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Measures of dispersion and normal distribution with examples

- 1. Measures of Dispersion & Normal Distribution By Dr. Dinesh kumar Meena, Pharm.D Ph.D Research Scholar Department of Pharmacology, Jawaharlal Institute of Postgraduate Medical Education & Research (JIPMER)
- 3. Central Tendency Summarises data with a single value which can represent the entire data Mean, Median & Mode This value does not reveal the variability present in Data Measures of dispersion used to quantify the variability of data
- 4. • The average income of three families (Ram, Rahim & Maria) is same • But there are considerable differences in individual incomes (Ram’s family differences in income are comparatively lower , In Rahim’s family differences are higher and In maria’s family differences is highest) • Average tell only one aspect of distribution i.e. representative size of the values • Dispersion is the extent to which values in a distribution differ from the average of the distribution
- 5. Measures of dispersion 1. Range 2. Quartile Deviation 3. Mean Deviation 4. Standard Deviation
- 6. Range • Range (R) is the difference between the largest value (L) and smallest value (S) in a distribution. R = L – S Range of income in Ram’s family : 18000-12000 = 6000 Rahim’s family : 22000-7000 = 15000 Maria’s family = 50000 – 0 = 50000 • High value of range represent high dispersion Mariy’s family > Rahim’s family > Ram’s family
- 7. • Range is not based on all the values in data set • As long as minimum and maximum values remain unaltered, any change in other values doesn’t affect range. • Range can not be calculates for open-ended frequency distribution. Those in which either the lower limit of the lowest class or upper limit of the highest class or both are not specified.
- 8. Quartile Deviation • Entire data is divided into four equal parts, each containing 25% of the values. • The upper and lower quartile (Q3 & Q1 respectively) are used to calculate interquartile range (Q3-Q1). • Half of the interquartile range is called Quartile Deviation • Quartile Deviation = Q3-Q1 2
- 9. Calculate the quartile deviation of following observations: 20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70. Q. D. Q3-Q1 2 Q1 = n + 1 Q 3 = 3(n + 1) n = number of observations 4 4 Q1 = 11+1 Q3 = 3 (11 +1) 4 4 Q1 = 3th value i.e. 29 Q3 = 9th value i.e. 51 Q.D. = 51-29 = 11 2
- 10. Mean Deviation • Range and Quartile Deviation gives a good idea about dispersion but do not attempt to calculate how far the values are, from their average. • Mean Deviation and Standard Deviation are two measures of dispersion which are based upon deviation of values from their average. • Mean deviation is simply the arithmetic mean of the differences of the values from their averages ( Average used is either the arithmetic mean or median)
- 11. Objective is to find a location to establish a college so that the average distance travelled by students is minimum We found that student will have to travel more, on average , if the college is situated at town A or E. but if college is some where in the idle, student likely to travel less. if situated at town A : Average distance travelled by student will be 9.94 km if situated at town C : Average distance travelled by student will be 5.9 km
- 12. Calculating Mean Deviation For ungrouped data For grouped data From Arithmetic Mean From Arithmetic Mean From Median From Median
- 13. Mean Deviation from Arithmetic Mean for ungrouped data Steps : 1. Calculate the Arithmetic Mean (A.M.) of the values 2. Calculate the difference between each value and A.M. (called deviations and dented by d) 3. Calculate the A.M. of these differences (Mean Deviations).
- 14. Example: Calculate Mean Deviation of following values by using Arithmetic Mean 2,4,7,8,9 1. A.M. of values = 6 2. A.M. of differences (deviations) = Mean Deviations = 12/5 = 2.4
- 15. Mean Deviation from Median for ungrouped data Steps : 1. Calculate the Median of the values 2. Calculate the difference between each value Median (called deviations and dented by d) 3. Calculate the A.M. of these differences (Mean Deviations).
