| Classwise Concept with Examples | ||||||
|---|---|---|---|---|---|---|
| 6th | 7th | 8th | 9th | 10th | 11th | 12th |
| Content On This Page | ||
|---|---|---|
| Data & Related Terms | Pictograph | Bar Graphs |
Chapter 9 Data Handling (Concepts)
Welcome to Chapter 9: Data Handling! In our daily lives, we come across a lot of information – like the scores in a cricket match, the number of students who like different fruits, or the temperature each day. This collection of information, often in the form of numbers or observations, is called Data. Just having raw data isn't always very helpful; it can be messy and hard to understand. This chapter is all about learning simple but powerful ways to collect, organize, and show data so that we can easily understand it, compare different pieces of information, and draw simple conclusions. Think of it as learning how to tidy up information and present it clearly, like organizing your toys or books!
Imagine you ask everyone in your class about their favorite color. You'll end up with a long list of colors in no particular order – that's raw data. To make sense of it, we need to organize it. One simple way is to use Tally Marks. For each response (say, 'Red'), we make a vertical mark (|). We keep doing this, and when we get to the fifth response for 'Red', instead of making a fifth vertical line, we draw a diagonal line across the first four (like $\bcancel{||||}$). This makes counting in groups of five much easier! Using tally marks helps us create a Frequency Distribution Table. 'Frequency' just means how many times something occurs. So, the table lists each category (like each color) and its frequency (how many students chose that color), often showing the tally marks as well. This organized table is much easier to read than the original raw list.
Organized tables are great, but sometimes pictures tell the story even better! We will learn about two common ways to represent data visually:
- Pictographs: These use pictures or symbols to represent data. For example, we could use a smiley face symbol to represent 5 students. If 15 students like apples, we would draw 3 smiley faces next to 'Apples'. The most important part of a pictograph is the key or scale, which tells us what each symbol stands for (e.g., = 5 students). We'll practice reading information from pictographs by understanding the key and counting the symbols (sometimes we might see half a symbol, representing half the value!). We'll also learn how to draw simple pictographs ourselves.
- Bar Graphs (or Bar Charts): These use rectangular bars to show information. The bars have the same width, and their heights (or lengths, if drawn horizontally) represent the frequency or value of the data. For example, we could have bars representing different colors, and the height of each bar would show how many students chose that color. Bar graphs are excellent for comparing quantities easily. We'll learn how to read information from bar graphs by looking at the axes (the labels on the bottom and side) and comparing the bar heights. We will also practice constructing our own bar graphs from frequency tables, making sure to choose a good scale, label everything clearly, keep bar widths and spacing uniform, and give the graph a suitable title.
This chapter uses simple examples and clear visuals to help you develop the basic skills needed to handle data – collecting, organizing it using tally marks and tables, and representing it visually using pictographs and bar graphs. These are fundamental skills for understanding information presented in newspapers, books, and many other places!
Data & Related Terms
In our daily lives, we constantly interact with information. Whether it's knowing the number of students present in class today, the marks you scored in different subjects, the weather forecast, or the price of vegetables in the market, all this information helps us understand and make decisions about the world around us. In mathematics, especially in a field called Statistics, we deal with collecting, organising, and understanding such information. This raw, collected information is called Data.
What is Data?
Simply put, Data is a collection of facts or figures gathered to gain specific information or to answer a question. Data can be in the form of numbers, measurements, observations, or even simple descriptions.
Example: If you want to know which ice cream flavour is most popular in your class, you would ask each classmate their favourite flavour and record their answers. The list of favourite flavours you get from everyone is the data you collected.
Example: The attendance recorded in your class diary for each day of the week is also a form of data.
Example: The weights of all the students in your class form a set of data.
Related Terms in Data Handling
When we work with data, there are a few terms that are commonly used:
Observation: Each individual piece of information or value in a collection of data is called an observation. It's the response or measurement for a single item or person.
Example: If the marks obtained by 5 students in a test are 25, 30, 28, 30, and 25, then 25, 30, 28, 30, and 25 are the individual observations in this data set.
Raw Data: Data that is collected and recorded in its original, unorganised form is called raw data. It's just a list of observations as they were gathered.
Example: The list of marks {25, 30, 28, 30, 25} exactly as collected is raw data.
Organised Data: Raw data, especially when there are many observations, can be confusing and hard to analyse quickly. When we arrange the raw data in a systematic and orderly manner, it becomes organised data. Organising data makes it easier to understand, interpret, and find patterns or summaries.
