Statistics

Hello my dear students, welcome to today's mathematics lesson. I am so happy to be here with you to learn about Statistics, which is Chapter 12 of your NCERT textbook. Statistics is such an important topic because it helps us make sense of data around us. Whether it's about the marks we score in exams, the rainfall in our city, or the prices of vegetables in the market, statistics helps us understand and interpret this information in a meaningful way.

So students, let's begin our journey into the world of statistics. In this chapter, we are going to learn how to represent data graphically. You already know that data can be presented in tables, but today we will see how we can make pictures out of this data to understand it even better. As the saying goes, "a picture is worth a thousand words," and this is absolutely true when it comes to representing data graphically.

Let us start with the first method of graphical representation, which is called a Bar Graph.

Now students, you might have seen bar graphs before in your earlier classes. Let me remind you what a bar graph is. A bar graph is a pictorial representation of data in which we draw bars of uniform width with equal spacing between them. One axis, usually the horizontal axis, shows the variable or the categories, and the other axis, usually the vertical axis, shows the values or frequencies. The height of each bar depends on the value of the variable it represents.

Let me explain this with an example from your textbook. Consider Example 1, where in a particular section of Class IX, 40 students were asked about the months of their birth, and a bar graph was prepared. In this case, the variable is the 'month of birth' and the value of the variable is the 'number of students born' in that particular month.

Now, from this bar graph, we can answer questions very easily. For instance, if someone asks how many students were born in the month of November, we can simply look at the bar corresponding to November and read its height. According to the graph, 4 students were born in November. Similarly, to find out in which month the maximum number of students were born, we look for the tallest bar, and that is August. So students, you can see how easy it is to read information from a bar graph at a glance.

Now, let us learn how to construct a bar graph. Look at Example 2 in your textbook. A family with a monthly income of ₹20,000 had planned the following expenditures per month under various heads. The data is given in a table. We have grocery expenditure of ₹4,000, rent ₹5,000, education of children ₹5,000, medicine ₹2,000, fuel ₹2,000, entertainment ₹1,000, and miscellaneous expenses ₹1,000. Remember, the table shows values in thousands of rupees.

To draw a bar graph for this data, we follow these steps:

First, we represent the different heads, which are our variables, on the horizontal axis. We can choose any scale for this because the width of the bar is not important. But for clarity, we take equal widths for all bars and maintain equal gaps between them. Let one head be represented by one unit.

Second, we represent the expenditure, which is the value, on the vertical axis. Since the maximum expenditure is ₹5,000, we choose the scale as 1 unit equals ₹1,000.

Third, to represent our first head, that is grocery, we draw a rectangular bar with width 1 unit and height 4 units because the expenditure is ₹4,000.

Fourth, we similarly draw bars for other heads, leaving a gap of 1 unit between two consecutive bars.

And that's how we construct a bar graph, students. Now, when you look at this bar graph, you can easily visualize the relative characteristics of the data. For example, you can see at a glance that the expenditure on education is more than double that of medical expenses. This is why bar graphs sometimes serve as a better representation of data than the tabular form.

Now students, let us move on to the next type of graphical representation, which is called a Histogram.

A histogram is a form of representation similar to a bar graph, but it is used for continuous class intervals. This is very important, students. In a bar graph, we deal with separate categories that are not necessarily connected, but in a histogram, we deal with data that is grouped into continuous class intervals. For example, if we have data about the weights of students, where weights are grouped into intervals like 30.5 to 35.5 kg, 35.5 to 40.5 kg, and so on, we use a histogram.

Let me show you an example. Consider Table 12.2 in your textbook, which gives the weights of 36 students of a class. The data is grouped into class intervals: 30.5 to 35.5 kg has 9 students, 35.5 to 40.5 kg has 6 students, 40.5 to 45.5 kg has 15 students, 45.5 to 50.5 kg has 3 students, 50.5 to 55.5 kg has 1 student, and 55.5 to 60.5 kg has 2 students.

Now, let us represent this data graphically as a histogram.

