Measures of Central Tendency

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No. Biostat - 5 Date: 18.01.2009 MEASURES OF CENTRAL TENDENCY Lecture Series on Biostatistics Dr. Bijaya Bhusan Nanda, M. Sc (Gold Medalist) Ph. D. (Stat.) Topper Orissa Statistics & Economics Services, 1988 bijayabnanda@yahoo.com

CONTENTS:

CONTENTS Descriptive Measures Measure of Central Tendency (CT) Concept and Definition Mean Median Mode Uses of Different Measures of CT Advantages and Disadvantages of Different Measures of CT

Enabling Objectives:

Enabling Objectives To equip the trainee with skills to manipulate the data in the form of numbers that they encounter as a health sciences professional. The better able they are to manipulate such data, better understanding they will have of the environment and forces that generate these data.

Descriptive Statistics :

Descriptive Statistics Classification and Tabulation of Data Presentation of Data Measure of Central Tendency (CT) Measure of Shape Measure of Dispersion

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Concept Data, in nature, has a tendency to cluster around a central Value. That central value condenses the large mass of data into a single representative figure. The Central Value can be obtained from sample values (Called Statistics) and Population observations (Called Parameters) Measures of Central Tendency

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Definition A measure of central tendency is a typical value around which other figures congregate. Simpson and Kofka (Statistician) Average is an attempt to find an single figure to describe a group of figures. Clark and Schakade (Statistician) Measures of Central Tendency

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Mathematical Average Arithmetic Mean Simply Mean Geometric Mean Harmonic Mean Positional Average Median Mode Mean, Median and Mode are the Most Commonly used MCT in Health Science Different Measure of Central Tendency (MCT)

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It should be rigidly defined so that different persons may not interpret it differently. It should be easy to understand and easy to calculate. It should be based on all the observations of the data. It should be easily subjected to further mathematical treatment. It should be least affected by the sampling fluctuation . It should not be unduly affected by the extreme values. Characteristics of an Ideal MCT

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M ost representative figure for the entire mass of data Tells the point about which items have a tendency to cluster Unduly affected by extreme items (very sensitive to extreme values) The positive deviations must balance the negative deviations (The sum of the deviations of individual values of observation from the mean will always add up to zero) The sum of squares of the deviations about the mean is minimum Characteristics of Mean

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AM is obtained by summing up all the observations and dividing the total by the number of observation . Calculation of Mean The mean is calculated by different methods in three types of series. Individual Data Series Discrete Data Series Grouped Data Arithmetic Mean (AM) or Mean

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Individual data : X 1 , X 2 , X 3 , X 4 , …………X n (n = Total No. of observation) Example: Find the mean size of Tuberculin test of 10 boys measured in millimeters. 3,5,7,7,8,8,9,10,11,12. 3 + 5 + 7 +7 + 8 + 8 + 9 + 10 + 11 + 12 Mean = 10 = 80 / 10 = 8 mm Mean = n X 1 + X 2 + X 3 +…..

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Let x 1 ,x 2 ,x 3 …… x n be the variate values and f 1 ,f 2 ,f 3 ….f n be their corresponding frequencies, then their mean is given by Mean = f 1 +f 2 +f 3 +….+f n x 1 .f 1 + x 2 .f 2 +x 3 .f 3 +……+x n .f n Example: Find mean days of Confinement after delivery in the following series. Days of Confinement ( X) No. of patients (f) Total days of confinement of each group. Xf 6 5 30 7 4 28 8 4 32 9 3 27 10 2 20 Total 18 137 Ans. Mean days of confinement = 137 / 18 = 7.61 Discrete frequency distribution

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Let m 1 , m 2 ,…….m n be the mid-point of the class and f 1 , f 2 ,……….f n be the corresponding frequency Direct method:  f i m i Mean:  X =  f i Where, A is the assumed mean and d i = m i -A. Shortcut method:  f i d i N Mean = A + , Where m i refers to the mid point of the i th class f i frequency of the i th class and  f i = N Grouped frequency distribution