Example: Arranging the marks {25, 30, 28, 30, 25} in ascending order {25, 25, 28, 30, 30} is one way to organise it. Another way is to use tables, which we will discuss next.
Frequency: The frequency of a particular observation is the number of times that observation occurs in the data set. It tells us how often a specific value or category appears.
Example: In the marks data {25, 30, 28, 30, 25}, the observation 25 appears 2 times, so its frequency is 2. The observation 30 appears 2 times, so its frequency is 2. The observation 28 appears 1 time, so its frequency is 1.
Tally Marks: Tally marks are a simple system of counting frequencies, especially useful when dealing with raw data. They help keep track of counts as you go through the data. For every occurrence of an observation, you draw a vertical line. For the fifth occurrence, you draw a diagonal line across the previous four. This grouping of five makes counting easier.
- Count 1: $|$
- Count 2: $||$
- Count 3: $|||$
- Count 4: $||||$
- Count 5: $\bcancel{||||}$
- Count 6: $\bcancel{||||} |$
- Count 9: $\bcancel{||||} ||||$
- Count 10: $\bcancel{||||} \bcancel{||||}$
Organising Data using a Frequency Distribution Table
A common and effective way to organise raw data is by creating a Frequency Distribution Table. This table systematically lists the different observations or categories from the data and shows how many times each one appears (its frequency), often using tally marks as an intermediate step.
Let's take the example of marks obtained by 10 students in a test (out of 10). The raw data is: 7, 5, 6, 7, 8, 5, 6, 7, 9, 7.
We can organise this data using a table with three columns: 'Marks Obtained', 'Tally Marks', and 'Number of Students (Frequency)'.
Go through the raw data one by one and make a tally mark for each mark in the corresponding row:
- 7: $||||$
- 5: $||$
- 6: $||$
- 8: $|$
- 9: $|$
Now, count the tally marks to get the frequency for each mark:
| Marks Obtained | Tally Marks | Number of Students (Frequency) |
|---|---|---|
| 5 | $||$ | 2 |
| 6 | $||$ | 2 |
| 7 | $||||$ | 4 |
| 8 | $|$ | 1 |
| 9 | $|$ | 1 |
| Total | $2+2+4+1+1=10$ |
This table is much easier to read and understand than the raw data list. For instance, we can immediately see that the mark '7' has the highest frequency (4 students got 7 marks).
Example 1. A die was thrown 20 times and the outcomes were recorded as follows: 1, 6, 3, 5, 4, 2, 6, 1, 3, 5, 4, 6, 2, 3, 1, 6, 5, 4, 3, 6. Organise this data using tally marks and prepare a frequency table.
Answer:
The possible outcomes when a standard die is thrown are the numbers 1, 2, 3, 4, 5, and 6. We will list these outcomes and count how many times each appears in the given raw data using tally marks.
Let's go through the outcomes one by one and make the tally marks:
- 1, 6, 3, 5, 4, 2 ($| | | | | |$)
- 6, 1, 3, 5, 4 ($|| ||| || |||$)
- 6, 2, 3, 1, 6 ($|| || |||| ||| |||$)
- 5, 4, 3, 6 ($\bcancel{||||} |$ $\bcancel{||||} |$ $\bcancel{||||} |$ $\bcancel{||||} ||||$)
Let's clean up the tally marks and convert them to counts for each outcome:
- Outcome 1: $|||$ (3 times)
- Outcome 2: $||$ (2 times)
- Outcome 3: $||||$ (4 times)
- Outcome 4: $|||$ (3 times)
- Outcome 5: $|||$ (3 times)
- Outcome 6: $\bcancel{||||}$ (5 times)
Now, we can construct the frequency distribution table:
| Outcome | Tally Marks | Frequency |
|---|---|---|
| 1 | $|||$ | 3 |
| 2 | $||$ | 2 |
| 3 | $||||$ | 4 |
| 4 | $|||$ | 3 |
| 5 | $|||$ | 3 |
| 6 | $\bcancel{||||}$ | 5 |
| Total | $3+2+4+3+3+5 = 20$ |
The table shows that, for example, the outcome '3' appeared 4 times, and the outcome '6' appeared 5 times during the 20 throws.
Pictograph
In the previous section, we learned about collecting and organising data, particularly using frequency tables and tally marks. While frequency tables summarise data well, sometimes a visual representation can make the data even easier to understand at a glance. A Pictograph is one such visual method.
What is a Pictograph?
