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Knowledge about the calculation of mean,mode & median.

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- 1. Presented by:- Dr. Aarati vijaykumar 1st year M.D (K.C)
- 2. Introduction: Definition of statistics: It is the ‘science of collecting, classifying, presenting & interpreting data’ relating to any sphere of enquiry. Having learnt the methods of collection & presentation of data, we have to understand & grasp the application of mathematical techniques involved in analysis & interpretation of the data. As medicos, we should learn to apply the formulae straight to our problems without worrying how they have been deduced. Application of methods for analysis is quite easy & we should become familiar with them so as to verify our preconceived ideas or to remove doubts which might arise at the first look of figures collected.
- 3. “ If a man will begin with certainties, he shall end in doubts’: but if he will be content to begin with doubts, he shall end in certainties.” - Francis Bacon Characteristics of frequency distribution is of two types, 1. Measures of central tendency ( Location, Position, Average) 2. Measures of dispersion ( Scatterdness, Variability, Spread)
- 4. Definition: It refers to a single central number or value that condenses the mass data & enables us to give an idea about the whole or entire data. Types: 1. Arithmetic Mean 2. Median Q2 3. The mode Z x
- 5. Introduction: It is the most commonly used measure of central tendency. It is also called as ‘Average’. Definition: It is defined as additional or summation of all individual observations divided by the total number of observation.
- 6. Types of series 1. Ungrouped series ( Ungrouped data, Unclassified data, Raw data ) : Includes individual observations without frequency. 2. Grouped series ( Classified data ) : Includes individual observation with frequency & class frequency. Calculation : 1. Direct method 2. Indirect method
- 7. Merits of Arithmetic Mean 1. Easy to understand & to calculate. 2. It is correctly or rigidly defined. 3. It is based on each & every observation. 4. Every set of data has one & only one A.M. 5. Used for further mathematical calculations like standard deviation. Demerits of Arithmetic Mean 1. Affected by extreme values ( either low or high) 2. It can not be obtained even if a single value is missing.
- 8. Introduction : It is called Q2 because it denotes 2nd quartile or positional value. It is the 2nd measure of central tendency. Here there are 3 quartile Q1 , Q2 , Q3 which divides the distribution into 4 parts or equal. A Q1 Q2 Q3 B .
- 9. Definition : Median divides the distribution into two equal parts i.e. 50% of the distribution is below the median & 50% is above the median. Q1 = n/4, Q2 = 3 x n/4 Ungrouped data: When ‘n’ is odd if the total number of observations are even, then arrange the observations either in ascending or descending order & calculate the median by formula. Q3 = n+1/ 2
- 10. Definition : Dictionary meaning of mode is common or fashionable. Mode is the value which occurs more frequently in a given set of data. There are 3 types Type 1 Ex: Selection of mode : Observation having the highest repetition. 10,11,12,26,20,40,20,10,12,10 Mode = 10
- 11. Type 2 : Selection of mode: Observation containing highest frequency. Ex: Number of children per family. No.of children/Family No.of families 0 13 1 24 2 25 3 13 4 14 25 is highest frequency so ‘2’ is mode. Type 3: Class containing highest frequency.
- 12. Merits of Mode: 1. Easy to calculate & understand. 2. Not affected by extreme value. 3. Mode can be found by both qualitative & quantitative data. Demerits of Mode: 1. Some times no mode or more then one mode in a given set of distribution. 2. Not used for further mathematical calculation. 3. Not commonly used.
