Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results: Number of observations = 25, mean = 18.2 seconds, standard deviation = 3.25 seconds. Further, another set of 15 observations

The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results:

Number of observations = 25, mean = 18.2 seconds, standard deviation = 3.25 seconds.
Further, another set of 15 observations x1, x2, ..., x15, also in seconds, is now available and we have
\\ \sum_{i=1}^{15}{x_{i}}=279\: \: and\: \: \sum_{i=1}^{15} \mathrm{x}_{i}=5524.Calculate the standard derivation based on all 40 observations.

 

 

Answers (1)

\\\\ Number~of observations=25, mean=18.2 seconds, standard deviation=3.25 seconds. \\\\ \\ \text{ There are another set of 15 observations }x_{1},x_{2}, \ldots ..,x_{15}\text{~ also in seconds is } \sum _{i=1}^{15}x_{i}=279~and~ \sum _{i=1}^{15}x_{i}^{2}=5524~ \\\\ \\ \text{ To find that the standard derivation based on all 40 observations } \\\\ \\ \text{ As per the given criteria in the first set Number of observations , }n_{1}=25 \\\\


\\ \\ Mean=18.2 \\\\ \\ \text{ And standard deviation } \sigma _{1}=3.25~ \\\\ \\ \text{ And in second set number of observations }n_{2}=15 \\\\ \\ \text{~ For the first set we have }\overline{x_{1}}=18.2=\frac{ \Sigma x_{i}}{25}~~~~~ \\\\ \\ ~ \Sigma x_{i}=25\ast18.2=455~ \\\\
\\ \\ \text{ Therefore the standard deviation becomes } \sigma _{1}^{2}=\frac{ \Sigma x_{i}^{2}}{25}- \left( 18.2 \right) ^{2}~ \\\\ \\ \text{ Substituting the values, we get } \left( 3.25 \right) ^{2}=\frac{ \Sigma x_{i}^{2}}{25}-331.24~~~ \\\\ \\ 10.5625+331.24=\frac{ \Sigma x_{i}^{2}}{25}=341.8025 \\\\ \\ ~~ \Sigma x_{i}^{2}=25\ast341.8025=8545.06 \\\\ \\ \text{For~the combined standard deviation of the 40 observation, n=40 and } \Sigma x_{i}^{2}=8545.06+5524=14069.69~~ \\\\ \\ ~ \Sigma x_{i}=455+279=734 \\\\ \\ \text{Therefore the standard deviation can be written as,} \sigma =\sqrt[]{\frac{ \Sigma x_{i}^{2}}{n}- \left( \frac{ \Sigma x_{i}}{n} \right) ^{2}}~ \\\\
\\ \\ \text{~~Substituting~the values, we get Therefore the standard deviation can be written as } \\\\ \\ \sigma =\sqrt{\frac{14069.69}{40}- \left( \frac{734}{40} \right) ^{2}}= \sqrt{351.7265-336.7225}=\sqrt{15.004}=3.87 \\\\

Posted by

infoexpert21

View full answer

Similar Questions

Latest Question