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CHAPTER
MEASURES OF CENTERAL TENDENCY
Objectives
At the end of this chapter students will be able to:
 identify types of measure of central tendency
 define and calculate the mean, mode, median, with their interpretation the result.
 summarize an aggregate of statistical data by using single measure and make comparison
3.1 Introduction and objectives of measures of central tendency
When we want to make comparison between a groups of number it is good to have a single value that is
considered to be a good representative of each group.
This single value is called the average of the group. Averages are also called measures of central tendency
A measure of central tendency (measures of center or central location) is a summary measure that attempts
to describe a whole set of data with a single value that represents the middle or center of its distribution.
Conti.
Summery Statistic that Represent center point.
Measure of central tendency are single value around which a large number of values of distribution
are condensed in the center of the distribution(These measures indicate where most values in a
distribution fall)
3.2.2 objective of central tendency
To get single value that describes the characteristics of the entire population
To facilitate group compression
For further statistical studies
Conti…
3.3. General characteristics of measure of central tendency
 It should be based on all observation
It should be as little as affected by extreme observations
It should be defined rigidly(have definite value)
3.4. The summation Notation( )
The symbol is used mathematical short hand for operation addition to denote “the sum of” suppose
you have 𝑛 observation 𝑥1,𝑥2,…𝑥𝑛 then the sum 𝑥1 +𝑥2 +,…+𝑥𝑛 can be written as 𝑖
𝑛
𝑥𝑖 and read as
the sum of all 𝑥𝑖, 𝑖 = 1,2,….,𝑛
Properties of summation
 𝑖
𝑛
𝑘 = 𝑛 ∗ 𝑘 , where k is any constant
 𝑖
𝑛
𝑘𝑋𝑖 = 𝑘 𝑖
𝑛
𝑋𝑖 , where k is any constant
 𝑖
𝑛
𝑎 + 𝑏𝑋𝑖 = 𝑎𝑛 + 𝑏 𝑖
𝑛
𝑋𝑖 , where a and b are any constant
 𝑖
𝑛
𝑋𝑖 + 𝑌𝑖 = 𝑖
𝑛
𝑋𝑖 + 𝑖
𝑛
𝑌𝑖
 𝑖
𝑛
𝑋𝑖 ∗ 𝑌𝑖 ≠ 𝑖
𝑛
𝑋𝑖 ∗ 𝑖
𝑛
𝑌𝑖
Types of Measures of Central Tendency
There are Three different measures of central tendency; each has its advantage and disadvantage.
 The Mean (Arithmetic, Geometric, Weighted and Harmonic)
 The Mode
 The Median
Mean
a) Arithmetic Mean
Is defined as the sum of the magnitude of the items divided by the number of items
Is denoted by A.M or 𝑋
𝑋 =
𝑥1+𝑥2+𝑥3,…….𝑥𝑛
𝑛
= 𝑖=1
𝑛
𝑋𝑖
𝑛
The mean of 𝑋, if 𝑥1 occurs 𝑛1 times, 𝑥2 occurs 𝑛2 times, and 𝑥𝑛 occurs 𝑓𝑛 times then
The mean will be 𝑥 = 𝑖=1
𝑛
𝑓𝑖𝑥𝑖
𝑓𝑖
Example: obtain the mean of the following number 2,7,8,2,7,3,7
Solution:
Arithmetic mean for grouped data
 If data are given in the shape of a continuous frequency distribution, then the mean is obtained as
follows
 The mean will be 𝑥 = 𝑖=1
𝑘
𝑓𝑖𝑥𝑖
𝑖
𝑘
𝑓𝑖
,where 𝑥𝑖 the class mark of the 𝑖𝑡ℎ class,𝑓𝑖 the frequency of the 𝑖𝑡ℎ
class
Special properties of arithmetic mean
1. The sum of deviation a set of items from their mean is always zero. i.e. 𝑖
𝑛
𝑋𝑖 − 𝑋 = 0
2. The sum of the squared deviations of a set of items from their mean is the minimum i.e.
𝑖
𝑛
𝑋𝑖 − 𝑋 2
< 𝑋𝑖 − 𝐴 2
, 𝐴 ≠ 𝑋
3. If 𝑋1 is the mean of 𝑛1 observation, if 𝑋2 is the mean of 𝑛2 observations,…if 𝑋𝑘 is the mean of
𝑛𝑘 observations, then then the mean of all observation in all groups often called the combined mean
is given by:
𝑋𝑐 =
𝑋1𝑛1+𝑋2𝑛2+⋯𝑋𝑘𝑛𝑘
𝑛1+𝑛2+⋯𝑛𝑘
= 𝑖
𝑘
𝑋𝑖𝑛𝑖
𝑖
𝑘
𝑛𝑖
Conti..