A Pictograph (also called a pictogram) is a way of representing data using relevant pictures, symbols, or icons. Instead of just listing numbers or using tally marks, a pictograph uses drawings or images where each image represents a certain number of items or units of data. This makes the data very easy to read and interpret quickly by visually comparing the rows or columns of pictures.
Components of a Pictograph
A well-made pictograph includes a few essential parts to make it clear and informative:
Title: This is a brief heading that tells us what the pictograph is about (e.g., "Number of Cars Sold", "Favourite Colours of Students").
Categories: These are the different items, groups, or classifications for which the data is being presented (e.g., names of days, types of fruits, names of classes).
Symbol / Picture: This is the image or icon chosen to represent the items being counted (e.g., a picture of a car for counting cars, a picture of a person for counting students). This symbol is repeated to show the quantity for each category.
Key (or Scale): This is the most important part of a pictograph. The key tells us exactly what quantity each single symbol represents. The key helps us to convert the number of symbols into the actual count of items. For example, "🍎 = 10 students" means every apple symbol drawn represents 10 students.
Interpreting a Pictograph
Reading and understanding information from a pictograph is called interpreting a pictograph. Follow these steps:
- Read the Title of the pictograph to know what information is being presented.
- Look at the Key (or scale) carefully. Understand what quantity each symbol represents. Also, see if any parts of a symbol are used (like half a symbol) and what quantity they represent.
- For each Category, count the number of symbols drawn.
- Multiply the number of symbols counted in Step 3 by the quantity represented by a single symbol (from the key). This will give you the total number of items for that category.
- Use this information to compare quantities across different categories and draw conclusions.
Example of Interpreting a Pictograph:
Let's look at an example of a pictograph showing the number of students who like different fruits in a class:
Title: Favourite Fruits of Students
Key: 🍎 = 5 students
| Mango: | 🍎 🍎 🍎 🍎 |
| Apple: | 🍎 🍎 🍎 |
| Orange: | 🍎 🍎 |
| Banana: | 🍎 🍎 🍎 🍎 🍎 |
From this pictograph:
- The Title is "Favourite Fruits of Students".
- The Key tells us that 1 apple symbol (🍎) represents 5 students.
- Now let's interpret for each category (fruit):
- Mango: There are 4 apple symbols. So, $4 \times 5 = 20$ students like Mango.
- Apple: There are 3 apple symbols. So, $3 \times 5 = 15$ students like Apple.
- Orange: There are 2 apple symbols. So, $2 \times 5 = 10$ students like Orange.
- Banana: There are 5 apple symbols. So, $5 \times 5 = 25$ students like Banana.
We can easily see that Banana is the most liked fruit (25 students) and Orange is the least liked (10 students).
Drawing a Pictograph
To draw a pictograph from given data, follow these steps:
- Make sure your data is collected and organised (a frequency table is very useful here).
- Choose a simple and appropriate symbol that represents the items in your data. For example, if you are counting people, a stick figure might be suitable. If counting cars, a car icon is good.
- Decide on a Scale (the key). This means deciding how many items each symbol will represent. The scale should be chosen carefully:
- It should be a number that can divide most or all of the frequencies evenly (like 5, 10, 100, etc.).
- It should make the number of symbols for each category manageable. Too many symbols make the pictograph crowded; too few might not show differences clearly.
- Consider if you need to use parts of symbols (like half a symbol) to represent quantities that are not whole multiples of the scale value.
- Draw the appropriate number of symbols for each category based on your chosen scale. If a quantity is, for example, 25 and your scale is 10 items per symbol, you would need 2 full symbols (20 items) and half a symbol (5 items).
- Write a clear Title for the pictograph.
- Write the Key clearly below the pictograph, explaining what one full symbol and any parts of symbols represent.
Example of Drawing a Pictograph:
The number of cars sold by a dealer in 5 days are given in the table:
| Day | Number of Cars Sold |
|---|---|
| Monday | 10 |
| Tuesday | 15 |
| Wednesday | 20 |
| Thursday | 5 |
| Friday | 10 |
Let's draw a pictograph for this data.
- The data is organised in a table.
- Choose a symbol. Let's use a simple car icon 🚗.
- Choose a scale. The numbers are 10, 15, 20, 5, 10. All these numbers are multiples of 5. Let's choose the key: 1 🚗 represents 5 cars sold.
- Calculate the number of symbols needed for each day:
- Monday: 10 cars $\implies 10 \div 5 = 2$ symbols.
