Dr Hidayathulla Shaikh

At the end of the lecture students should be able to  Enumerate various measures of central tendency  Enumerate various measures of dispersion  Explain various measures of central tendency  Explain various measures of dispersion

 The measures of central tendency are central value around which the other values are distributed.  The main objective of measure of central tendency is to condense the entire mass of data and to facilitate comparison.  Measure of dispersion helps to know how widely the observations are spread on either side of the average.  Dispersion is the degree of spread or variation of variable about a central value.

3.Standard Deviation 2.Mean Deviation 1.Range (b) Measures of Dispersion 3.Mode 2.Median 1.Mean (a) Measures of Central Tendency

1) Mean 2) Median 3) Mode

Mean is obtained by summing up all the observations and dividing the total by the number of observations. It is given by X 1,X 2,X 3,………….,X n,n observations Where, Ʃ (sigma) is sum of X is variable or value of each observation in data, n is sample size or number of observations in data 1) MEAN X/n= n _ MEAN (x) = X 1 +X 2 +X 3 +………+X n

Merits:(Mean) 3.It can’t be calculated by inspection. 2.If one observation is missed, mean can’t be calculated 1.It is affected by extreme values Demerits: 4.It is amenable to algebraic treatment 3.It based upon all the observations. 2.It is easy to understand and easy to calculate. 1.It is rigidly defined.

Ex:The Ex:The following data gives the plaque scores in 5 students. Calculate Mean plaque score. (Mean plaque score of 5 students) = = / 5 Mean Solution: , , , , = [ ] / 5

Median is the middle value value (frequencies) after arranging them either either in the ascending or in the descending order. If n is even number, Median is the MEAN of the middle two terms/numbers If n is odd number, Median divides the observations EXACTLY into half (Middle Term). 2) MEDIAN

Merits:(Median) 3.It is affected much by fluctuations of sampling. 2.It is not amenable to algebraic treatment. 1.It is not based upon all the observations. Demerits: 3.It can be located graphically also. 2.It is not affected by extreme values. 1.It is easy to understand and easy to calculate.

Ex: The following data gives the plaque cores in 5 students. Calculate Median plaque score. = Median , , , , Ascending order: , , , ,

Mode is the one which is the most repeated in the particular series of observations. It is the value of the variable which occurs most frequently in a series of observation. OR 3) MODE

1.25, 3.10, 0.95, 0.75, 1.81, 2.72, 2.50 Ex: Calculate Mode from the following Plaque scores 1.81 repeated 2 times, The value of mode is 1.81 Therefore, the frequency of 1.81 is 2

Merits:(Mode) 2.In some cases mode is ill defined. 1.It is not based on all the observations. Demerits: 4.It can be calculated both from qualitative and quantitative data. 3.It is not affected by fluctuations of sampling. 2.It can be calculated by graphically also. 1.It is easy to understand and calculate.

Measures of Dispersion 3. Standard Deviation 2. Mean Deviation 1. Range

It is the difference between the highest and lowest values in the series. it is affected by sampling fluctuationsit is affected by sampling fluctuations It is based on extreme valuesIt is based on extreme values It is not based upon all the observationsIt is not based upon all the observations It is simplest measure of dispersionIt is simplest measure of dispersion = Maximum value - Minimum value RANGE 1) RANGE = Highest - Lowest

Ex:Find Ex:Find range of incubation period of small pox when we administered in 9 patients it was found to be Range = = 8 Maximum Value = 15 Here, Minimum Value = 7 14, 13, 11, 15, 10, 7, 9, 12, 10 (in days)

It is the average of deviations from the mean. It is given by where Ʃ(sigma) is sum of, X which is value of each observation, X which is mean, n is the number of observations and I I indicates ignoring negative signs. 2) MEAN DEVIATION (M.D) I x-x I / nM.D =

Steps involved in calculation of mean deviation: Ix-x I /n Divide sum by n, i.e.Divide sum by n, i.e. Ix-xI find the sum of absolute deviations,find the sum of absolute deviations, Find absolute deviation of (x-x) that is Ix-xIFind absolute deviation of (x-x) that is Ix-xI Which means ignore the negative sign Find deviations, i.e.(x-x)Find deviations, i.e.(x-x) Calculate mean.Calculate mean.

Merits for (Mean.Deviation) 2.It is not amenable to algebraic treatment 1.Mathematically illogical Demerits 3.It is based upon all the observations 2.Easy to understand. 1.Simple and easy to calculate.

Ex. The following data gives the respiration rate per minute. Find the Mean Deviation. Mean Deviation Respiration Rate per min (x) ( x- x) I x- xI Total ( x = 20) 0 22 Total ( x = 20) , 22, 24, 16, 17, 18, 19, 21, 20 = = 22/9 = I x - x I /n

3) Standard Deviation Deviation (S.D) is the square root of means. OR  Greater  Greater the Standard Deviation greater will be magnitude of dispersion from mean  A  A small Standard Deviation means a higher degree of uniformity of observations. (x -x )2 )2 )2 )2 / n for large samples = (x -x )2 )2 )2 )2 / n-1 for small samples S.D. = x = mean of the observations

STEPS INVOLVED IN CALCULATING STANDARD DEVIATION 6. Take square root n for large samples n-1 for small samples 5. Divide sum by n-1 or n 4. Find sum of the squares, (x -x )2 3. Find Deviations square, i.e. (x -x ) 2 2. Find Deviations, i.e.(x -x ) 1. Calculate Mean

Ex. The following data gives the respiration rate per minute. Find the Standard Deviation. Standard Deviation Respiration Rate per min (x) ( x- x) (x- x) Total ( x = 20) 0 60 Total ( x = 20) , 22, 24, 16, 17, 18, 19, 21, 20 = = 60/9-1 = (x -x )2 / n-1 =

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