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Example: Calculate overall fatality rate in smallpox from the age wise fatality rate given below. Age Gr in yr 0-1 2-4 5-9 Above 9 No. of Cases 150 304 421 170 Fatality rate 35.33 21.38 16.86 14.17

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Age Gr. in year No. of Cases (f) Fatality Rate (x) fx 0-1 150 35.33 5299.5 2-4 304 21.38 6499.52 5-9 421 16.86 7098.06 Above 9 170 14.17 2408.9 Total 1045 87.74 21305.98 Solution : Mean fatality rate= = ∑fx ∑f 21305.98 1045 =20.39

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Weight in Kg 2.0-2.4 2.5-2.9 3.0-3.4 3.5-3.9 4.0-4.4 4.5+ No. of infant 17 97 187 135 28 6 Problem: Find the mean weight of 470 infants born in a hospital in one year from following table.

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Solution : Weight in Kg. X Continuous CI in kg. Mid Value d i = m i -A f fd Sum of fd 2.0-2.4 1.95-2.45 2.2 -1.0 17 -17 -65.5 2.5-2.9 2.45-2.95 2.7 -0.5 97 -48.5 3.0-3.4 2.95-3.45 3.2 0 187 0 3.5-3.9 3.45-3.85 3.7 0.5 135 67.5 +104.5 4.0-4.4 3.95-4.45 4.2 1 28 28 4.5+ 4.45+ 4.7 1.5 6 9 Total 470 +39 Mean Weight = w + ( Σ fx / Σ f ) Mean Weight = 3.2 +( 39 / 470 ) = 3.28 Kg

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CI-Wt) Cont. CI f d fd 90 - 99 89.5 – 99.5 1 -5 -5 100 – 109 99.5 – 109.5 1 -4 -4 110 – 119 109.5 – 119.5 9 -3 -27 120 – 129 119.5 – 129.5 30 -2 -60 130 – 139 129.5 – 139.5 42 -1 -42 140 – 149 139.5 – 149.5 66 0 0 150 – 159 149.5 – 159.5 47 1 47 160 – 169 159.5 – 169.5 39 2 78 170 – 179 169.5 – 179.5 15 3 45 180 – 189 179.5 – 189.5 11 4 44 190 -199 189.5 -199.5 1 5 5 200 – 209 199.5 – 209.5 3 6 18 Total 265 99 N=265 Step deviation method X= A + Σ fd N X h = 145 + (0.3736) ( 10 ) = 145 +0 3.74 = 148.74 Ex:- Calculation of the mean from a frequency distribution of weights of 265 male students at the university of Washington d = X-A h X= 145 + 99 265 x 10

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Merits of Mean: It can be easily calculated. Its calculation is based on all the observations. It is easy to understand. It is rigidly defined by the mathematical formula. It is least affected by sampling fluctuations. It is the best measure to compare two or more series of data. It does not depend upon any position. Merits, Demerits and Uses of Mean

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Demerits of Mean : It may not be represented in actual data so it is theoretical. It is affected by extreme values. It can not be calculated if all the observations are not known. It can not be used for qualitative data i.e. love, beauty , honesty, etc. It may lead to fallacious conditions in the absence of original observations. Uses of Mean : 1. It is extremely used in medical statistics. 2. Estimates are always obtained by mean.

