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Variance and Standard Deviation



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Chapter 4 : Variance and Standard Deviation



Variability arrow_upward


  • The goal for variability is to obtain a measure of how the scores spread out in a distribution. 
  • A measure of variability accompanies a measure of central tendency as basic descriptive statistics for a set of scores.
  • Variability can be measured with:
    • Range.
    • Interquartile range.
    • Standard deviation.
    • Variance.

    Standard Deviation arrow_upward


  • A measure of variability that describes an average distance of every score from the mean is known as standard deviation.
  • The square root of the variance is also standard deviation.
  • It shows variation about the Mean.
  • Population Standard Deviation:
    • If x1 , x2 , …….. xN denotes all N values from a population, then the population standard deviation is given by:

        Where,

         is the mean of population.

      

  • Sample Standard Deviation (S):
    • If x1 , x2 , …….. xN denotes all N values from a population, then the sample standard deviation is given by:

       Where,

        is the mean of sample.


    Steps to Calculate Standard Deviation of a Sample arrow_upward


  • Calculate the mean of the sample.
  • Find the difference between each entry (x) and the mean. Square the deviations from the mean.
  • Sum the squares of the deviations from the mean.
  • Divide the sum by (n – 1) to get the variance.
  • Take the square root of the variance to get the standard deviation.
  • Example:
  • Find the Standard deviation of the data set given below:
  • 1, 2, 3, 4, 5
  • Solution:
    Step 1:

    X

    1

    3

    -2

    4

    2

    3

    -1

    1

    3

    3

    0

    0

    4

    3

    1

    1

    5

    3

    2

    4


    Step 2:
  • Find the sum of  
  • 4 + 1 + 0 + 1 + 4 = 10 
  • Step 3:
  • Find n – 1.
  • n = 5
  • n – 1 = 5 – 1 = 4
  • Step 4:
  • Now we get the Standard Deviation using the formula:
  •    


    Variance arrow_upward


  • The square of the standard deviation, it is a description of how much each score varies from the mean.
  • Example:
  • Determine the standard deviation and variance from following data:
  •     30, 26 and 22

  • Here, n = 3.
  • Solution:

    30

    4

    16

    26

    0

    0

    22

    -4

    16

    Mean = 26

    0

    32


  • The Variance:
  •     = 32 ÷ 2 = 16

  • The Standard Deviation:

  • Understanding Standard Deviations arrow_upward


  • Greater  means more dispersion of data.
  • Below are the two data sets with same mean but different standard deviations.

  • Coefficient of Variation arrow_upward


  • It is the measure of relative variation.
  • It shows variation relative to the mean.
  • It is used to compare two or more sets of data measured in different units.
  • Where,
  •     = Sample Standard Deviation.

         = Sample Mean.


    Z – Score arrow_upward


  • Z – Score is the difference between the value and the mean, divided by the standard deviation.
    • The value of the z-score tells exactly where a raw score is located relative to all the other scores in the distribution.
    • A score that is located two standard deviations above the mean will have a z-score of +2.00.
  • The z-score is often called the Standardized Value.
  • Useful in identifying outliers (extreme values).
    • Outliers are values in a data set that are located far from the mean.
    • The larger the Z – Score, the larger the distance from the mean.
    • A Z – Score is considered an outlier if it is

           Or


    Negative Z – Score arrow_upward


  • Z- Score is negative if a data value is less than the sample mean.

  • Positive Z – Score arrow_upward


  • Z- Score is positive if a data value is greater than the sample mean.

  • Zero Z – Score arrow_upward


  • Z- Score is zero if the data value is equal to the sample mean.

  • Chebyshev’s Theorem arrow_upward


  • Chebyshev's Theorem enables us to state that a proportion of data values must be within a specified number of standard deviation of the mean.
  • This theorem applies to any data set regardless of the shape of the distribution of the data.

  • Chebyshev’s Rule:

  • At least   of the data values must be within z standard deviation of the mean where z is any value greater than 1.
  • For z = 2, 3 and 4, the theorem states that:
    • At least  75% of the observations must be contained within the distances of 2 Standard Deviation around the mean.
    • At least 88.89% of the observations must be contained within the distances of 3 Standard Deviation around the mean.
    • At least 93.75% of the within the distances of 4 Standard Deviation around the mean.


    Bell-Shaped Distribution arrow_upward


  • “Bell curve” refers to the shape that is created when a line is plotted using the data points for an item that meets the criteria of “normal distribution”.
  • For most data sets:
    • Approximately 68% of the observations fall within  Standard Deviation (SD) around the mean.


    • Approximately 95% of the observations fall within  Standard Deviation SD around the mean.

    • Approximately 99.7% of the observations fall within  SD around the mean.

     

    Applications:
  • Normal distributions are used in statistics to make inferences about the population mean when the sample size n is large.


  • Thank You from Kimavi arrow_upward


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