Measure of Central tendency

Economic theory is made up of definitions and assumptions about human behaviors. The definition and assumptions are usually represented in words, mathematically or graphically. This topic is devoted to some of the tools of economic analysis. These are tables, charts, graphs, means, mean deviation, standard deviations, geometric mean and variance etc 
    The basic tools of economic analysis are categorized as follows:
  1. Measures of central tendency
  2. Measures of dispersion
  3. Graphical presentation of data. 

MEASURES FOR CENTRAL TENDENCY

Measures of central tendency is also known as measure of location .This describes the center or average of a set of data. In fact it provides the average value around which a set of data revolves search averages are arithmetic mean the media the mode geometric mean and of course and harmonic mean .Measure of Central tendency can also be described as the statistical measures that pick the middle or Central value ,single quantity or member in the majority. When it appears ,that is the most rational thing to do from a given set of numerical data.The commonest measure of Central tendency are the arithmetic mean,the median and the mode.

What is Central Tendency?

Central tendency is a descriptive summary of a dataset through a single value that reflects the center of the data distribution. Along with the variability (dispersion) of a dataset, central tendency is a branch of descriptive statistics.The central tendency is one of the most quintessential concepts in statistics. Although it does not provide information regarding the individual values in the dataset, it delivers a comprehensive summary of the whole dataset.

Measures of Central Tendency

The central tendency of a dataset can be described using the following measures:
  1. (a) Mean / Arithmetic Mean (Average): Represents the sum of all values in a dataset divided by the total number of the values.E.g
EXAMPLES
1. The number of hours a certain students spent on his studies in a particular week are as follows: 3,5,4,3, 5,6,2. Find the arithmetic mean.

SOLUTION
X=£×/n
=3+5+4+3+6+2=28/7
=4

2.  Find the mean of the data below: 5,7,9,3,4

SOLUTION
 X=£×/n
i.e 5+7+9+3+4/5
X=28/5=5.6


(b) Harmonic mean
Harmonic mean of a set of observation is define as the reciprocal  of the arithmetic mean of the reciprocal of the numbers. Harmonic mean is denoted by H.It can be calculated by the use of the formular
H=N/£ 1/x

EXAMPLE

1. Find the harmonic mean of a set of number 6,9.

SOLUTION
2/ 1/6+ 1/9
2/3+2/18
2/5/18
H=7

Advantages of harmonic mean
  • It does not affect the extreme values.
  • All values in the observation are represented
  • Harmonic mean is determinate

Disadvantages of Harmonic mean
  • It is difficult to calculate
  • It's scope is limited
  • Harmonic mean,s principle is difficult to understand
Quadratic mean
This is the square root of the arithmetic mean of their square.It is denoted by R .M.S and the formular is expressed as


         2
√£X
   ---
  N

Example

1. Find the quadratic mean of these numbers 1,2,4
SOLUTION

√1x1+2x2+4x4/3
√1+4+16/3
√21/3
√7=2.6


Advantage of quadratic Mean 
  • All values of the data are taken into account
Disadvantage of quadratic mean
  • It is not easily understood

2. Median: The middle value in a dataset that is arranged in ascending order (from the smallest value to the largest value). If a dataset contains an even number of values .Median of the dataset is the mean of the two middle  values.

EXAMPLE
1. Given the following data calculate the median
3,4,7,6,2,5,11

To solve to solve the above problem the number is usually arranged in order of magnitude from the lowest to the highest or vice versa using the above examples these are arranged in order of magnitude following the lowest as follow:
2,3,4,5,6,7,11
The median=5

The general formular or principle in determining the median for a set of odd number of observation is given as 
(n+1)th/2
For us to actually apply this general principle let us make use of the example, below

2,3,4,5,6,7,11
(n+1)th/n
(7+1)/2
=4
Then the median is 4th items again if we have even number of the observations then the name is the average of the values of the central number when they are arranged in order of magnitude
.
Advantages of median
  • The median is easy to compute
  •  The solution can be obtained at a glance.
  • The median does not require serious population in order to obtain.
  • The media can usually be understood
  • It can be obtained using graphical method
  • The median is not affected by the value of the extreme numbers.
Disadvantages of median
  • It may pose a lot of problem in calculation when large volume of number is involved.
  • The arrangement of numbers may prove cumbersome.
  • The median may not be required for further statistical calculation.
3. Mode: Defines the most frequently occurring value in a dataset. In some cases, a dataset may contain multiple modes while some datasets may not have any mode at all.           

  Even though the measures above are the most commonly used to define central tendency, there are some other measures, including, but not limited to, geometric mean, harmonic mean, midrange, and geometric median.The selection of a central tendency measure depends on the properties of a dataset. For instance, the mode is the only central tendency measure for categorical data, while a median works best with ordinal data.Although the mean is regarded as the best measure of central tendency for quantitative data, that is not always the case. For example, the mean may not work well with quantitative datasets that contain extremely large or extremely small values. The extreme values may distort the mean. Thus, you may consider other measures.

Example

1. Given the following set of numbers 3,4, 4 ,6,9 ,12 find the mode.

Solution
You will recall that the mode is the number with the highest frequency hence in this case the mode is 4 that importantly the mode exist here we are as there are some cases that the mode will not exist.

Advantages of the Mode
  1. The mode is commonly used for the treatment of Central tendency.
  2. Who is the mode is always the value of a set of data it is realistic to stop therefore it can appear as a good representative.
  3. The mode gives quick model information.
  4. Computation of the mode does not pose any problem today and numerator.
Disadvantages of the Mode
  1. It lacked accuracy toss it can be used for for the mathematical manipulations 
  2. The mode is not very good measure of Central tendency 
  3. It may be difficult to calculate when it involves a large set of data
Use of statistical analysis in economics 
  1. Useful for decision making: The use of statistical helps in decision making in economics.
  2. Easy interpretation of economic theories: Tools of economic analysis make it very easy for economic theories to be interpreted .
  3. Helps to establish relations among variables:It also helps to estimates the magnitude of relations among variables.
  4. Testing of Economic theories: Thevarious tools for analysing economic theories to test the prediction of economic theories against evidence.
  5. Help to analyse data attempts to summarise and analyse large volume of data.





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