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Definition: Median is defined as the middle most or the central value of the variable in a set of observations, when the observations are arranged either in ascending or in descending order of their magnitudes. It divides the series into two equal parts. It is a positional average, whereas the mean is the calculated average. Median

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A positional value of the variable which divides the distribution into two equal parts, i.e., the median is a value that divides the set of observations into two halves so that one half of observations are less than or equal to it and the other half are greater than or equal to it. Extreme items do not affect median and is specially useful in open ended frequencies. For discrete data, mean and median do not change if all the measurements are multiplied by the same positive number and the result divided later by the same constant As a positional average, median does not involve values of all items . The median is always between the arithmetic mean and the mode Characteristic of Median

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For Individual data series: Arrange the values of X in ascending or descending order. When the number of observations N, is odd, the middle most value –i.e. the (N+1)/2 th value in the arrangement will be the median. When N is even, the A.M of N/2 th and (N+1)/2 th values of the variable will give the median. Computation of Median

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Example: ESRs of 7 subjects are 3,4,5,6,4,7,5. Find the median. Ans : Let us arrange the values in ascending order. 3,4,4,5,5,6,7. The 4 th observation i.e. 5 is the median in this series. Example: ESRs of 8 subjects are 3,4,5,6,4,7,6,7. Find the median. Ans : Let us arrange the values in ascending order. 3,4,4,5,6,6,7,7. In this series the median is the a.m of 4 th and 5 th Observations

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Arrange the value of the variable in ascending or descending order of magnitude. Find out the cumulative frequencies (c.f.). Since median is the size of (N+1)/2 th . Item, look at the cumulative frequency column find that c.f. which is either equal to (N+1)/2 or next higher to that and value of the variable corresponding to it is the median. Discrete Data Series

Grouped frequency distribution:

Grouped frequency distribution N/2 is used as the rank of the median instead of (N+1)/2. The formula for calculating median is given as: Median = L+ [(N/2-c.f)/f]  I Where L = Lower limit of the median class i.e. the class in which the middle item of the distribution lies. c.f = Cumulative frequency of the class preceding the media n class f = Frequency of the median class. I= class interval of the median class. The class whose cumulative frequency is equal to N/2 or next higher to it in the distribution is the median class.

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Example : Calculation of the median (x). data represent weights of 265 male students at the university of Washington Class – interval ( Weight) f Cumulative frequency f “less than” 90 - 99 1 1 100 – 109 1 2 110 – 119 9 11 120 – 129 30 41 130 – 139 42 83 140 – 149 66 149 150 – 159 47 196 160 – 169 39 235 170 – 179 15 250 180 – 189 11 261 190 -199 1 262 200 - 209 3 265 N = 265 N/2=132.5 Median for grouped or interval data X= L + N/2-Cf f  I X= 140 + 132.5-83 66 x 10 = 140 + 49.5 66 x 10 = 140 + (0.750) ( 10 ) = 140 + 7.50 = 147.5

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Definition: Mode is defined as the most frequently occurring measurement in a set of observations, or a measurement of relatively great concentration, for some frequency distributions may have more than one such point of concentration, even though these concentration might not contain precisely the same frequencies. Mode

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Mode of a categorical or a discrete numerical variable is that value of the variable which occurs maximum number of times and for a continuous variable it is the value around which the series has maximum concentration. The mode does not necessarily describe the ‘most’( for example, more than 50 %) of the cases Like median, mode is also a positional average. Hence mode is useful in eliminating the effect of extreme variations and to study popular (highest occurring) case (used in qualitative data) The mode is usually a good indicator of the centre of the data only if there is one dominating frequency. Mode not amenable for algebraic treatment (like median or mean) Median lies between mean & mode. For normal distribution, mean, median and mode are equal (one and the same) Characteristics of Mode

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In case of discrete series, quite often the mode can be determined by closely looking at the data. Discrete Data Series Quantity of glucose (mg%) in blood of 25 students 70 88 95 101 106 79 93 96 101 107 83 93 97 103 108 86 95 97 103 112 87 95 98 106 115 Example. Find the mode - First we arrange this data set in the ascending order and find the frequency.

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Quantity of glucose (mg%) in blood Frequency Quantity of glucose (mg%) in blood Frequency 70 1 97 2 79 1 98 1 83 1 101 2 86 1 103 2 87 1 106 2 88 1 107 1 93 3 108 1 95 2 112 1 96 1 115 1 This data set contains 25 observations. We see that, the value of 93 is repeated most often. Therefore, the mode of the data set is 93.