First, we represent the weights on the horizontal axis on a suitable scale. We can choose the scale as 1 cm equals 5 kg. Also, note that the first class interval starts from 30.5 and not from zero. So we show this on the graph by marking a kink or a break on the axis to indicate that we have skipped the values from zero to 30.5.

Second, we represent the number of students, which is the frequency, on the vertical axis on a suitable scale. Since the maximum frequency is 15, we need to choose a scale that can accommodate this maximum frequency.

Third, we draw rectangles or rectangular bars of width equal to the class size and lengths according to the frequencies of the corresponding class intervals. For example, the rectangle for the class interval 30.5 to 35.5 will have a width of 1 cm (representing 5 kg) and a length of 4.5 cm (representing 9 students, if we choose our scale appropriately).

Fourth, in this way, we obtain the histogram. Now students, observe that since there are no gaps between consecutive rectangles, the resultant graph appears like a solid figure. This is why it is called a histogram. It is a graphical representation of a grouped frequency distribution with continuous classes.

Now here is a very important point, students. Unlike a bar graph, where the width of the bar does not carry any meaning, in a histogram, the width of the bar plays a significant role. The areas of the rectangles are proportional to the corresponding frequencies. However, since the widths of the rectangles are all equal in this case, the lengths of the rectangles are proportional to the frequencies. That is why we draw the lengths according to the frequencies.

Now students, let us consider a different situation. What happens when the class intervals have different widths? This is a crucial concept, so pay attention.

Consider Example 3. A teacher wanted to analyze the performance of two sections of students in a mathematics test of 100 marks. She found that a few students got under 20 marks and a few got 70 marks or above. So she decided to group them into intervals of varying sizes: 0 to 20, 20 to 30, and so on up to 60 to 70, and 70 to 100. Notice that the first interval is of width 20, most intervals in the middle are of width 10, and the last interval is of width 30.

Now, when a histogram is prepared for this data with the class intervals as they are, we get a misleading picture. This is because the areas of the rectangles must be proportional to the frequencies in a histogram. Earlier, when all rectangles had equal widths, this problem did not arise because the lengths were directly proportional to the frequencies. But here, since the widths are varying, the histogram does not give a correct picture. For example, it might show a greater frequency in the interval 70 to 100 than in 60 to 70, which is not actually the case.

So, we need to make certain modifications in the lengths of the rectangles so that the areas are again proportional to the frequencies. Let me explain the steps to do this.

First, we select a class interval with the minimum class size. In this example, the minimum class size is 10.

Second, the lengths of the rectangles are modified to be proportionate to this minimum class size of 10.

For instance, for the class interval 0 to 20, the class size is 20 and the frequency is 7. So when the class size is 10, the length of the rectangle will be 7 divided by 20 multiplied by 10, which equals 3.5.

Similarly, for the class interval 20 to 30, the class size is 10 and the frequency is 10. So the length becomes 10 divided by 10 multiplied by 10, which equals 10.

We continue this process for all class intervals. For the class interval 70 to 100, the class size is 30 and the frequency is 8. So the length becomes 8 divided by 30 multiplied by 10, which equals approximately 2.67.

Now students, these lengths are calculated for an interval of 10 marks in each case, so we may call these lengths as "proportion of students per 10 marks interval." This gives us the correct histogram with varying widths.

Now, let us move on to the third type of graphical representation, which is called a Frequency Polygon.

A frequency polygon is another visual way of representing quantitative data and its frequencies. It is a polygon, which means a many-sided figure. Let me explain how to draw a frequency polygon.

Consider the histogram we drew earlier for the weights of students. To draw a frequency polygon from this histogram, we join the mid-points of the upper sides of the adjacent rectangles by line segments. Let me explain this more clearly. In the histogram, each rectangle has a top side. We find the mid-point of the top side of each rectangle. Then we join these mid-points of adjacent rectangles by straight lines. When we do this, we get a shape that looks like a polygon.

But to complete the polygon, we need to add two more points. We assume that there is a class interval with zero frequency before the first class interval and one after the last class interval. We find the mid-points of these imaginary class intervals and join them to the first and last mid-points respectively. This ensures that the area of the frequency polygon is the same as the area of the histogram. This is an important property, students.