- Tuesday: 15 cars $\implies 15 \div 5 = 3$ symbols.
- Wednesday: 20 cars $\implies 20 \div 5 = 4$ symbols.
- Thursday: 5 cars $\implies 5 \div 5 = 1$ symbol.
- Friday: 10 cars $\implies 10 \div 5 = 2$ symbols.
- Write the title: Cars Sold by a Dealer.
- Write the key: 🚗 = 5 Cars.
- Draw the pictograph using the calculated number of symbols for each day:
Title: Cars Sold by a Dealer
Key: 🚗 = 5 Cars
| Monday: | 🚗 🚗 |
| Tuesday: | 🚗 🚗 🚗 |
| Wednesday: | 🚗 🚗 🚗 🚗 |
| Thursday: | 🚗 |
| Friday: | 🚗 🚗 |
Example 1. The number of cycles produced in a factory during five consecutive weeks is as follows:
| Week | Number of Cycles |
|---|---|
| 1st Week | 600 |
| 2nd Week | 950 |
| 3rd Week | 800 |
| 4th Week | 700 |
| 5th Week | 650 |
Prepare a pictograph of the number of cycles produced using the symbol of a cycle 🚲, where 1 symbol represents 100 cycles.
Answer:
The data is given, and we are instructed to use the symbol 🚲 with a key: 1 🚲 represents 100 cycles. This means every symbol drawn counts as 100 cycles produced.
To find the number of symbols needed for each week, we divide the number of cycles produced by the value of one symbol (100):
- 1st Week: 600 cycles $\implies 600 \div 100 = 6$ symbols.
- 2nd Week: 950 cycles $\implies 950 \div 100 = 9.5$. This means we need 9 full symbols and a representation for $0.5 \times 100 = 50$ cycles. We will represent this as 'Half 🚲' in the key. So, 9 full symbols and half a symbol.
- 3rd Week: 800 cycles $\implies 800 \div 100 = 8$ symbols.
- 4th Week: 700 cycles $\implies 700 \div 100 = 7$ symbols.
- 5th Week: 650 cycles $\implies 650 \div 100 = 6.5$. This means we need 6 full symbols and a representation for $0.5 \times 100 = 50$ cycles. So, 6 full symbols and half a symbol.
Now, we can draw the pictograph with the title and key.
Title: Cycles Produced in a Factory
Key: 🚲 = 100 Cycles, (Half symbol represents 50 Cycles)
| 1st Week: | 🚲 🚲 🚲 🚲 🚲 🚲 |
| 2nd Week: | 🚲 🚲 🚲 🚲 🚲 🚲 🚲 🚲 🚲 🚲 |
| 3rd Week: | 🚲 🚲 🚲 🚲 🚲 🚲 🚲 🚲 |
| 4th Week: | 🚲 🚲 🚲 🚲 🚲 🚲 🚲 |
| 5th Week: | 🚲 🚲 🚲 🚲 🚲 🚲 🚲 |
From this pictograph, we can quickly compare the cycle production across different weeks visually.
Bar Graphs
Just like pictographs, bar graphs are another visual tool used to represent and compare data. They are widely used because they are easy to create and interpret, providing a clear picture of the data at a glance.
What is a Bar Graph?
A Bar Graph (also known as a bar chart) is a graphical way of representing data where quantities are shown using bars. These bars are usually rectangular and have a uniform width. They can be drawn either vertically (standing upright) or horizontally (lying on their side). There is equal spacing between the bars.
The key idea is that the length (for horizontal bars) or the height (for vertical bars) of each bar is proportional to the value or frequency it represents. This allows for easy comparison of quantities by simply looking at the relative lengths of the bars.
Components of a Bar Graph
A standard bar graph consists of the following parts:
Title: A clear and concise heading that describes the data being presented in the graph.
Axes: A bar graph has two perpendicular lines called axes. Typically, these are the horizontal axis (usually drawn left-to-right) and the vertical axis (usually drawn bottom-to-top).
- One axis is used to represent the categories (e.g., names of items, days, months, groups).
- The other axis is used to represent the quantity or frequency of each category (e.g., number of students, count of items, sales figures).
Scale: A scale is chosen for the axis that represents the quantity or frequency. It indicates how much each unit of length along that axis represents. Choosing an appropriate scale is important so that the graph is easy to read and fits within the available space. For example, a scale could be '1 unit length on the axis = 10 students' or '1 division = $\textsf{₹}50$'. The scale must start from 0.
