Hello, and welcome to today's mathematics lesson. We are going to explore Chapter Twenty-Three: Data Handling, which is also known as Statistics. By the end of this lesson, you will understand what data means, how we organize it using frequency tables, and how we can represent data visually through bar graphs and pie charts. Let us begin.
First, let us understand the word statistics itself. This word is used in two different ways. In the singular sense, statistics refers to the entire subject — the branch of knowledge that deals with collecting, analyzing, presenting, and interpreting numerical information. In the plural sense, statistics means the actual collection of numerical facts themselves, gathered systematically for a specific purpose. For example, we might talk about population statistics, tax statistics, or examination statistics in your school.
Now, what exactly is data? Data is a set of numerical facts collected with a definite purpose in mind. Imagine you measure the heights of six children in your class. You might get values like one hundred fifty-three centimetres, one hundred fifty centimetres, one hundred fifty-four centimetres, and so on. This collection of numbers — one hundred fifty-three, one hundred forty-seven, one hundred fifty, one hundred fifty-two, one hundred fifty-four, and one hundred fifty-three — is called a set of data.
Data can come from various sources. Individuals can collect it by going person to person, asking about income, age, or savings. Government agencies collect data too, such as birth rates or price rises over time. But once data is collected, it must be arranged systematically to make sense of it. This systematic arrangement, usually in table form, is called tabulation.
Let us move to an important concept: frequency. Consider this set of numbers: five, seven, three, eight, seven, five, five, three, five, eight, seven. Look carefully — how many times does the number five appear? It appears four times. So we say the frequency of five is four. Similarly, three appears twice, so its frequency is two. Seven appears three times, giving it a frequency of three. And eight appears twice, so its frequency is also two.
Frequency, therefore, is simply a number that tells us how many times a particular value occurs in a data set.
Now let us distinguish between three ways of presenting data: raw data, arrayed data, and frequency distribution.
Suppose thirty students in a class score these marks out of ten in a test: nine, eight, six, ten, five, six, eight, seven, ten, five, and so on. When data is recorded exactly as collected, in its original jumbled form, it is called raw data.
If we arrange this same data in ascending order — that is, from smallest to largest — we get: four, four, four, five, five, five, five, five, six, six, six, six, six, six, six, seven, seven, eight, eight, eight, eight, eight, eight, nine, nine, nine, ten, ten, ten, ten. This ordered arrangement is called arrayed data, or simply an array. We could also arrange it in descending order if we wished.
Now, we can present this arrayed data more neatly in a table showing each mark and how many students obtained it. Mark four appears three times, mark five appears five times, mark six appears seven times, and so on. This tabular arrangement, showing the frequency of each observation, is called a frequency distribution. The table itself is called a frequency distribution table, or simply a frequency table.
Let us see how to construct a frequency table step by step.
Imagine you have recorded minimum temperatures in Delhi over thirty days. The values include eleven point eight, eleven point six, eleven point six, eleven point four, eleven point three, and so on. Here is how you organize this:
First, list all the different temperatures in ascending order in the first column. Second, use tally marks in the second column to count occurrences. Make vertical strokes for each occurrence. When you reach four strokes, do not add a fifth the same way. Instead, draw a diagonal stroke across the previous four, forming a bundle of five. This makes counting easier. Finally, count the tally marks for each temperature and write the total in the frequency column.
For example, eleven point three degrees Celsius occurs seven times, eleven point four occurs five times, eleven point five occurs six times, and so on. This type of table, where each individual value is listed separately, is called an ungrouped frequency distribution.
But what happens when data covers a wide range? Listing every single value becomes impractical. This is where grouped frequency distribution comes in.
Consider marks obtained by forty students in an examination. The marks range from twelve to fifty-nine. Instead of listing each mark separately, we group them into class intervals: 10-20, 20-30, 30-40, 40-50, and 50-60.
Here is an important rule: when a value equals the upper limit of one class and the lower limit of the next, it belongs to the higher class. So twenty goes into twenty to thirty, not ten to twenty. Similarly, thirty goes into thirty to forty, not twenty to thirty.
Using tally marks, we find that ten to twenty contains five students, twenty to thirty contains ten students, thirty to forty contains six students, forty to fifty contains eight students, and fifty to sixty contains eleven students. This grouped arrangement makes the data much easier to understand.
Let us clarify some terminology used in grouped distributions.
Each group like ten to twenty is called a class interval. The two numbers that bound it — ten and twenty — are called class limits. The smaller number, ten, is the lower class limit. The larger number, twenty, is the upper class limit.
