In Class IX, you have studied the classification of given data into ungrouped aswell asgrouped frequency distributions. You have also learnt to represent the datapictoriallyin the form of various graphs such as bar graphs, histograms (including thoseof varyingwidths) and frequency polygons. In fact, you went a step further by studyingcertainnumerical repr esentatives of the ungroup ed data, also called measures ofcentraltendency, namely, mean, median and mode. In this chapter, we shall extendthe studyof these three measures, i.e., mean, median and mode from ungrouped data tothat ofgrouped data. We shall also discuss the concept of cumulativ e frequency, thecumulative frequency distribution and how to draw cumulative frequencycurves, called ogives.
The mean (or average) of observations, as we know, is the sum of the values of all theobservations divided by the total number of observations. From Class IX, recall that ifx1, x2,. . ., xn are observations with respective f requencies f1, f2, . . ., fn, then thismeans observation x1 occurs f1 times, x2 occurs f2 times, and so on.Now, the sum o f the values of all the observations = f1x1 + f2x2 + . . . + fnxn, andthe number of observations = f1 + f2 + . . . + fn.So, the mean x of the data is given byRecall that we can write this in short form by using the Greek letter Σ (capitalsigma) which means summation. That is,
The marks obtained by 30 students of Class X of a certain school in aMathematics paper consisting of 100 mark s are presented in table below. Find themean of the marks obtained by the students.
Now, for each class-interval, we require a point which would serve as therepresentative of the whole class. It is assumed that the frequency of each classintervalis centred around its mid-point. So the mid-point (or class mark) of eachclass can be chosen to represent the observations falling in the class. Recall that wefind the mid-point of a class (or its class mark) by finding the average of its upper andlower limits. That is,
The sum of the values in the last column gives us Σ fi xi. So, the mean x of thegiven data is given byThis new method of finding the mean is known as the DirectMethod.
So, from Table 14.4, the mean of the deviations,Now, let us find the relation between d and x .Since in obtaining di, we subtracted ‘a’ from each xi, so, in order to get themeanx , we need to add ‘a’ to d . This can be explained mathematically as:
The method discussed above is called the AssumedMean Method.