4. if wrong figure has been used when calculating the mean of the correct mean can be obtained
with out repeating the whole processes
Correct mean = wrong mean +
𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑣𝑎𝑙𝑢𝑒 −𝑤𝑟𝑜𝑛𝑔 𝑣𝑎𝑙𝑢𝑒
𝑛
Where n is total number of observations.
Conti.
Example: an average weight of 10 students was calculated to be 65. later on it was discovered that
one weight was misread as 40 instead 80kg. Calculate the correct average weight
5. The effect of transforming original series on the mean.
a) If constant k is added/subtract to or from every observation then the new mean will be the
𝑋𝑚𝑒𝑎𝑛 ± 𝑘 respectively
b) If every observations are multiplied by a constant 𝑘 then the new mean will be 𝑘 ∗ 𝑜𝑙𝑑 𝑚𝑒𝑎𝑛
Conti…
b) weighted arithmetic mean
 While calculating simple arithmetic mean, all item were assumed to be equally importance(each value in the data set has
equal weight).
 When the observation are different weight we use weighted average.
 Weights are assumed to each item in proportion to items relative importance.
 If 𝑥1, 𝑥2,…..,𝑥𝑛 represents value of item, and 𝑤1 ,𝑤2,…., 𝑤𝑛 are the corresponding weights, then the weighted mean,(𝑥𝑤)
given by
(𝑥𝑤) =
𝑥1𝑤1+𝑥2𝑤2+,…..,𝑥𝑛𝑤𝑛
𝑤1+𝑤2+,….,𝑤𝑛
= 𝑖
𝑛
𝑥𝑖𝑤𝑖
𝑖
𝑛
𝑤𝑖
Conti..
Merits and demerits of arithmetic mean
Merits:
 Its based on all observation.
 It is suitable for further mathematical treatment
 It is stable average, i.e. Its not affected by fluctuation of sampling
 It is easy to calculate and simple to understand
Conti…
Demerits:
Its affected by extreme observations
Its can not be used in the case of open ended classes
It can not be used when dealing with qualitative characteristics, such as beauty.
Geometric mean
The geometric mean like arithmetic mean is calculated average. It is used when observed values are
measured as ratios, percentages, proportions, or growth rates.
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Presentation1.pptx

  • 1. CHAPTER MEASURES OF CENTERAL TENDENCY Objectives At the end of this chapter students will be able to:  identify types of measure of central tendency  define and calculate the mean, mode, median, with their interpretation the result.  summarize an aggregate of statistical data by using single measure and make comparison 3.1 Introduction and objectives of measures of central tendency When we want to make comparison between a groups of number it is good to have a single value that is considered to be a good representative of each group. This single value is called the average of the group. Averages are also called measures of central tendency A measure of central tendency (measures of center or central location) is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or center of its distribution.
  • 2. Conti. Summery Statistic that Represent center point. Measure of central tendency are single value around which a large number of values of distribution are condensed in the center of the distribution(These measures indicate where most values in a distribution fall) 3.2.2 objective of central tendency To get single value that describes the characteristics of the entire population To facilitate group compression For further statistical studies
  • 3. Conti… 3.3. General characteristics of measure of central tendency  It should be based on all observation It should be as little as affected by extreme observations It should be defined rigidly(have definite value) 3.4. The summation Notation( ) The symbol is used mathematical short hand for operation addition to denote “the sum of” suppose you have 𝑛 observation 𝑥1,𝑥2,…𝑥𝑛 then the sum 𝑥1 +𝑥2 +,…+𝑥𝑛 can be written as 𝑖 𝑛 𝑥𝑖 and read as the sum of all 𝑥𝑖, 𝑖 = 1,2,….,𝑛
