Measures of Central Tendency

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Measures of Central Tendency : 

Measures of Central Tendency S.Madhan SRM School of Teacher Education SRM University

Measures of Central Tendency : 

Measures of Central Tendency A measures of central of tendency may be defined as single expression of the net result of a complex group There are two main objectives for the study of measures of central tendency To get one single value that represent the entire data To facilitate comparison

Measures of Central Tendency : 

Measures of Central Tendency There are three averages or measures of central tendency Arithmetic mean Median Mode

Arithmetic Mean : 

Arithmetic Mean The most commonly used and familiar index of central tendency for a set of raw data or a distribution is the mean The mean is simple arithmetic average The arithmetic mean of a set of values is their sum divided by their number

Merits of the Use of Mean : 

Merits of the Use of Mean It is easy to understand It is easy to calculate It utilizes entire data in the group It provides a good comparison It is rigidly defined

Limitations : 

Limitations In the absence of actual data it can mislead Abnormal difference between the highest and the lowest score would lead to fallacious conclusions A mean sometimes gives such results as appear almost absurd. e.g. 4.3 children Its value cannot be determined graphically

Calculation of Arithmetic Mean : 

Calculation of Arithmetic Mean For Ungrouped Data Mean= Sum of observations Number of observations = X1+X2+…….+Xn n

Slide 8: 

Calculate mean for 40, 45, 50, 55, 60, 68 Mean= Sum of observations Number of observations = 40+45+50+55+60+68 6 = 318 6 Mean = 53

Slide 9: 

For Grouped Data Mean = AM + (Σfd) i N d(deviation) = X-AM i X = Midpoint, AM = Assumed Mean i = Class Interval size fd = Product of the frequency and the corresponding deviation

Slide 11: 

Mean = AM + (Σfd) i N = 65 + (-25) * 10 40 = 65 – 6.25 Mean = 58.75

Median : 

Median When all the observation of a variable are arranged in either ascending or descending order the middle observation is Median. It divides whole data into equal portion. In other words 50% observations will be smaller than the median and 50% will be larger than it.

Merits of Median : 

Merits of Median Like mean, median is simple to understand Median is not affected by extreme items Median never gives absurd or fallacious results Median is specially useful in qualitative phenomena

Limitations : 

Limitations It is not suitable for algebraic treatment The arrangement of the items in the ascending order or descending order becomes very tedious sometimes It cannot be used for computing other statistical measures such as S.D or correlation

Calculation of Median : 

Calculation of Median Un grouped data When there is an odd number of items Median = The middle value item When there is an even number of items Median = Sum of middle two scores 2

Slide 16: 

Calculate Median 7, 6, 9, 10, 4 Arrange the given data in ascending order: 4, 6, 7, 9, 10 N = 5 (odd number) Median = Middle term Median = 7 Calculate Median 6, 9, 3, 4, 10, 5 Arrange the given data in ascending order: 3, 4, 5, 6, 9, 10 N = 6 (even number) Median= Sum of the middle two scores = 5+6 2 2 Median = 5.5

Slide 17: 

Grouped Data Median = l + (N/2 – F) i fm Where, l = exact lower limit of the CI in which Median lies F = Cumulative frequency up to the lower limit of the CI containing Median fm = Frequency of the CI containing Median i = Size of the class interval

Slide 18: 

N/2=20

Slide 19: 

Median = l + (N/2 – F) i fm Here, l = 50, F = 14, fm = 10, i = 10 Median = 50 + (20 – 14) * 10 10 = 50 + 6 Median = 56

Mode : 

Mode The observation which occurs most frequently in a series is Mode

Merits of Mode : 

Merits of Mode It can be easily located by mere inspection It eliminates extreme variations It is commonly understood Mode can be determined graphically

Limitations : 

Limitations It is measure having very limited practical value It is not capable of further mathematical treatment It is ill-defined and indefinite and so trustworthy

Calculation of Mode : 

Calculation of Mode Ungrouped Data Mode = largest number of times that item appear Grouped Data Mode = 3*Median – 2*Mean

Slide 24: 

Calculate Mode for 100, 120, 120, 100, 124, 132, 120 Mode = 120, since 120 occurs the largest number of times (3 times) Calculate Mode for 100, 101, 110, 111, 113, 101, 113, 115 Mode = 101 & 113, since 101 and 113 occurs twice

Slide 25: 

For grouped data let we consider the previous problem that we solved in Mean and Median We have, Mean = 58.75 & Median = 56 Mode = 3*Median – 2*Mode = 3*56 – 2*58.75 = 168 – 117.5 Mode = 50.5

References : 

References Evaluation, Test and Measurement - J.C.Aggarwal Teaching of Mathematics - Dr. Anice James

Thank You : 

Thank You