Measures of central tendency

Measures of central tendency

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  Jagdish Powar Statistician cumtutor Communicate Medicine SMBT, IMSRC, Nashik.
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  Competency & Learningobjectives Competency Learning Objectives CM6.4 Enumerate, discuss and demonstrate Common sampling techniques, simple statistical methods, frequency distribution, measures of central tendency and dispersion The student should be able to  Define measures of central tendency-Mean, Mode, Median  Calculate the different measures of central tendency from given dataset,  Comment on skewness of given dataset 2PSM, SMBT, Nashik
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  What is measureof central tendency? • A measure of central tendency is a summary statistic that represents the center point or value of a dataset. These measures simply summarize a set of data in a single value. These are representative of data and around which large portion of observations gathered. • In statistics, the three most common measures of central tendency are the mean, median and mode. 3PSM, SMBT, Nashik
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  Requirement of agood measure of central tendency 1. It should be rigidly defined. 2. It should be based on all observations (values). 3. It should be easy to calculate and understand. 4. It should have sampling stability i.e. it should not be affected by sampling fluctuations. 5. It should be capable for the further mathematical calculations. 6. It should not be affected by extreme values. 4PSM, SMBT, Nashik
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  Mean is verysimple and commonly used measure of central tendency. Mean is sum of all observations divided by number of observations. For ungrouped data:- Mean ( 𝑥) = 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑛𝑜.𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 = 𝛴𝑥 𝑛 For grouped data:- Mean ( 𝑥) = 𝛴𝑓∗𝑥 𝑁 N = 𝛴𝑓 x= midpoints of class interval f= frequency 5PSM, SMBT, Nashik
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  Central values whenobservations are arranged in ascending or descending order. • For ungrouped data:- Arrange the observations in ascending or descending order n = Odd value Median= 𝑛+1 2 th value n = Even value Median = 𝑛 2 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+ 𝑛 2 +1 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 2 • For grouped data:- Median = L + ( 𝑁 2 −𝐶𝐹) 𝑓 ∗ 𝑖 L- lower class limit of median class CF- lcf of pre-median class f- Frequency of median class i- width of median class The class with LCF is just greater than N/2 is known median class 6PSM, SMBT, Nashik
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  MODE The observation whichoccurs most frequently in series is known as mode.  For ungrouped data Mode= frequently occurred observation.  For grouped data Mode = L + (𝑓𝑚−𝑓1) (2∗𝑓𝑚−𝑓1−𝑓2) ∗ 𝑖 L- lower class limit of modal class fm- frequency of modal class f1- frequency of pre-modal class f2- frequency of post-modal class i - Width of modal class. The class with maximum frequency is model class 7PSM, SMBT, Nashik
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  SKEWNESS Positive Skewed (Righttailed): – Mean > Median > Mode Negative Skewed (Left tailed): – Mean < Median < Mode Symmetric (Normal): – Mean = Median = Mode Relation between mean, median and mode Mean – Mode = 3(Mean – Median) Mode= 3*Median - 2* Mode 8PSM, SMBT, Nashik
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  1)The following numbergives the incubation period of 10 consecutive patients of hepatitis. Calculate mean, mode, median and comment on its skewness 24, 22, 35, 15, 26, 19, 24, 18, 22, 25 Sol- Mean:- Mean ( 𝑥) = 𝛴𝑥 𝑛 = 24+22+⋯……………..+22+25 10 = 230 10 =23 Median:- Ascending order 15 18 19 22 22 24 24 25 26 35 n = 10, Even value Median = 𝑛 2 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+ 𝑛 2 +1 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 2 = 5 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+6 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 2 = 22+24 2 = 23 9PSM, SMBT, Nashik
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  Mode Mode= frequently occurredobservation The observations 22 & 24 occurred most frequently in given data set Mode=22, 24 (Bi-model distribution) Skewness:- Here Mean = Median= 23 Symmetric or Normal distribution 10PSM, SMBT, Nashik
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  2) Following tablegives the age distribution of patients from RHTC, SMBT- Ekdara Calculate Mean, median, mode and comment on skewness. Sol- Mean:- Mean ( 𝑥) = 𝛴𝑓∗𝑥 𝑁 N = 𝛴𝑓, x= midpoints of class interval, f= frequency Mean ( 𝑥) = 𝛴𝑓∗𝑥 𝑁 = 6400 160 =40 Mean age= 40 years Age (Years) 10-20 20-30 30-40 40-50 50-60 60-70 No. of patients 15 25 30 55 25 10 Class Interval f x (midpoint) f*x 10.-20 15 15 225 20-30 25 25 625 30-40 30 35 1050 40-50 55 45 2475 50-60 25 55 1375 60-70 10 65 650 160 6400 11PSM, SMBT, Nashik
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  Median:- CI f LCF 10.-2015 15 20-30 25 40 30-40 30 70 40-50 55 125 50-60 25 150 60-70 10 160 160 Median class N/2=160/2 =80 LCF=125 > N/2(80) hence median class is 40-50 Median = L + ( 𝑁 2 −𝐶𝐹) 𝑓 ∗ 𝑖 L- lower class limit of median class=40 CF- lcf of pre-median class=70 f- Frequency of median class =55 i- width of median class=10 Median = 40 + (80−70) 55 ∗ 10 =40+1.81=41.81 Median age= 41.81 years 12PSM, SMBT, Nashik
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  Mode CI f 10.-20 15 20-3025 30-40 30 40-50 55 50-60 25 60-70 10 The class interval 40-50 is with maximum frequency hence model class is 40-50. Mode = L + (𝑓𝑚−𝑓1) (2∗𝑓𝑚−𝑓1−𝑓2) ∗ 𝑖 L- lower class limit of modal class=40 fm- frequency of modal class=55 f1- frequency of pre-modal class=30 f2- frequency of post-modal class=25 i - Width of modal class=10 Mode =40 + (55−30) (2∗55−30−25) ∗ 10 =40+4.55=44.55 Mode age=44.55 years Mean (40)< Median(41.81) < Mode(44.55) Negative skewed distribution. 13PSM, SMBT, Nashik
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  PSM, SMBT, Nashik14