Calculation of Mode::

Calculation of Mode: L = lower limit of the modal class i.e.- the class containing mode f 1 = Frequency of modal class f 0 = Frequency of the pre-modal class f 2 =Frequency of the post modal class I = Class length of the modal class Mode= L + {(f 1 – f 0 ) [(f 1 – f 0 ) + (f 1 – f 2 )] x I Grouped frequency distribution

Example: Find the value of the mode from the data given below :

Example: Find the value of the mode from the data given below Weight in Kg Exclusive CI No of students Weight in Kg No of students 93-97 92.5-97.5 2 113-117 14 98-102 97.5-102.5 5 118-122 6 103-107 102.5-107.5 12 123-127 3 108-112 107.5-112.5 17 128-131 1 Solution: By inspection mode lies in the class 108-112. But the real limit of this class is 107.5-112.5 Where, L = 107.5,f 1 = 17, f 0 = 12, f 2 = 14, I = 5 Mode= L + {(f 1 – f 0 ) [(f 1 – f 0 ) + (f 1 – f x )] x I = 110.63

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Merits: Mode is readily comprehensible and easy to calculate. Mode is not at all affected by extreme values, Mode can be conveniently located even if the frequency distribution has class-intervals of unequal magnitude provided the modal class and the classes preceding and succeeding it are of the same magnitude, Merits, demerits and use of mode

Demerits: :

Demerits: Mode is ill-defined. Not always possible to find a clearly defined mode. It is not based upon all the observations, It is not amenable to further mathematical treatment As compared with mean, mode is affected to a great extent by fluctuations of sampling, When data sets contain two, three, or many modes, they are difficult to interpret and compare. Uses Mode is useful for qualitative data.

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The mean is rigidly defined. So is the median, and so is the mode except when there are more than one value with the highest frequency density. The computation of the three measures of central tendency involves equal amount of labour. But in case of continuous distribution, the calculation of exact mode is impossible when few observations are given. Even though a large number of observations grouped into a frequency distribution are available, the mode is difficult to determine. The present formula discussed here for computation of mode suffers from the drawback that it assumes uniform distribution of frequency in the modal class. As regards median, when the number of observations are even it is determined approximately as the midpoint of two middle items. The value of the mode as well as the median can be determined graphically where as the A.M cannot be determined graphically. The A.M is based upon all the observations. But the median and, so also, mode are not based on all the observations Comparison of mean, median and mode

Contd……:

Contd…… When the A.Ms for two or more series of the variables and the number of observations in them are available, the A.M of combined series can be computed directly from the A.M of the individual series. But the median or mode of the combined series cannot be computed from the median or mode of the individual series. The A.M median and mode are easily comprehensible. When the data contains few extreme values i.e. very high or small values, the A.M is not representative of the series. In such case the median is more appropriate. Of the three measures, the mean is generally the one that is least affected by sampling fluctuations, although in some particular situations the median or the mode may be superior in this respect. In case of open end distribution there is no difficulty in calculating the median or mode. But, mean cannot be computed in this case. Thus it is evident from the above comparison, by and large the A.M is the best measure of central tendency.

REFERENCE:

REFERENCE Applied Biostatistical Analysis, W.W. Daniel Biostatistical Analysis, J.S. Zar Mathematical Statistics and data analysis, John. A. Rice Fundamentals of Applied Statistics, S.C Gupta and V. K Kapoor. Fundamental Mathematical Statistics, S.C. Gupta, V.K. Kapoor

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THANK YOU

Outlier :

Outlier An observation (or measurement) that is unusually large or small relative to the other values in a data set is called an outlier . Outliers typically are attributable to one of the following causes: The measurement is observed, recorded, or entered into the computer incorrectly. The measurements come from a different population. The measurement is correct, but represents a rare event.

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