5 is added to all the data values then the new standard deviation is ​

Answer 1

SOLUTION

GIVEN

• The standard deviation of a data is 2.8

• 5 is added to all the data values

TO DETERMINE

The new standard deviation

EVALUATION

We first prove that Standard deviation ( S.D ) is not affected by the change of origin

Let

$\sf{x_1,x_1,.....,x_n}$

be n sample values and each of them be increased by the same amount a ( which is known as change of origin) and if the new variables are

$\sf{y_1,y_1,.....,y_n}$

Then

$\sf{y_i = x_i + a} \: \: \: where \: \: \: i = \sf{1,2,.....,n}$

Let S.D ( x ) & S. D (y) represents the corresponding standard deviations for the variables x and y respectively

S. D(x)

$\displaystyle \sf{ = \sqrt{ \: \frac{ \sum \: {(x_i - \bar{x} \: )}^{2} }{n} } }$

S. D(y)

$\displaystyle \sf{= \sqrt{ \: \frac{ \sum \: {(y_i - \bar{y} \: )}^{2} }{n} } }$

$\displaystyle \sf{ = \sqrt{ \: \frac{ \sum \: { \{(x_i + a)- ( \bar{x} + a) \}}^{2} }{n} } }$

$\displaystyle \sf{= \sqrt{ \: \frac{ \sum \: {(x_i - \bar{x} \: )}^{2} }{n} } }$

= S. D (x)

Hence Standard deviation ( S.D ) is not affected by the change of origin

Now we are coming to the question

The standard deviation of a data is 2.8

5 is added to all the data values

Since Standard deviation ( S.D ) is not affected by the change of origin

Hence the new standard deviation = 2.8

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