- 13. Examples of Ungrouped series : 1. Direct method = ∑x/n x = Individual observation n = Number of observation Ex: Systolic BP of the patients, calculate mean, mode & median. 1. 110mmHg x1 2. 100mmHg x2 3. 150mmHg x3 4. 140mmHg x4 5. 140mmHg x5 6. 120mmHg x6 x
- 14. Mean ( Average ) : = ∑x/n ∑ = Summation n = Number of samples x = Individual observation. ∑x = x1+ x2+ x3 + x4 + x5 + x6 = 760/6 = 126.6mmHg Mode : Most repeated number in the data: 140mmHg Median : 100, 110, 120, 140, 140,150 = 120+140 = 260/2 = 130mmHg x
- 15. Step deviation method of calculation mean : Ex: Height of the school children's given below find out the mean. 1. 148cm x1 2. 143cm x2 3. 160cm x3 4. 152cm x4 5. 157cm x5 6. 150cm x6 7. 155cm x7 Working origin ( w ) = 150cm
- 16. Formula : = ∑ ( x – w ) / n 148 -150 = -2 143 -150 = -7 160 - 150 = 10 152 - 150 = 2 157 - 150 = 7 150 -150 = 0 155 -150 = 5 = 15/7 = 2.1 = w + = 150 + 2.1 = 152.2 x x x
- 17. Find mean days of confinement after delivery in the following? Mean = ∑fx/n , ∑f = n = 137/18 = 7.61 Days of confinement x No. of patients grouped f Total days of each group fx 6 5 30 7 4 28 8 4 32 9 3 27 10 2 20 18 137
- 18. Definition: Measures of variability describes the spread or scatterdness of the individual observation around the central tendency. Significance : 1. Gives complete idea/picture of data 2. Helps in comparison of distribution. 3. Useful for further calculations 4. Gives idea about the reliability of average value.
- 19. Methods of dispersion 1. Range ( R ) 2. Inter quartile range ( IQR ) 3. Quartile deviation / Semi inter quartile range 4. Mean deviation / Average deviation (MD) 5. Standard deviation (SD)
- 20. Range : Definition: Is defined as the difference between the highest & lowest values in a set of data. R = H – L Ex: Weight of an adult person 50 -100kg Merits: Easy to calculate & understand Has got a well defined formula gives first hand information about variation Demerits: It is not based on all the values Affected by extreme value
- 21. Definition: It is the interval between the value of upper quartile ( the value above which 25% observation falls) & lower quartile ( the values which fall below the 25% ). So the measures gives us the range of middle 50% of observation & it is very helpful when the observations are not homogenous & extreme in nature. It is the superior measure over the range in such conditions.
- 22. Ex: Weight of the persons 1. 40kg 2. 45kg 3. 50kg 4. 55kg Q1 5. 60kg 6. 65kg 7. 70kg 8. 75kg 9. 80kg Q3 10. 85kg IQR = 55kg – 80kg 11. 90kg 12. 95kg
- 23. Merits of IQR: Easy & simple to understand Easy to calculate Not affected by extreme values Demerits of IQR : It is a positional value which is based on two quartile Based on first & last values
- 24. Definition : It is an average amount of scatter of the items in a distribution from any measures of the central tendency by ignoring the mathematical signs. Formula: M.D = ∑ |x – | / nx
- 25. Example: Average marks obtained in 5 internals by a student. x x - 25 25- 22 = 3 15 15- 22 = -7 25 25-22 = 3 25 25-22 = 3 20 20- 22 = -2 = ∑x/n = 110/5 = 22 x x
- 26. M.D = ∑ |x – | / n = 18/5 M.D = 3.6% Co-efficient of average/Mean deviation: CAD = MD/Mean × 100 = 3.6/22 × 100 = 180/11 = 16.36% x
- 27. Introduction: It is most widely used, best method of calculating deviation. Though in AD it takes into consideration of all the observation & it ignores the mathematical signs, but SD overcomes this problem by squaring the deviation. Definition: SD is the square root of summation of square of deviation of given set of observation from the AM divided by the total number of observation.
- 28. Formula : Ungrouped series Standard deviation = ∑( x- )2 / n n ˃ 30 Grouped series Standard deviation = ∑f (x - )2 / n n ˂ 30 Where, ∑ – is Summation of, x – is Individual observation, – is Arithmetic mean, n – is Total number of observation x x x
- 29. Average marks obtained in 5 internals by a student S.D = ∑ ( X - ) 2 / n = 80/5 = 16 = 4 Marks obtained x x - ( x - )2 25 25 – 22 = 3 9 15 15 – 22 = -7 49 25 25 – 22 = 3 9 25 25 – 22 = 3 9 20 20 – 22 = -2 4 = 110 = 80 x x x
- 30. Co – efficient of SD = SD/ Mean x 100 = 4 / 22 x 100 = 400 / 22 = 18.1 % Significance of SD : Based on all observations. Best method of calculation without ignoring mathematical signs. Useful for further statistical calculations. (i.e. Test of Significance etc.) Useful for calculation of standard error. Lesser the standard deviation, better the estimation of population mean.

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