Now, what if there is no class preceding the first class? Let me explain with Example 4. Consider the marks obtained by 51 students in a test, given in Table 12.5. The class intervals are 0 to 10, 10 to 20, and so on up to 90 to 100. Here, the first class is 0 to 10, so there is no class before it.

To draw a frequency polygon for this data, we first draw a histogram and mark the mid-points of the tops of the rectangles. Then, to find the class preceding 0 to 10, we extend the horizontal axis in the negative direction and find the mid-point of the imaginary class interval negative 10 to 0. We join the first mid-point to this point with zero frequency. The point where this line segment meets the vertical axis is marked as a starting point. Similarly, we find the mid-point of the class succeeding the last class and join it to complete the polygon.

Now students, frequency polygons can also be drawn independently without drawing a histogram. For this, we need to find the class-marks of the class intervals. The class-mark is the midpoint of a class interval. To find the class-mark, we add the upper limit and the lower limit of the class and divide by 2. So, class-mark equals upper limit plus lower limit, divided by 2.

Let me show you an example of drawing a frequency polygon without a histogram. Consider Example 5. The data is about the weekly observations made in a study on the cost of living index in a city. The class intervals are 140 to 150, 150 to 160, 160 to 170, and so on up to 190 to 200.

To find the class-mark for the class 140 to 150, we add 150 and 140, which gives us 290, and divide by 2, which gives us 145. Similarly, we find the class-marks for all other classes: 155, 165, 175, 185, and 195.

Now, to draw the frequency polygon, we plot the class-marks along the horizontal axis and the frequencies along the vertical axis. We plot points for each class-mark and its corresponding frequency. Then we join these points by line segments. But we must not forget to plot two additional points: one with zero frequency before the first class and one with zero frequency after the last class. This gives us the complete frequency polygon.

So students, the points we plot are A at 135 with frequency 0, B at 145 with frequency 5, C at 155 with frequency 10, D at 165 with frequency 20, E at 175 with frequency 9, F at 185 with frequency 6, G at 195 with frequency 2, and H at 205 with frequency 0. When we join these points, we get the frequency polygon.

Now, why do we use frequency polygons, students? Frequency polygons are particularly useful when the data is continuous and very large. They are also very useful for comparing two different sets of data of the same nature. For example, if we want to compare the performance of two different sections of the same class, we can draw both frequency polygons on the same graph and easily see which section performed better.

So students, in this chapter, we have learned three important ways to represent data graphically: bar graphs, histograms, and frequency polygons.

Let me summarize what we have learned today.

First, we learned about bar graphs. A bar graph is a pictorial representation of data where bars of uniform width are drawn with equal spacing between them on one axis, depicting the variable, and the values are shown on the other axis. The heights of the bars depend on the values of the variable. Bar graphs are used to compare different categories.

Second, we learned about histograms. A histogram is used for continuous class intervals. It is similar to a bar graph, but there are no gaps between the rectangles. The important point to remember is that the areas of the rectangles are proportional to the frequencies. When the class intervals have equal widths, the lengths of the rectangles are directly proportional to the frequencies. But when the class intervals have different widths, we need to adjust the lengths so that the areas remain proportional to the frequencies.

Third, we learned about frequency polygons. A frequency polygon is drawn by joining the mid-points of the upper sides of the rectangles in a histogram. It can also be drawn independently using class-marks. The class-mark is calculated by adding the upper and lower limits of a class interval and dividing by 2. Frequency polygons are useful for comparing different sets of data and are especially helpful when we have large amounts of continuous data.

Now students, these are the key concepts from Chapter 12 on Statistics. You have learned how data can be presented graphically in the form of bar graphs, histograms, and frequency polygons. Each of these representations has its own use and helps us understand data in different ways. Bar graphs are great for comparing categories, histograms are perfect for continuous grouped data, and frequency polygons are excellent for comparing different datasets and for visualizing the distribution of data.

Remember, the choice of graphical representation depends on the type of data you have and what you want to show. Practice drawing these graphs, and you will become comfortable with representing data visually.

That's all for today, students. Thank you for your attention. In our next lesson, we will continue exploring more about statistics. Until then, keep practicing and stay curious!