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Measures of Central Tendency



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Chapter 2 : Measure of Central Tendency



Measure of Central Tendency arrow_upward


  • A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data.
  •  It can be measured by:
    • Mean (Arithmetic Mean)
    • Median
    • Mode
    • The Geometric Mean
  • Variation can be measured by:
    • Range
    • Quartile
    • Variance
    • Standard Deviation
    • Coefficient of Variation

    Mean arrow_upward


  • It is the most common measure of central tendency.
  • The mean is equal to the sum of all the values in the data set divided by the number of values in the data set.
  •  

  • Example: Find the mean of the following:
  • {66, 72, 83, 89}
    • Solution:
    • Mean = (66 + 72 + 83 + 89) / 4
    • => Mean = 310 / 4
    • => Mean = 77.5

    Sample Mean

  • If n is the sample size then we have:
  • Where,
    • Sum of all values =
    • If the data represents a sample, the number of entries = n

    Population Mean :

  • If N is population size then we have:
  • Where,
  •      Sum

         Sum of all values of

    • If the data represents an entire population, the number of entries =N

    Mean from a Frequency Distribution arrow_upward


  • It is used when the only source of data is a frequency distribution.
  • Where,
  • n = Sample size,

    c = Number of classes in the frequency distribution,

    mj = Midpoint of the jth class,

    fj = Frequency of the jth class


    Median arrow_upward


  • The median is the middle score for a set of data that has been arranged in ascending or descending order.

  • Steps to find Median:

    Step: 1
  • Arrange numbers in ascending order.
  • Step: 2A
  • For odd numbers:
  • Step: 2B
  • For even numbers median is the average of the (n/2)th observation and the  observation.
  • Example 1: Find the median of the following:
  • 72, 65, 81, 89, 83
    • Solution: Arrange the numbers in ascending order.

    65, 72, 81, 83, 89

  • For odd numbers:
    • Here the median is 3rd observation that is 81.
  • Example 2: Find the median of the following:
    • {31, 57, 12, 22, 43, 50}
  • Solution: Arrange the numbers in ascending order.
  • 12, 22, 31, 43, 50, 57
  • For even numbers:
    • The median is the average of the middle two values,
  • Here median is 37.

  • Outliers arrow_upward


  • Outliers are numbers in a data set that are either way bigger or way smaller than the other numbers in a data set.
  • Example: In 1, 2, 3, 4, 4, 6 and 31 data set the number 31 is the outlier.

  • Mean is impacted by Outliers:

  • If the outlier is a high value, it will cause the mean value to shift to the higher side, while a low valued outlier will drop the mean value to a lower number.
  • Below example is showing how the mean is impacted by outliers:

  • Median is not impacted by Outliers:

  • Median is not impacted by outliers as shown below:
  •  


    Mode arrow_upward


  • Mode is a number that occurs most frequently in the data set.
  • Steps to find Mode:

  • Step: 1
    • Arrange the numbers in ascending or descending order.
  • Step: 2
    • Find the number which is occurring maximum number of times in the set.
    Note:
  • If there are two such numbers which occur maximum number of times then there is NO mode.
  • Example 1: Find the mode of the following:
  • {9, 3, 3, 7, 8, 15, 3, 9}

    • Solution: Arrange the numbers in ascending order:
    • 3, 3, 3, 7, 8, 9, 9, 15
    • As 3 is occurring maximum number of times, 3 is the mode.
  • Example 2: For data:
  • 6, 7, 2, 5, 3, 4, 9, 8

    • As no item in the data set given above is repeating itself so there will be No mode.

    Range arrow_upward


  • The difference between the largest and the smallest values of a distribution is known as range.
  • Example 1: The range of 10, 13, 17, 17, 18 will be:

  • Percentiles arrow_upward


  • Percentile is the value of a variable below which a certain percent of observations fall.
  • Example: The 30th percentile is the value below which 30 percent of the observations may be found.
  • The percent falling above the percentile Pth will be (100 – P)%.
  •  


    Quartiles arrow_upward


  • Quartiles split ordered data into 4 equal portions.
  • Each Quartile has position and value.
  • With the data in an ordered array, the position of is:
  • The value of is the value associated with that position in the ordered array.

  • Interquartile Range (IQR) arrow_upward


  • It is also known as mid-spread or middle fifty.
  • It is equal to the difference between upper quartile Q1 and lower quartile Q3 .
    • The middle is 50% of the values.
    • Resistant to extreme values.
  • Interquartile range = IQR = Q3 - Q1


  • Thank You from Kimavi arrow_upward


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