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Central tendency and Variation or Dispersion

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- 1. Central Tendency and Variation By; Mr. Johny Kutty Joseph Asstt. Professor
- 2. Measures of Central Tendency • An average value within the range of the entire data that is used to represent all of the values of the series. Such indexes are called measures of central tendency. It is of three types; • A. Arithmetic Mean or Mean • B. Median • C. Mode
- 3. Mean • It is the most common method of central tendency. • Mean is computed by dividing the sum of all the values by the total number of values. It is represented by X • The Mean can be of (Discrete data, discrete frequency data, continuous frequency data. • It is calculated by the formula (Discrete data) X = sum of the values (Σx) number of the values (n)
- 4. Mean (Discrete) • Example: The hemoglobin levels of 10 women are given. (12.5, 13,10,11.5, 11,14,9,7.5,10,12) Calculate the Mean (X) • X = 110.5/10 = 11.05. • Calculate Mean from Discrete frequency table. • X = Σfx/Σf (where “x” = corresponding value variable, “f” = frequency of the variable.)
- 5. Mean (Frequency) • Example: Given data gives age of 100 girls. Find the mean age. • X = Σfx/Σf (where “x” = corresponding value variable, “f” = frequency of the variable.) • X = 1713/100 = 17.13 years. Age in years (x) No. of students (f) 16 35 17 31 18 20 19 14 Σf=100 (n) xf 560 527 360 266 Σxf=1713
- 6. Mean (Grouped Data) • Calculate Mean from Continuous frequency Table. • X = Σxf /Σf (where “x” = mid point of class interval. “f” is the frequency of class interval. Mid point of the class interval is calculated by “lower limit + upper limit/2”) • Calculate the Mean age of following people. • Mean = 3762.5/135=27.87 years Age in years (x) No. of people (f) 15-20 15 20-25 20 25-30 40 30-35 60 Σf=135 (n) Mid point (x) 17.5 22.5 27.5 32.5 xf 262.5 450 1100 1950 Σxf=3762.5
- 7. Merits & Demerits of Mean Merits • It is simple average and easy to compute. • It is affected by values of each item in the series. • It is fixed mathematical formula and always gives the same answer. Demerits • Very small and very large items usually affect the value of average. • Mean cannot be computed in case of open ended classes (with no upper/lower limit). • Not a good measure of central tendency.
- 8. Median • It is the middle most value when the data is arranged in ascending order of magnitude. • It divides the observation into two equal parts. • One half will have items larger than the Median value. • One half will have items smaller than the Median value • It is denoted by “M”.
- 9. Median for Discrete Data • Formula: M = n + 1/ 2 th observation . • The following data gives 7 person weight in pounds: 158,167,143,169,172,146,151. Calculate the Median. • Arrange the data: 143,146,151,158,167,169 and 172. • M=n + 1/2 = 7 + 1/2 = 4th observation = 158 • If “n” is an even number: Add the values of the observation before and after the fraction result and divide with 2 (ie. 3.5; 3 + 4 /2).
- 10. Median for Discrete Frequency Data • Formula: M = n + 1/2 th observation . • First calculate cumulative frequency. • Consider the final cumulative frequency as “n”. Calculate “M” • The value of the observation will be in reference to the cumulative frequency column. • In case the calculated “M” does not exactly match with the “cf” in the table, the next higher cumulative frequency to the calculated “M” may be considered as observation for median.
- 11. Median for Discrete Frequency Data • Example: Calculate median for the following frequency table data. • M= n+1/2 = 53+1/2 = 54/2 = 27th observation = M= 200 Income/day (x) 100 150 200 250 300 350 Number of People (f) 5 19 03 11 06 09 Cumulative Frequency (cf) 5 24 27 38 44 53
- 12. Median for Grouped Data • In case of continuous frequency table, Median can be calculated using the formula: • Where; “n” = Σf, “cf” (some time denoted as “m”)= the cumulative frequency of the class just before the median class, f = frequency of the median class, w = the width of the class and Lm = lower limit of the median class.
- 13. Median for Grouped Data • Calculate the median for the following grouped/ frequency data. • First calculate the cumulative frequency. Income/da y (x) 100- 150 150 - 200 200 - 250 250 - 300 300 - 350 350 - 400 Number of People (f) 5 19 03 11 06 09 Income/da y (x) 100- 150 150 - 200 200 - 250 250 - 300 300 - 350 350 - 400 Number of People (f) 5 19 03 11 06 09 Cum. frequency (cf) 5 24 27 38 44 53
- 14. Median for Grouped Data • Find the median class using the formula n + 1/2th observation where n = cf. • In the given example; n m+ 1 /2th = 53 + 1 / 2 = 27th class of observation ie 200 – 250. • Apply the formula • M = (53/2 – 24) / 3 x 50 + 200. • M= 2.5/3 x 50 + 200 = 2.5/3x50+200 = 241.7 • Therefore M = 241.7. The value of median always falls in the median class. • Now, n = 54 = median class will be 54 + 1 /2 th class; i.e. 27.5th class; In this case the class that includes 27.5 is cumulative frequency of 38.
- 15. Merits & Demerits of Median Merits • It is useful in case of open ended and unequal classes. • Extreme values do not affect the median. • Most appropriate in case of qualitative data. • Can be represented graphically. Demerits • Arranging data in frequency is required. • Median is not generally calculated for quantitative data since it is not useful in further algebraic treatments.
