Variance and Standard Deviation

 Standard deviation is the positive square root of the mean of the squared deviation from the arithmetic mean.

The square of standard deviation is called variance.

Three methods for calculating variance and standard deviation are direct method, short-cut method or assumed mean method and step deviation method.

Variance and standard deviation from an ungrouped data can be computed using

    direct method

    short-cut method

Variance and standard deviation from grouped data can be computed using step deviation method.

Variance and standard deviation from grouped data within class can be computed using step deviation method.

The measure of variability which is independent of units is called coefficient of variation (denoted as C.V.).

C.V. = (S. D./Mean) × 100, where mean is not equal to 0.

For two series with equal means, the series with greater standard deviation (or variance) is more variable or dispersed than the other. Also, the series with lesser value of standard deviation (or variance) is said to be more consistent than the other.

Combined mean is the collective arithmetic mean of several sets of data combined into a single arithmetic mean.

Combined Standard deviation is the collective standard deviation of several sets of data combined into a single standard deviation.

To Access the full content, Please Purchase

  • Q1

    The variance of the first ‘n’ natural numbers is

    Marks:1
    Answer:

    (n2 – 1)/12.

    Explanation:

    View Answer
  • Q2

    Megha takes a test of 25 marks. The observations of marks of 5 students are 10, 9, 14, 17 and 20. Now, the standard deviation is

    Marks:1
    Answer:

    4.15.

    Explanation:

    View Answer
  • Q3

    The standard deviation of the following series is

    Marks:1
    Answer:

    9.

    Explanation:

    Let us assume an arbitrary mean a = 25
    Class interval h = 10
    Construct the following table:

     

    View Answer
  • Q4

    If the standard deviation of a set of observation is 4 and if each observation is divided by 4, the standard deviation of the new set of observations will be

    Marks:1
    Answer:

    1.

    Explanation:
    We know that, if y = x/h, then Sy = Sx/|h|

            

    Since each observation is divided by 4.

    Therefore, the S.D of new set of observation will be 4/4 = 1.

    View Answer
  • Q5

    The standard deviation for the following frequency distribution is

    Marks:1
    Answer:

    2.53.

    Explanation:

    CI

    Frequency (fi)

    Mid point xi

    fi xi

    0 – 4

    4 – 8

    8 – 12


    12 -16

    4

    8

    2

    1

    2

    6

    10

    14

    8

    48

    20

    14

    – 4

    0

    4

    8

    16

    0

    16

    64

     Total           

            15

     

     90 

    8

    96 

        

          

    View Answer