In Class IX, you have studied the classification of given data into ungrouped as well as grouped frequency distributions. You have also learnt to represent the data pictorially in the form of various graphs such as bar graphs, histograms (including those of varying widths) and frequency polygons. In fact, you went a step further by studying certain numerical representatives of the ungrouped data, also called measures of central tendency, namely, mean, median and mode. In this chapter, we shall extend the study of these three measures, i.e., mean, median and mode from ungrouped data to that of grouped data. We shall also discuss the concept of cumulative frequency, the cumulative frequency distribution and how to draw cumulative frequency curves, called ogives.

The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations. From Class IX, recall that if x1, x2, ..., xn are observations with respective frequencies f1, f2, ..., fn, then the sum of the values of all the observations = f1x1 + f2x2 + ... + fnxn, and the number of observations = f1 + f2 + ... + fn. So, the mean x of the data is given by x = (f1x1 + f2x2 + ... + fnxn) / (f1 + f2 + ... + fn). Recall that we can write this in short form by using the Greek letter Σ (capital sigma) which means summation.

In most of our real life situations, data is usually so large that to make a meaningful study it needs to be condensed as grouped data. So, we need to convert given ungrouped data into grouped data and devise some method to find its mean. Let us convert the ungrouped data of Example 1 into grouped data by forming class-intervals of width, say 15.

Now, for each class-interval, we require a point which would serve as the representative of the whole class. It is assumed that the frequency of each class-interval is centred around its mid-point. So the mid-point (or class mark) of each class can be chosen to represent the observations falling in the class. Recall that we find the mid-point of a class (or its class mark) by finding the average of its upper and lower limits.

Let us apply this formula to find the mean in the following example. Example 1: The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting of 100 marks are presented in a table below. Find the mean of the marks obtained by the students.

This new method of finding the mean is known as the Direct Method. We observe that Tables 13.1 and 13.3 are using the same data and employing the same formula for the calculation of the mean but the results obtained are different. Can you think why this is so, and which one is more accurate? The difference in the two values is because of the mid-point assumption in Table 13.3, 59.3 being the exact mean, while 62 an approximate mean. Sometimes when the numerical values of x and f are large, finding the product of x and f becomes tedious and time consuming. So, for such situations, let us think of a method of reducing these calculations.