The class mark is the value exactly midway between the lower and upper limits. It is calculated as: lower class limit plus upper class limit, divided by two. For the interval ten to twenty, the class mark is ten plus twenty, divided by two, which equals fifteen. For fifty to sixty, the class mark is fifty plus sixty, divided by two, which equals fifty-five. The class mark represents the entire interval when we need a single value.
Now we turn to graphical representation of data. Graphs have a powerful effect on our minds — they make patterns visible instantly. A well-made graph should always have a clear title and proper labels so anyone can understand what it shows.
We will study three types: the bar graph, the double bar graph, and the pie chart.
A bar graph is the simplest and most widely used way to show numerical data. It uses rectangular bars of equal width but differing heights.
Here are the key features to remember. All bars must have the same width — the width itself carries no meaning. The height of each bar is directly proportional to the quantity it represents. Equal spaces are left between consecutive bars.
Let us construct a bar graph showing speeds of different vehicles. A bicycle travels at ten kilometres per hour, a scooter at forty, a car at sixty, a bus at fifty, and a train at eighty.
First, draw two perpendicular lines intersecting at point O. The horizontal line is called the x-axis, and the vertical line is the y-axis. On the x-axis, mark equal spaces and label them with vehicle names: Bicycle, Scooter, Car, Bus, Train. On the y-axis, mark a suitable scale for speed. Then draw vertical bars rising from each vehicle name, with heights matching their speeds. The bar for the bicycle rises to ten, the scooter to forty, the car to sixty, and so on. All bars have the same width, and equal gaps separate them.
Once a bar graph is drawn, we can answer questions from it. Which store sold the most articles? By what percentage is one store's sales greater than another's? What is the ratio between highest and lowest sales? How much must a store increase sales to match its rival? All these become visible at a glance.
A double bar graph allows us to compare two sets of data side by side. For example, we might show both girls and boys in each class using adjacent bars in different colours or patterns. This makes comparisons immediate — which class has fewest students, which has most, what is the ratio of girls to boys in a particular class. The two bars for each category sit right next to each other, making the comparison effortless.
Finally, we come to the pie chart, also called a pie graph. This represents data as sectors of a circle.
Recall that an angle whose vertex is at the centre of a circle is called a central angle. The region enclosed by this angle is called a sector. In a pie chart, each category gets a sector whose angle is proportional to its value.
Since a complete circle contains three hundred sixty degrees, we divide three hundred sixty in proportion to the data values. Let us return to our vehicle speeds: ten, forty, sixty, fifty, and eighty kilometres per hour. The total is two hundred forty.
The bicycle's share is ten out of two hundred forty. So its central angle is ten divided by two hundred forty, multiplied by three hundred sixty degrees, which equals fifteen degrees. The scooter gets forty divided by two hundred forty times three hundred sixty, which equals sixty degrees. The car receives ninety degrees, the bus seventy-five degrees, and the train one hundred twenty degrees. Check: fifteen plus sixty plus ninety plus seventy-five plus one hundred twenty equals three hundred sixty degrees.
To draw the pie chart, first draw a circle of suitable radius. Then, using a protractor, measure and draw each sector with its calculated central angle. Label each sector clearly.
Pie charts are excellent for showing proportions and percentages. If you know the total number of items, you can find how many belong to each category. Conversely, if you know the angles, you can find percentages directly without knowing individual values. The central angle for any category divided by three hundred sixty, multiplied by one hundred, gives its percentage of the total.
Let us now recap the key points from today's lesson.
First, statistics is both a subject and a collection of numerical facts called data. Data must be organized systematically through tabulation.
Second, frequency tells us how many times a value appears in a data set. Raw data can be arranged into arrays and then into frequency tables.
Third, when data ranges are large, we use grouped frequency distributions with class intervals, class limits, and class marks. The class mark equals lower limit plus upper limit, divided by two.
Fourth, bar graphs use equal-width bars with heights proportional to values. Double bar graphs compare two data sets side by side.
Fifth, pie charts represent data as sectors of a circle, with central angles proportional to the values. The angle for each category equals its value divided by total value, multiplied by 360 degrees.
Sixth, graphical representation makes patterns visible and helps us answer questions about data quickly and intuitively.
That brings us to the end of our lesson on Data Handling. You have learned how to organize raw numbers into meaningful tables and transform those tables into powerful visual representations. These skills will serve you not just in mathematics, but in understanding the world around you — from news reports to scientific studies. Keep practising, stay curious, and I look forward to seeing you in the next lesson. Goodbye!