Find the sample variance and standard deviation. 20, 16, 2, 7, 12

Question

Find the sample variance and standard deviation. 20, 16​, 2​, 7​, 12

constanzma · Accepted Answer

Step 1Given data set is 20,16,2,7,12Let the given data set be       x    1    ,      x    2    ,      x    3    ,      x    4    ,      x    5        x    1    =  20        x    2    =  16        x    3    =  2        x    4    =  7        x    5    =  12It is clear that, the total number of data points   (  x  )  =  5The formula for the sample variance is given bySample variance   (      σ    2    )  =            (              x        i            −              x        ¯                    )        2                    n      −      1       Formulawhere       x    ¯    =  m  e  a  n.which is given by       x    ¯    =      Sum of all data points    Total number of datapoints   Formula.Calculation of mean   (      x    ¯    )We know   m  e  a  n  (      x    ¯    )  =      Sum of all data points    Total number of data points    ⇒      x    ¯    =                    x        1            +              x        2            +              x        3            +              x        4            +              x        5              n  Step 2  ⇒      x    ¯    =            20      +      16      +      2      +      7      +      12        5    ⇒      x    ¯    =      57    5    ⇒      x    ¯    =  11.4mean   (      x    ¯    ) of the given data set is       x    ¯    =  11.4We know   ⇒      σ    2    =            (              x        1            −              x        ¯                    )        2            +      (              x        2            −              x        ¯                    )        2            +      (              x        3            −              x        ¯                    )        2            +      (              x        4            −              x        ¯                    )        2            +      (              x        5            −              x        ¯                    )        2                    5      −      1        ⇒      σ    2    =            (      20      −      11.4              )        2            +      (      16      −      11.4              )        2            +      (      2      −      11.4              )        2            +      (      7      −      11.4              )        2            +      (      12      −      11.4              )        2              4    ⇒      σ    2    =            (      8.6              )        2            +      (      4.6              )        2            +      (      −      9.4              )        2            +      (      −      4.4              )        2            +      (      0.6              )        2              4    ⇒      σ    2    =            73.96      +      21.16      +      88.36      +      19.36      +      0.36        4    ⇒      σ    2    =      203.2    4    ⇒      σ    2    =  50.8Step 3We obtainedVariance   (      σ    2    )  =  50.8Standard Deviation   (  σ  )  =      V    a    r    i    a    n    c    e    ⇒  σ  =      50.8    ⇒  σ  =  7.127Standard deviation   (  σ  )  =  7.127We obtainedSample variance   (      σ    2    )  =  50.8Sample standard deviation   (  σ  )  =  7.127

Elsa Brewer · Accepted Answer

Step 1:Add the data  20  +  16  +  2  +  7  +  12  =  57Divide this by number of data   =  5Mean   =  57      /    5  =  11.4Step 2:  20  −  11.4  =  8.6  (  8.6      )    2    .     73.96  16  −  11.4  =  4.6.  (  4.6      )    2    .     21.16  2  −  11.4  =  −  9.4  (  −  9.4      )    2    .     88.36  7  −  11.4  =  −  4.4.  (  −  4.4      )    2    .     19.36  12  −  11.4  =  0.6.  (  0.6      )    2    .     0.36Step 3Sum of all the squares in the third column   =  203.20Step 4Sample Variance   =  203.20      /    5  −  1  =  203.20      /    4  =  50.80Step 5Standard deviation   =      sample variance          1              /            2        =      50.80          1              /            2        =  7.1274So,the standard deviation is 7.1274.

Find the sample variance and standard deviation. 20, 16, 2, 7, 12

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