- 16. Mode • It is the value which is has the highest frequency. • It is noted by “ Z”. • Calculate the mode of the following discrete data. (1,7,4,1,4,5,3,4,7,9,6,5,4,3,2,4,5,4,4,3) • In the given series Z = 4
- 17. Mode Discrete Frequency Data • Calculate “Z” for the given data. • Highest frequency is 19 and hence the corresponding figure “ 150” is the mode. Income/da y (x) 100 150 200 250 300 350 Number of People (f) 5 19 03 11 06 09
- 18. Mode for Grouped Data • In case of continuous frequency data table, Mode can be calculated using the formula: • Z = l + (f – f1)w 2f – f1 – f2 Where; l = lowest limit of the modal class f = frequency of the modal class w = width of the modal class f1= frequency of the class just before the modal class. f2 = frequency of the class just after the modal class. • Modal class is the one which has the highest frequency.
- 19. Mode for Grouped Data • Example: Calculate the mode for the following; • Apply the formula; Z = l + (f – f1)w 2f – f1 – f2 • l = 150, f = 19, w = 50, f1=5, f2=3 • 150 + (19-5)x50/2x19-5-3 • 150 + 14x50/38-8 ; 150+ 700/30 = 150 + 23.33 • Z = 173.33 Income/da y (x) 100- 150 150 - 200 200 - 250 250 - 300 300 - 350 350 - 400 Number of People (f) 5 19 03 11 06 09
- 20. Mode for Grouped Data • There may be no mode if no value appears more than any other. • There may also be two modes (bimodal), three modes (trimodal), or four or more modes (multimodal). • If there are more classes with same frequency then the mode will be more.
- 21. Mode by Grouping Method • A common approach to this is grouping method or finding the relation of mean, median and mode. • The relation can be explained as Mode = 3 * Median – 2 * Mean • This method can be used calculate accurate mode for bimodal, tri-modal type of data.
- 22. Mode by grouping method. X f 1 15 5 16 2 17 7 18 6 19 7 20 2 21 3 22 4 f 2 7 13 9 7 f 3 9 13 5 f 6 20 9 f 5 15 12 f 4 14 15
- 23. Mode by Grouping method • Arrange the data in a table for analysis. • Mode is = 18 X f1 f2 f3 f4 f5 f6 15 16 17 18 19 20 21 22
- 24. Measures of Dispersion/Variation • The observation deviating from the central value • It is also called as variability of the data. • The Means are same. • The blue and green lines are more heterogeneous and red is homogenous. • Variability or dispersion is the measure to express the extent to which the scores are different from each other.
- 25. Measures of Dispersion • The different measures of variability or dispersion are the following • Range • Mean Deviation • Standard Deviation • Quartile Deviation
- 26. Range • Range is the difference between the highest and lowest value in the data. • R = H – L (H = highest value, L = Lowest Value). • Example: 3,5,6,8,9,3,4,5,6,9,12,13: R = 13-3=10. • Example: 0-10, 10-20, 20-30, 30-40; R = 40-0=40 • Merits of Range: simple to understand and easy to calculate. • Demerits of Range: not suitable for deep analysis and in case of extreme values.
- 27. Standard Deviation (SD) • It is the positive square root of mean of the squared deviations of values from the arithmetic mean. • Most commonly used and denoted by “SD” or σ (sigma) • Formula for Discrete Data • Formula for Grouped/continuous Data
- 28. Standard Deviation (SD) • In most of the cases n-1 is used instead of n as denominator in SD formula. • It is called as Bessel's correction as it gives an unbiased estimator of the population variance. • N is appropriate for sample but n-1 creates more accuracy in terms or generalization to population.
- 29. Standard Deviation (SD) • Given are the data of blood cholesterol levels of 10 persons. Calculate SD. • ( 240, 260, 290, 245, 255, 288, 272, 263, 277, 257) • Prepare a table • Calculate the Mean of the Data. • In this case Mean = 264.7. • Compute Σ (X – x )2
- 30. Standard Deviation (SD) x 240 245 255 257 260 263 272 277 288 290 Σx = 2647 (x - x) 240 – 264.7 = - 24.7 245 – 264.7 = - 19.7 255 – 264.7 = - 9.7 257 – 264.7 = - 7.7 260 – 264.7 = - 4.7 263 – 264.7 = - 1.7 272 – 264.7 = 7.3 277 – 264.7 = 12.3 288 – 264.7 = - 23.3 290 – 264.7 = - 25.3 X = 264.7 (x – x )2 610.09 388.09 94.09 59.29 22.09 2.89 53.29 151.29 542.89 640.09 Σ(x – x )2 = 2564.1 Compute SD by using the formula SD = √ 2564.1 10 = √256.41 SD= 16.01
- 31. Standard Deviation (SD) • Given are the data of weight of adolescent girls. Calculate SD for the continuous frequency data/grouped data. • Prepare the table and compute Σ (X – x )2 • Calculate the Mean using the formula Σxf/Σf. (can be performed in the table) Weight 60-64 64 - 68 68 - 71 71 - 74 No. of subjects 10 09 07 02
- 32. Standard Deviation (SD) x f 60 - 64 10 64 - 68 09 68 – 72 07 72 – 76 02 Σf = 28 (x - x) - 4.14 00 3.86 7.86 (x – x )2 17.14 00 14.9 61.78 Midpoint x 62 66 70 74 xf 620 594 490 148 Σxf = 1852 f(x – x )2 171.40 00 104.3 123.56 Σf(x – x )2 = 399.26 Compute SD by using the formula SD = √ 399.26 28 = √14.26 SD= 3.78
- 33. Characteristics of Good variation • It should be rigidly defined. • Easy to understand and calculate. • It should be based on all observations. • It should be amenable for further algebraic treatment. • It should be affected by sampling fluctuations. • Considering all the above factors SD is considered as the good measure of variation.

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