Question

Calculate the range,average deviation,variance,standard deviation, of the following set of observation 3,4,5,5,6,7,7,7,8,8


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Answer 1

Step-by-step explanation:

Given = 3,4,5,5,6,7,7,7,8,8

Range(R) = Highest value(H) - lowest value(L)

            R  = 8 - 3

            R  =  5

Average deviation = 1/n∑|x-Mean|

 here, n = total number of observation

           x = set of observation

   x               |x-Mean|           Deviation           x-assumed mean (A)      A²

   3                      3                                                          -4                         16

   4                      2                                                          -3                          9

   5                       1                                                          -2                          4

   5                       1                                                          -2                          4

   6                       0                                                         -1                            1

   7                        1          assumed mean = 7                 0                           0

   7                        1                                                           0                           0

   7                        1                                                          0                             0

   8                        2                                                          1                             1

   8                        2                                                          1                             1

 60                       14                                                                                      36

 Mean = ∑x/n

             = 60/10

             = 6

Average deviation = (1/10)14

                               = 1.4

Variance = A²/(n-1)

               =  36/(10-1)

               =   4

Standard Variance = √variance

                               = √4

                               = 2

           

Answer 2

Answer:

The range, average deviation, variance, and standard deviation of the given set of observations are 5, 2.33, 2.6, and 1.61 respectively.

Step-by-step explanation:

Range:

The range is the difference between the largest value and the smallest value of a given set of data.

Range = Max. value - Min. value

Range = 8 - 3 = 5.

Therefore, the range of the given data set is 5.

Average Deviation:

Average deviation gives the measure of the dispersion from a central value.

It is calculated as the average of the differences between each value and the mean value.

$AD = \frac{1}{n}$Σ$|x-mean|$

Substituting the values, we get

$AD=\frac{14}{6}$

$AD = 2.33$

Therefore, the average deviation of the given data is 2.33

Variance:

Variance gives the measure of the variability of the observations. It tells us how far the data is spread out.

$S^{2} =$Σ$({x-mean})^{2}$ / n -1, where n is the number of observations in the data set.

Substituting the values, we get

$S^{2} = \frac{26}{10}$

$S^{2} = 2.6$

Therefore, the variance of the given data is 2.6

Standard Deviation:

Standard deviation gives the measure of how dispersed the data is relative to the mean value. It is simply given as the square root of the variance.

σ = $\sqrt{S^{2} }$

σ = $\sqrt{2.6}$

σ = 1.61

Therefore, the standard deviation of the given data is 1.61

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