- 16. Example: Calculate Mean Deviation of following values by using Median 2,4,7,8,9 1. Median of values = 7 2. A.M. of differences (deviations) = Mean Deviations = 11/5 = 2.2
- 17. Mean Deviation from Arithmetic Mean for grouped data Steps : 1. Find the mid point of each class (m.p.) 2. Calculate the sum of frequencies (Ʃ f) 3. Calculate the difference (deviation) of the class mid point from mean (d) 4. Multiply each d value with its corresponding frequency to get f(d) values. 5. Sum all f(d) values. 6. Apply the following formula M.D. = Ʃ f (d) Ʃ f
- 18. C.I f m.p. d (f-m.p.) f x d 10-20 5 15 25 125 20-30 8 25 15 120 30-50 16 40 0 0 50-70 8 60 20 160 70-80 3 75 35 105 Ʃ f = 40 Ʃ = 510 M.D. = 510 / 40 = 12.75
- 19. Mean Deviation from Median for grouped data 1. M.D> M.D. = 665/50 = 13.3
- 20. • Mean Deviation is based on all values thus a change in one value will affect it. • It the least when calculated from Median. • It is highest when calculated from Mean
- 21. Standard Deviation Standard Deviation is the positive square root of the mean of squared deviations from mean. Steps: If there are five values X1, X2, X3, X4, X5 1. Mean of these values is calculated 2. Deviations of the values from mean is calculated 3. These deviations then are squared 4. Mean of these squared deviations is the variance. 5. Positive squared root of the variance is the standard deviations. S.D. = Ʃ d2 n
- 22. Calculating Standard Deviation For ungrouped data For grouped data 1. Actual Mean Method 1. Actual Mean Method 2. Direct Method
- 23. Calculating Standard Deviation for ungrouped data by Actual Mean Method Calculate SD of following values : 5, 10, 25, 30, 50 Mean of values Ʃ X= 24 X - Ʃ X is calculated for each value 1270/5 = 15.93 S.D. = Ʃ d2 n
- 24. Calculating Standard Deviation for ungrouped data by Direct Method Calculate SD of following values : 5, 10, 25, 30, 50 = 15.93
- 25. Calculating Standard Deviation for grouped data by Actual Mean Method
- 26. 1. Calculate the mean of distribution 2. Calculate the Deviation of mid values from Mean 3. Multiply deviations from their corresponding frequencies 4. Calculate Values by multiplying with “d” 5. Apply formula as following
- 27. • SD is most widely used measure of dispersion • SD is based on all values therefore a change in one value affects the value of SD. • It is independent of origin but not of scale.
- 29. Frequency Distribution (Classifying the raw data of quantitative variables) Observed Theoretical/ Probability Which are observed by actual observations or experiments Ex.. Frequency distribution of marks obtained by 50 students in test. Which are not obtained by actual experiments Ex. What are the possibilities of getting head if I toss the coin 150 times. 1. Binomial distribution 2. Poisson distribution 3. Normal distribution
- 30. Normal Distribution 1. It is a frequency curve based upon large no. of observations and small intervals 2. Also known as Gaussian curve or probability curve 3. Normal distribution is most commonly encountered distribution in statistics. 4. It describes the empirical distribution of certain measurements like weight and height of individuals, blood pressure etc.
- 31. Characteristics of normal distribution 1. Continuous 2. Bilateral symmetrical 3. Smooth, Bell shaped curve 4. Mean = Median = Mode 5. Central part is convex and it has two inflections and two lines never touch the base line, since based on infinite number of observations. 6. Shape of normal curve determined by : Mean and SD
- 32. Normal Distribution curve 50 % 50 % Mode Median Mean
- 33. Example : Summer job income of 16 students Income (INR) Frequency ( No. of students) 500 1 1000 2 1500 3 2000 4 2500 3 3000 2 3500 1 1 2 3 4 2 0 0 0 1 5 0 0 1 0 0 0 5 0 0 2 5 0 0 3 0 0 0 3 5 0 0
- 34. Skewness • Asymmetric distribution is also know as skewed distribution • It represent an imbalance and asymmetry from mean of a data distribution. • It occurs when the frequency curve is distorted either left or right side which indicated presence of skewness. • Left side skewness: frequency curve is distorted left side (Left tail) • Right side skewness: frequency curve distorted right side (Right tail).
- 35. Right side Skewness Mean lies extreme Right side of the curve Mean >Median >Mode Tail lies at the right side of curve Left side Skewness Mean lies extreme Left side of the curve Mean <Median <Mode Tail lies at the Left side of curve
- 36. Measurement of skewness: 1. With mean and median Skewness = 3 (Mean –Median) / Standard Deviation 2. With Mean and Mode Skewness = Mean – Mode/Standard Deviation 3. With Quartile values Skewness = Q3 + Q1 – 2 (Median) / Q3 – Q1
- 37. Calculate the skewness for following observations Skewness = Mean – Mode / SD Mean = 40.5 SD = 17.168 Mode = ? Mode = L1 + F1-F0 x 10 = 35 2F1-F0-F2 L1: Lower limit of class interval i.e. 30 F1: Highest frequency i.e. 16 F0: Previous frequency of highest frequency i.e. 8 F2: Next frequency of highest frequency i.e. 8 Skewness = 40.5 – 35 / 17.165 = 0.32
- 38. Z Score (Standard Score) • Deviation from the mean in a normal distribution or curve is called relative or standard normal deviate and • It is denoted by the symbol Z. • It measured in terms of SDs and indicates how much an observation is bigger or smaller than mean in units of SD. Formula = • Z core = + ve value : it is above the mean (Right side) • Z score = -ve value : it is below the mean (Left side )
- 39. Example: Result of a practice exam of student A and group of 100 student is given as following. Practice exam result of student A Practice exam result of group of 100 student Z score = Observation – Mean / SD Z score for Technique : 15-18/4.8 = - 0.62 (Below the mean) for Research & Techniques : 12-9/1.2 = 2.5 (Above the mean) for Subject : 34-32/1.8 = 1.1 (Above the mean) Techniques 15 Research & Biostatistics 12 Subject 34 Techniques 18 4.8 Research & biostatistics 9 1.2 Subject 32 1.8
- 40. Thank You