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##### Measures of Central Tendency

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**Measures of Central Tendency**Mean Median Mode Measurement of Central Tendency By Dr.Anil Jain**Type Of Measures of Central Tendency**Measurement of Central Tendency By Dr.Anil Jain**Mean, median, and mode are three kinds of "averages". There**are many "averages" in statistics, but these three are the most common and certain.These are also known as central tendency measurement. Measurement of Central Tendency By Dr.Anil Jain**Mean**Definition of Mean:-The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers. Importance of Mean:- • It’s the most simplest method of Central tendencies measurement. • In this calculation importance is given to every individual value. • Importance is given to every data’s value. Measurement of Central Tendency By Dr.Anil Jain**Merits of Mean:-These are the merits of arithmetic mean in**scientific studies and researches:- • Simplicity • Certainty • Algebraic Discussion Possible • Permanence • Order Not Necessary • Test is Possible • Comparison • Rigidly defined. • It is easy to calculate and simple to follow. • It is based on all the observations. • It is determined for almost every kind of data. • It is finite and not indefinite. • It is readily put to algebraic treatment. • It is least affected by fluctuations of sampling. Measurement of Central Tendency By Dr.Anil Jain**Limits of Mean:-These are the limitations of a mean:-**• Non representative • Non Realistic • Misleading Conclusion • Superficial • The arithmetic mean is highly affected by extreme values. • It cannot average the ratios and percentages properly. • It is not an appropriate average for highly skewed distributions. • It cannot be computed accurately if any item is missing. • It can neither be determined by inspection or by graphical location • Arithmetic mean cannot be computed for qualitative data like data on intelligence honesty and smoking habit etc. • It is too much affected by extreme observations and hence it is not adequately represent data consisting of some extreme point. • Arithmetic mean cannot be computed when class intervals have open ends. Measurement of Central Tendency By Dr.Anil Jain**USES OF MEAN**1) Mean is used in fields such as business, engineering and computer science. 2) It is used in report card or in our population. 3)In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent. For example, per capita GDP gives an approximation of the arithmetic average income of a nation's population. 4)While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers. 5) It’s used to compute the variance and SD (Standard Deviation). It’s good for inferential statistics. Measurement of Central Tendency By Dr.Anil Jain**Formulas for Mean**Formula to individual series=∑x/n Formula to discrete series=∑fx/n Formula to continue series=A+ ∑fd/n X i ∑=addition of required column x=total of value f= frequency d=deviation =x-a/i n= number of Value A=assumed mean i= class interval Measurement of Central Tendency By Dr.Anil Jain**MedianThe "median" is the "middle" value in the list of**numbers. To find the median, your numbers should be listed in numerical order.It divides the group into two equal parts, one part comprising all values greater than the median value and the other part comprising all the values smaller than the median value. Merits of Median • Simplicity • Free from the effect of extreme values • Certainty Measurement of Central Tendency By Dr.Anil Jain**Real value.**• Graphic presentation. • Possible even when data is incomplete. • It is rigidly defined as in the case of Mean. • Even if the value of extreme item is much different from other values, it is not much affected by these values e.g. median in case of 4, 7, 12, 18, 19 is 12 and if we add two values equal to 450 10000, new median is 18. • It can also be used for the Quantities, those can't give A.M; as is in case of intelligence etc. • For open end intervals, it is also suitable one. As taking any value of the intervals, value of Median remains the same. • It can be easily calculated and is also easy to understand. • Median is also used for other statistical devices. • It can be located by inspection in some cases. • Extreme items may not be available to get Median. Measurement of Central Tendency By Dr.Anil Jain**Demerits of Median**• Lack of representative character. • Unrealistic. • Lack of algebraic treatment. • Even if the value of extreme items is too large, it does not effect too much, but due to this reason, sometimes median does not remain the representative of the series. • It is affected much more by fluctuations of sampling than A.M. • Unlike mean we can neither find total of terms as in case of A.M. nor median of some groups when combined. • In a continuous series it has to be interpolated. We can find its true-value only if the frequencies are uniformly spread over the whole class interval in which median lies. • If the number of series is even, we can only make its estimate; as the A.M. of two middle terms is taken as Median. Measurement of Central Tendency By Dr.Anil Jain**Formulas of Median**Formula to individual series= for odd n+1 and for even n+1/2 + n/2 Formula to continue series= L+N/2-F/f x i L= Lower limit of the median class F= CF just before entering in the median class f= frequency of median class N=addition of Frequency A=assumed mean i= class interval Measurement of Central Tendency By Dr.Anil Jain**Mode**The value of the variable which occurs most frequently in a distribution is called the Mode . Merits of mode Following are the various merits of mode:- • Simple and popular • Less effect of marginal values • Graphic presentation • Best representative • No need of knowing all the items or frequencies • commonly understood. • Mode is a value which exists in the series whereas arithmetic average may be a figure which may not be found in the series. It is the most common item of a series and is not an isolated example like the median. • It is not affected by the value of extreme items if the distribution follows the natural law relating to extremes. Usually, there is little concentration of items around extreme values. • It can be correctly calculated in open-end classes. • In a continuous series, mode can, be calculated even if all the item values are not given. Only the modal class and the frequencies of its adjoining classes are required to compute mode. Measurement of Central Tendency By Dr.Anil Jain**Demerits of mode**Following are the various demerits of mode: • Uncertain and vague • Not capable of algebraic treatment • Difficult • Complex procedure of grouping • Ignores extreme marginal frequencies. • Mode is ill-defined in case of bi-modal, multi-modal series. • It is not a representative average as it is not based on all the items of the distributions, If in a series of 1000 items 20 have a particular value and other values have frequency less than 20, mode becomes 20. But certainly 20 is not the typical or representative average. • Mode is affected to a great extent by the fluctuations of sampling. Measurement of Central Tendency By Dr.Anil Jain**Formulas of Mode**Formula to individual series=value of most frequent data Formula to continue series= L=f0-f1/2f0-f1-f2 X i L= Lower limit of the mode class f0= frequency of mode class f1= frequency of just before the mode class f2=frequency of just after the mode class i= class interval Measurement of Central Tendency By Dr.Anil Jain**Let’s ReviewAbout the only hard part of finding the mean,**median, and mode is keeping straight which "average" is which. Just remember the following: Mean:-The average of a set of data. Median:-The middle number of a set of data, when arrange in ascending order. Mode:-The most frequent value of a set of data. Measurement of Central Tendency By Dr.Anil Jain**Thanks**Measurement of Central Tendency By Dr.Anil Jain