Practical.3 Measures Of Central Tendency – Mean, Median .

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Practical.3Measures of central tendency – mean, median, mode, geometric mean and harmonicmean for grouped dataArithmetic mean or meanGrouped DataThe mean for grouped data is obtained from the following formula:Where x the mid-point of individual classf the frequency of individual classN the sum of the frequencies or total frequencies.Short-cut methodWhereA any value in xN total frequencyc width of the class intervalExample 1Given the following frequency distribution, calculate the arithmetic meanMarks: 646362616059:1812976Number ofStudents81

SolutionXffxd -2-14596354-3-18603713-7Direct methodShort-cut methodHere A 62Example 2For the frequency distribution of seed yield of sesamum given in table calculate the meanyield per plot.Yield per 64.5-84.5plot in(ing)84.5-104.5104.5-124.5124.5-144.5No ofplots572032

SolutionYield ( in g)No of Plots (f)Mid 114.517124.5-144.520134.5240Total3544A 94.5The mean yield per plot isMedianGrouped dataIn a grouped distribution, values are associated with frequencies. Grouping can bein the form of a discrete frequency distribution or a continuous frequency distribution.Whatever may be the type of distribution, cumulative frequencies have to be calculated toknow the total number of items.Cumulative frequency: (cf)Cumulative frequency of each class is the sum of the frequency of the class and thefrequencies of the pervious classes, ie adding the frequencies successively, so that the lastcumulative frequency gives the total number of items.3

Discrete SeriesStep1: Find cumulative frequencies.Step2: FindStep3: See in the cumulative frequencies the value just greater thanStep4: Then the corresponding value of x is median.Example 3The following data pertains to the number of members in a family. Find median size ofthe family.Number of12345678910111213561013 953221members 604cf149152538475255575960

Median size of size of 30.5th itemThe cumulative frequency just greater than 30.5 is 38.and the value of x corresponding to38 is 6.Hence the median size is 6 members per family.Continuous SeriesThe steps given below are followed for the calculation of median in continuousseries.Step1: Find cumulative frequencies.Step2: FindStep3: See in the cumulative frequency the value first greater than, Then thecorresponding class interval is called the Median class. Then apply the formulaMedian Wherel Lower limit of the median classm cumulative frequency preceding the medianc width of the median classf frequency in the median class.N Total frequency.Example 4For the frequency distribution of weights of sorghum ear-heads given in tablebelow. Calculate the median.5

Weights of earNo of earCumulativeheads ( in g)heads (f)frequency 16020160Total160SolutionMedian Here100, N 160, f 45, c 20, m 60Median Geometric meanGrouped DataFor grouped dataGM AntilogExample 5Find the Geometric mean for the following6

Weight of sorghum (x)No. of ear head(f)50465675168089571004SolutionWeight ofsorghum (x)No. of earhead(f)Log xflog 1431.71135152.13031.95Total509.55599.21Here N 50GM Antilog Antilog Antilog 1.9842 96.43Continuous distributionExample 6For the frequency distribution of weights of sorghum ear-heads given in tablebelow. Calculate the Geometric mean7

Weights of earheads ( in g)No of earheads 160SolutionWeights of earheads ( in g)No of earheads (f)Mid xLog xf log 43.52Total160324.2Here N 160GM Antilog Antilog Antilog 106.238

Harmonic meanFor a frequency distributionH.M Example 7The marks secured by some students of a class are given below. Calculate the harmonicmean.Marks202122232425Number ofStudents427131SolutionMarksXNo ofStudents .04350.04352430.04170.12512510.04000.040018H.M 0.8216 9

Learning ExerciseFor the following frequency distribution find the(i)Mean(ii)Median(iii)Mode(iv)Harmonic mean(iv)Geometric meanWeight of earheads in gmsNo. of earhead40 - 60660 - 80880 – 10035100 -12055120 -14030140 – 16015160 – 18012180 – 200910

Measures of central tendency – mean, median, mode, geometric mean and harmonic mean for grouped data Arithmetic mean or mean Grouped Data The mean for grouped data is obtained from the following formula: Where x the mid-point of i

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