Introduction Arithmetic mean Median Mode Geometric mean Harmonic mean Relationship bw mean median amp mode Arithmetic mean Calculation of arithmetic mean in the following ways ID: 1018838
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1. Measurements of central tendency
2. Introduction Arithmetic mean Median Mode Geometric mean Harmonic mean Relationship b/w mean, median & mode
3. Arithmetic meanCalculation of arithmetic mean in the following ways:In a series of individual observationsIn a discrete seriesIn a continuous series
4. i) In a series of individual observations Equation: X̅ x̅ = AM ∑X = Sum of all the observations n = number of observationsFor example: calculate the arithmetic mean of following observations x = 2, 4, 6, 8, 10, 20, 12, 20
5. ii) In a series of discrete observations Equation: X̅ x̅ = AM ∑fX = sum of values of variables and corresponding frequencies ∑f = sum of frequenciesFor example: calculate the arithmetic mean of following observations x2461012f35224
6. iii) In a series of continuous series The histogram can be constructed in two ways depending up class- interval: a) For distributions that have equal class intervalsb) For distributions that have unequal class intervals Equation: X̅ x̅ = AM ∑fm = sum of values of mid-point and corresponding frequencies ∑f = sum of frequenciesFor example: calculate the arithmetic mean of following observations CI10-2020-3030-4040-5050-6060-70f352244
7. Median The histogram can be constructed in two ways depending up class- interval: a) For distributions that have equal class intervalsb) For distributions that have unequal class intervals The median is usually defined as that value which divides a distribution so that an equal number of items occur on either side of it. 50% of the observations will be smaller than the median, if the number of value is odd, the median will be middle value. The data should be arranged in ascending or descending order
8. Median The histogram can be constructed in two ways depending up class- interval: a) For distributions that have equal class intervalsb) For distributions that have unequal class intervals i) Calculation of median in a series of individual observations a) Arrange the data in ascending or descending order b) Median is calculated by finding the value of n+1/2th item M For example: calculate the median of following observations x = 2, 4, 6, 8, 10, 20, 12, 20, 2, 13, 45, 5, 16, 14, 11
9. Median The histogram can be constructed in two ways depending up class- interval: a) For distributions that have equal class intervalsb) For distributions that have unequal class intervals i) Calculation of median in a discrete series a) Arrange the data in ascending or descending order b) Find out the CF c) Median: the value of n+1/2th item d) It can be found by first locating the cumulative frequency which is equal to n+1/2 or the next higher to this and then determine the value corresponding to it. It will be M For example: calculate the median of following observations x214610221216f3522458
10. i) Calculation of median in a discrete series Solution: M = size of M = = 15th item (more close to 15th item) M = 22 XFFcf2323314562562102710212512224145171251682516822429
11. Median The histogram can be constructed in two ways depending up class- interval: a) For distributions that have equal class intervalsb) For distributions that have unequal class intervals ii) Calculation of median in a continuous series M x i L = lower limit of the median class cf = cumulative frequency of the class preceding the median class F = frequency of the median class i = class interval For example: calculate the median of following observations CI214610221216f3522458CI10-2020-3030-4040-5050-6060-70f352244
12. ii) Calculation of median in continuous series Solution: M = size of M = = 10th item (median lies in the class = 30-40) L = 30, cf = 8, i= 10, f= 2 M x 10 = ? CIFcf10-203320-305830-4021040-5021250-6041660-70420
13. Exercise no. 10 Calculate the median in following observationCI10-2020-3030-4040-5050-6060-7070-80F2342458Exercise no. 11 Calculate the median in following observationCI10-2020-3030-6060-8080-9090-100F23121048
14. CIFCF10-202220-303530-404940-5041350-6041760-7052270-8052780-9043190-100839Exercise no. 11 Calculate the median in following observation
15. Graphic location of median Median can be located graphically by applying any one of the following methods:Draw only one ogive curve by “less than & more than” methodDraw two ogives curve – one by “less than & other by more than” methodFor example: Determine the value of median graphically CI10-2020-3030-4040-5050-6060-70f352244
16. Graphic location of medianCIfcf10-203320-305830-4021040-5021250-6041660-70420Less than ogiveMore than ogiveVariablecfVariableCf20310203082016401030125012401060165087020603700
17. Mode The “mode” is another measure of central tendency Mode is the most typical value of a distribution because it is repeated the highest number of items in the series The mode of a distribution is the value at the point around which the items tend to be most heavily concentrated Two typesUnimodal Bimodal or multimodal
18. Mode The “mode” is another measure of central tendency Mode is the most typical value of a distribution because it is repeated the highest number of items in the series The mode of a distribution is the value at the point around which the items tend to be most heavily concentrated Two typesUnimodal Bimodal or multimodalFor example: x = 8,9,10,2,5,8,16,18,14,7 x = 8,9,10,2,5,8,16,18,14,10 x = 6,7,8,9,10,11,12,15,4,5,20
19. Mode in symmetrical and skewed distributions In perfectly symmetrical distribution, there is only one mode, the three measures of central tendency – the mode, median & mean – coincide with the highest point There are three distributions of central tendency Symmetrical distribution (Mo=Me=Mean)Positively skewed distribution (Mo>Me>Mean)Negatively skewed distribution
20. Mode in symmetrical and skewed distributions a) Symmetrical distribution In such there is only one mode, the three measures of central tendency – the mode, median & mean – coincide with the highest point Positively skewed distribution The mode is at the highest point and the median lies to the right of this point, but mean falls to the right of the median Negatively skewed distribution The mode will have the largest value, the median lies to the left of the mode and the mean falls to the left of the median
21. Mode in symmetrical and skewed distributions X = 2,4,8,10,8,6,7,9,12
22. Mode b) Calculation of mode in a continuous series In continuous series, the modal class should be ascertained first which is defined as the class having maximum frequency Equation: Mode x i L = Lower limit of the modal class = Difference b/w the frequency of the modal class and the frequency of the preceding class (ignore sign) = Difference b/w the frequency of the modal class and the frequency of the succeeding class (ignore signs) i = class interval
23. b) Calculation of mode in a continuous series For example: Calculate the mode for following distribution Solution: Mode lies in the class = 20 – 30 L = 20, = (5 - 3) = 2, Mode x i ∆2 = (5 – 2) = 3, i = 10 = x 10 = ? CI10-2020-3030-4040-5050-6060-70f352244
24. c) Graphic location of modeFor example: Calculate the mode for following distribution CI10-2020-3030-4040-5050-6060-70f352244
25. Relationship between Mean, Median and Mode Equation: Mean - Mode = 3(Mean – Median) Mode = 3 Median – 2 MeanFor exp: Calculate the mean if the mode is 11 and median is 14. Mean = 15.5
26. THANKS