  • 4. Properties of summation  𝑖 𝑛 𝑘 = 𝑛 ∗ 𝑘 , where k is any constant  𝑖 𝑛 𝑘𝑋𝑖 = 𝑘 𝑖 𝑛 𝑋𝑖 , where k is any constant  𝑖 𝑛 𝑎 + 𝑏𝑋𝑖 = 𝑎𝑛 + 𝑏 𝑖 𝑛 𝑋𝑖 , where a and b are any constant  𝑖 𝑛 𝑋𝑖 + 𝑌𝑖 = 𝑖 𝑛 𝑋𝑖 + 𝑖 𝑛 𝑌𝑖  𝑖 𝑛 𝑋𝑖 ∗ 𝑌𝑖 ≠ 𝑖 𝑛 𝑋𝑖 ∗ 𝑖 𝑛 𝑌𝑖 Types of Measures of Central Tendency There are Three different measures of central tendency; each has its advantage and disadvantage.  The Mean (Arithmetic, Geometric, Weighted and Harmonic)  The Mode  The Median
  • 5. Mean a) Arithmetic Mean Is defined as the sum of the magnitude of the items divided by the number of items Is denoted by A.M or 𝑋 𝑋 = 𝑥1+𝑥2+𝑥3,…….𝑥𝑛 𝑛 = 𝑖=1 𝑛 𝑋𝑖 𝑛 The mean of 𝑋, if 𝑥1 occurs 𝑛1 times, 𝑥2 occurs 𝑛2 times, and 𝑥𝑛 occurs 𝑓𝑛 times then The mean will be 𝑥 = 𝑖=1 𝑛 𝑓𝑖𝑥𝑖 𝑓𝑖 Example: obtain the mean of the following number 2,7,8,2,7,3,7 Solution:
  • 6. Arithmetic mean for grouped data  If data are given in the shape of a continuous frequency distribution, then the mean is obtained as follows  The mean will be 𝑥 = 𝑖=1 𝑘 𝑓𝑖𝑥𝑖 𝑖 𝑘 𝑓𝑖 ,where 𝑥𝑖 the class mark of the 𝑖𝑡ℎ class,𝑓𝑖 the frequency of the 𝑖𝑡ℎ class
  • 7. Special properties of arithmetic mean 1. The sum of deviation a set of items from their mean is always zero. i.e. 𝑖 𝑛 𝑋𝑖 − 𝑋 = 0 2. The sum of the squared deviations of a set of items from their mean is the minimum i.e. 𝑖 𝑛 𝑋𝑖 − 𝑋 2 < 𝑋𝑖 − 𝐴 2 , 𝐴 ≠ 𝑋 3. If 𝑋1 is the mean of 𝑛1 observation, if 𝑋2 is the mean of 𝑛2 observations,…if 𝑋𝑘 is the mean of 𝑛𝑘 observations, then then the mean of all observation in all groups often called the combined mean is given by: 𝑋𝑐 = 𝑋1𝑛1+𝑋2𝑛2+⋯𝑋𝑘𝑛𝑘 𝑛1+𝑛2+⋯𝑛𝑘 = 𝑖 𝑘 𝑋𝑖𝑛𝑖 𝑖 𝑘 𝑛𝑖
  • 8. Conti.. 4. if wrong figure has been used when calculating the mean of the correct mean can be obtained with out repeating the whole processes Correct mean = wrong mean + 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑣𝑎𝑙𝑢𝑒 −𝑤𝑟𝑜𝑛𝑔 𝑣𝑎𝑙𝑢𝑒 𝑛 Where n is total number of observations.
  • 9. Conti. Example: an average weight of 10 students was calculated to be 65. later on it was discovered that one weight was misread as 40 instead 80kg. Calculate the correct average weight 5. The effect of transforming original series on the mean. a) If constant k is added/subtract to or from every observation then the new mean will be the 𝑋𝑚𝑒𝑎𝑛 ± 𝑘 respectively b) If every observations are multiplied by a constant 𝑘 then the new mean will be 𝑘 ∗ 𝑜𝑙𝑑 𝑚𝑒𝑎𝑛
  • 10. Conti… b) weighted arithmetic mean  While calculating simple arithmetic mean, all item were assumed to be equally importance(each value in the data set has equal weight).  When the observation are different weight we use weighted average.  Weights are assumed to each item in proportion to items relative importance.  If 𝑥1, 𝑥2,…..,𝑥𝑛 represents value of item, and 𝑤1 ,𝑤2,…., 𝑤𝑛 are the corresponding weights, then the weighted mean,(𝑥𝑤) given by (𝑥𝑤) = 𝑥1𝑤1+𝑥2𝑤2+,…..,𝑥𝑛𝑤𝑛 𝑤1+𝑤2+,….,𝑤𝑛 = 𝑖 𝑛 𝑥𝑖𝑤𝑖 𝑖 𝑛 𝑤𝑖
  • 11. Conti.. Merits and demerits of arithmetic mean Merits:  Its based on all observation.  It is suitable for further mathematical treatment  It is stable average, i.e. Its not affected by fluctuation of sampling  It is easy to calculate and simple to understand
  • 12. Conti… Demerits: Its affected by extreme observations Its can not be used in the case of open ended classes It can not be used when dealing with qualitative characteristics, such as beauty. Geometric mean The geometric mean like arithmetic mean is calculated average. It is used when observed values are measured as ratios, percentages, proportions, or growth rates.