Solution:

The data in this form is called raw data.
By looking at it in this form, can you find the highest and the lowest marks?
Did it take you some time to search for the maximum and minimum scores? Wouldn’t
it be less time consuming if these scores were arranged in ascending or descending
order? So let us arrange the marks in ascending order as

25 36 42 55 60 62 73 75 78 95

Now, we can clearly see that the lowest marks are `25` and the highest marks are `95.`
The difference of the highest and the lowest values in the data is called the range of the
data. So, the range in this case is `95 – 25 = 70.`

Presentation of data in ascending or descending order can be quite time consuming,
particularly when the number of observations in an experiment is large, as in the case
of the next example.

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Solution:

Recall that the number of students who have obtained a certain number of marks is
called the frequency of those marks. For instance, `4` students got `70` marks. So the
frequency of `70` marks is `4.` To make the data more easily understandable, we write it
in a table, as given below:

Table 14.1 is called an ungrouped frequency distribution table, or simply a frequency
distribution table. Note that you can use also tally marks in preparing these tables,

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Solution:

To present such a large amount of data so that a reader can make sense of it easily,
we condense it into groups like `20-29, 30-39, . . ., 90-99` (since our data is from
`23` to `98`). These groupings are called ‘classes’ or ‘class-intervals’, and their size is
called the class-size or class width, which is 10 in this case. In each of these classes,
the least number is called the lower class limit and the greatest number is called the
upper class limit, e.g., in `20-29, 20` is the ‘lower class limit’ and `29` is the ‘upper class
limit’.

Also, recall that using tally marks, the data above can be condensed in tabular form as follows:

Presenting data in this form simplifies and condenses data and enables us to observe
certain important features at a glance. This is called a grouped frequency distribution
table. Here we can easily observe that `50%` or more plants survived in `8 + 18 + 10 +
23 + 12 = 71` schools.

We observe that the classes in the table above are non-overlapping. Note that we
could have made more classes of shorter size, or fewer classes of larger size also. For
instance, the intervals could have been `22-26, 27-31`, and so on. So, there is no hard
and fast rule about this except that the classes should not overlap.

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Solution:

Now, if two new students of weights `35.5` kg and `40.5` kg are admitted in this class,
then in which interval will we include them? We cannot add them in the ones ending
with `35` or `40`, nor to the following ones. This is because there are gaps in between the
upper and lower limits of two consecutive classes. So, we need to divide the intervals
so that the upper and lower limits of consecutive intervals are the same. For this, we
find the difference between the upper limit of a class and the lower limit of its succeeding
class. We then add half of this difference to each of the upper limits and subtract the
same from each of the lower limits.
For example, consider the classes `31 - 35` and `36 - 40`.
The lower limit of `36 - 40 = 36`
The upper limit of `31 - 35 = 35`
The difference `= 36 – 35 = 1`
So, half the difference `= 1/2= 0.5`

So the new class interval formed from `31 - 35` is `(31 – 0.5) - (35 + 0.5), i.e., 30.5 - 35.5.`
Similarly, the new class formed from the class `36 - 40` is `(36 – 0.5) - (40 + 0.5), i.e.,`
`35.5 - 40.5.`
Continuing in the same manner, the continuous classes formed are:
`30.5-35.5, 35.5-40.5, 40.5-45.5, 45.5-50.5, 50.5-55.5, 55.5-60.5,`
`60.5 - 65.5, 65.5 - 70.5, 70.5 - 75.5.`

Now it is possible for us to include the weights of the new students in these classes. But, another problem crops up because `35.5` appears in both the classes `30.5 - 35.5` and `35.5 - 40.5`. In which class do you think this weight should be considered?
if it is considered in both classes, it will be counted twice. By convention, we consider `35.5` in the class `35.5 - 40.5` and not in `30.5 - 35.5.`
Similarly, `40.5` is considered in `40.5 - 45.5` and not in `35.5 - 40.5`.
So, the new weights `35.5 kg` and `40.5 kg` would be included in `35.5 - 40.5` and `40.5 - 45.5`, respectively. Now, with these assumptions, the new frequency distribution table will be as shown below:

Now, let us move to the data collected by you in Activity 1. This time we ask you to present these as frequency distribution tables.

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