Download Probability and Statstics - Mean Variance and more Study notes Economics in PDF only on Docsity!
Measures of Central Tendency For a given set of data, a value around which data are centered/ concentrated is called a measure of central tendency. Important measures of Central tendency: Arithmetic Mean, weighted Arithmetic Mean Geometric Mean, Harmonic Mean, Median, Mode. Main uses of measures of central tendency: ▲ It presents a concise picture of large and complicated set of data. ▲ It makes possible to compare two sets of data. ▲ It facilitates the interpretation of data. ▲ It helps in taking various decisions. For instance, one can observe if the income of a
family is below the average income or above the average income. Arithmetic mean (AM): For a given set of observations X 1 ,…,X (^) n , arithmetic mean is
defined as
If Xi occurs fi times (i=1,2,…,n), and N= F0 53 fi, then AM is given by For grouped data, xi is the mid point of i-th class interval. Median: The median of a set of observations arranged in ascending (or descending) order is the middle observation or mean of two middle observations. Calculation of median for grouped data: Calculate cumulative frequencies (less than type). The median is given by
Class interval having maximum frequency is modal class f 0 =frequency of modal class
f-1=frequency of the class preceding the modal class f 1 =frequency of the class succeeding the modal
class
Geometric Mean (GM): GM of n observations X 1 ,…,X (^) n, is given by GM=(X 1 …X^ n )1/n or log GM=(1/n)(logX 1 +…+logX (^) n)
Similarly we may define the GM for grouped data as log GM=(1/N)(f 1 logX^1 +…+f^ nlogX^ n )
Note: GM cannot be defined if some of the observations are negative. What happens when some observation is zero? Harmonic Mean (HM): For n observations X 1 , …,Xn , the HM is given by Ex: If mean of set of n (^) i observations is (i=1,
…,k), then what is the mean of combined set of n 1 +…+n (^) k observations.
Ex: The mean of 20 observations is 120. It has been found that by mistake an observation 150 is taken as 130 for calculating the mean. What is the correct mean?
Measures of Dispersion: The extent to which the numerical data tend to spread about an average value is called the variation or dispersion of data.
For grouped data
If we take , then we obtain mean deviation about mean. Ex: Mean deviation is minimum when taken about median (without proof). Mean Square Deviation : Mean square deviation about an arbitrary point a is given by
Mean square deviation about mean is called the variance. Hence formula for variance is
Standard deviation: SD(X)= F0 (^73) x = F0D6 Var(X)
Another expression for variance is as follows:
(Note: n should be taken as n-1 , as the degree of freedom, similarly N to be replaced by N-1. (See: Book by Sheldon Ross-3rd^ edition).
