Measuresof Central Tendency are used to describe the set of data. Mean Median Mode
1.Arithmetic Mean2.Shortcut Method to Mean3.Direct Method to mean4.Shortcut Method for Direct Method5.Calculation of Mean:- *Individual series *Discret series *continuous series
MeanThe mean is also known as the average of the set. Add all of the number is in the set and divide this sum by the number of items in the set.
1.Easy to understand.2.Simple to compute.3. Based on all items.4.Affected by extreme observation.5.Capable of further calculation.
1.Arithmetic Mean: X X N X X1 X 2 X 3 ..... Xn N =No. of observations/items in a series x 20 30 10
2. Shortcut method to Mean Ex:- X d= X-A d X A 20 N Assumed mean 30 A d X A 40 50
3.Direct Method:- Marks(X) f fx f x 20 4 X f 30 2 40 3 f N 50 1
4.Shortcut Method for Direct Method:- f d X A f d x A Marks f d = x-A f.d 20 8 30 12 40 20 50 10
Calculation of Mean in Continuous series:- f ( m) X f m mid point of class interval Marks (x) f m f (m) 10 – 20 8 20 – 30 12 30 – 40 20 40 – 50 10 50 - 60 6
Since its definition is precise and clear, calculation is easy and its value always determinate. Its chief merit consists in the fact that it take onto consideration every item in the data. NO special arrangement of data is necessary while calculating it, unlike median and mode where we have to arrange and group the data in a certain manner. It forms a good basis of comparison while comparing different groups of numerical data.
Omission of even a single item of the data gives an incorrect value of the A.M., unlike median and mode where extreme items can simply be discarded. It may not identical with any one of the items of the data. That is, it may not be one of the figures that comprise the data. The fact that it gives a large weight to extreme items is its handicap since it then fails to be a good representatives. That is, the value of A.M. of the data consisting of very and very small items may lead to wrong conclusions. It cannot be used in qualitative studies, unlike the median.
1.Definition2.Characteristics3.Calculating median:- *Individual series *Discret series *Continuous series4.Advantages & Disadvantages ofmedian.
The value of the middle item of a series that is ordered either in the ascending or descending order of magnitude is called the median.1.Measure of central tendency marks order marks order2.Middle value in ordered sequence 40 30 If n is odd, middle 60 value of sequence 20 10 If n is even, average of 50 2 middle values. (n= no 40 of items in a series) 10 20 30 50
Odd series:- middle value of sequence.(series must be in order) Even series:- median= N 1 th item 2 N No. of observation
Arrange the data in ascending or descending order Find c.f. Apply N 1 th 2 Look for the value in cf. corresponding figure to cf is the median. Income No. of cf person(f) 800 16 1000 24 1500 26 1800 30 2000 20 2500 6
Method:- same as discrete series but instead of N 1 2 Napply 2 to determine the median class.Then apply, N cfMedian = L 2 *i f L= lower limit of the median class cf= cf of the class preciding the median class f= frequency of median class i=class interval
Marks No.of cf student(f)0–5 4 N 25 – 10 5 N10- 15 10 cf L 2 *i f15 – 20 10
Most of the conditions of an ideal average are satisfied by it. The median is of immense use while estimating qualities such as honestly, intelligence, virtue, morality etc. and proves to be a good representative. It’s possible to find the median by knowing only the values of central items and the no. of items. That is, the values of extreme items are not necessary for finding the median.
An irregular series is characterized by extreme variations between the items, in the case of such a series the median fails to be representative. Arrangement of the data either in the ascending order of the magnitude or descending order is the chief requisite. This process is tedious when the data is vast. Sometimes the medians may exist between two values, thus involving the work of estimation. Unlike the A.M., it is not possible to find the total value of all items if we know the value of the median and the no. of items.
QUARTILE:- It divides the entire series into 4 equal parts.Quartiles in:- # Individual & discrete series # Continuous series
N 1 Q1 th 4 N 1 Q3 3 th 4 N NQ1 th Q3 3 th 4 4 N N cf 3 cf 4 4Q1 L *i Q3 L *i f f
The value of the item in avariable that is repeated thegreatest number of times iscalled the mode.
The number that repeats themost 3, 4, 4, 10, 15, 16, 18 4 is the mode
there may be more than one mode3, 3, 4, 4, 4, 7, 9, 9, 9 4 & 9 are the modes
when there is only single value(f) which is highest. f1 f M L i 2 f1 f f 2L= lower limit of model classf1= frequency of model classfo= frequency of class preciding the model classf2= frequency of class successiding the model class
Calculate mode f1 f marks No. of M L i student 2 f1 f f 20 – 10 1010 – 20 1220 – 30 1530 – 40 840 - 50 5
When there are two similarfrequencies/values which are highest.MODE = 3 MEDIAN – 2MEAN
1.It is the most predominant item in adiscrete series.2.If a continuous series is a regular one i.e.with the maximum frequency in the centrethe mode can be calculated easily withoutknowing the frequencies at the twoextremities of the series.3.It is very useful average in studying businessrelating to sales, profits etc.
1.It is quite possible that in certain types ofdata it may not be properly defined & hence itmay be indeterminate & indefinite.2.The value of the mode is not the valueobtained by considering the value of every itemof the data.3.In the case of an irregular series it is not veryeasy to determine the mode.4.It is quite possible that in certain instancesthere may be two or more values for the mode.
BOOK:- Quantitative Techniques inBusiness by : Dr. Aditham B.Rao