Bars: These are the rectangular shapes that represent the data.
- All bars must have the same width.
- The spacing between adjacent bars must be equal.
- The height (in a vertical bar graph) or length (in a horizontal bar graph) of each bar corresponds to the quantity or frequency of the category it represents, according to the chosen scale.
Interpreting a Bar Graph
Interpreting a bar graph means reading and understanding the information it presents and drawing conclusions. Follow these steps:
- Read the Title to understand the overall theme of the data.
- Look at the labels on the horizontal and vertical axes to understand what each axis represents (which one is for categories and which for quantities/frequencies).
- Carefully look at the Scale on the frequency/quantity axis. Understand the value represented by each unit or division on this axis.
- For each Bar, find the category it represents (from the category axis). Then, look at the top (for vertical bars) or end (for horizontal bars) of the bar and trace across to the frequency/quantity axis to read its value using the scale.
- Compare the heights/lengths of different bars to easily compare the quantities of different categories. The tallest/longest bar represents the category with the highest value/frequency, and the shortest bar represents the lowest value/frequency.
Example of Interpreting a Bar Graph:
Here is a bar graph showing the favourite colours of students in a class:
From this bar graph:
- Title: Favourite Colours of Students.
- Axes: The horizontal axis represents Colours (Red, Blue, Green, Yellow, Black). The vertical axis represents the Number of Students.
- Scale: The vertical axis is marked with intervals. Let's assume each marked division (e.g., from 0 to 5, 5 to 10, etc.) represents 5 students.
- Bars:
- The bar for Red goes up to the mark 20 on the vertical axis. So, 20 students like Red.
- The bar for Blue goes up to the mark 25. So, 25 students like Blue.
- The bar for Green goes up to the mark 15. So, 15 students like Green.
- The bar for Yellow goes up to the mark 10. So, 10 students like Yellow.
- The bar for Black goes up to the mark 5. So, 5 students like Black.
By looking at the heights of the bars, we can immediately see that the Blue bar is the tallest, meaning Blue is the most favourite colour (25 students). The Black bar is the shortest, meaning Black is the least favourite colour (5 students).
Drawing a Bar Graph
To construct a bar graph from a given set of data, follow these steps:
- Draw two perpendicular lines, one horizontal and one vertical. These are your axes. Their intersection point is usually marked as 0.
- Decide which axis will represent the categories (e.g., items, names) and which will represent the quantities or frequencies. Label the axes clearly.
- Mark the categories on the chosen axis (e.g., mark points for each day or each subject). Leave equal space between the points where the bars will be drawn. Also, decide on the width of the bars. The width of all bars should be the same, and the space between any two adjacent bars should also be the same.
- Choose a suitable Scale for the frequency/quantity axis. Look at the values you need to represent and find a number that divides them easily or allows the largest value to fit comfortably on the axis. Mark equal divisions along this axis according to your chosen scale (e.g., if the scale is 1 unit = 10 items, mark 10, 20, 30, ...). Write the scale clearly, usually near the axis (e.g., "Scale: 1 unit = 100 Bicycles").
- Draw the rectangular Bars. For each category, draw a bar starting from the category axis. The height (for vertical) or length (for horizontal) of the bar should correspond to the quantity of that category, as determined by the scale. Make sure the bars have uniform width and equal spacing.
- Give a clear Title to the bar graph, usually placed at the top.
Example of Drawing a Bar Graph:
The number of bicycles manufactured in a factory during five consecutive years is given in the table below:
| Year | Number of Bicycles Manufactured |
|---|---|
| 2016 | 800 |
| 2017 | 1000 |
| 2018 | 1200 |
| 2019 | 900 |
| 2020 | 1100 |
Let's draw a bar graph to represent this data.
- Draw the horizontal and vertical axes.
- Let the horizontal axis represent 'Year' (categories) and the vertical axis represent the 'Number of Bicycles Manufactured' (quantities). Label the axes.
- Mark the years 2016, 2017, 2018, 2019, and 2020 at equal distances along the horizontal axis. Decide on a uniform width for the bars and equal spacing between them.
- Choose a suitable scale for the vertical axis. The quantities are 800, 1000, 1200, 900, 1100. These are all multiples of 100. A good scale would be "1 unit = 100 Bicycles". Mark the vertical axis with divisions representing 0, 100, 200, ..., up to a value slightly greater than the maximum (e.g., 1300). Write "Scale: 1 unit = 100 Bicycles" or similar.
- Calculate the height of each bar based on the scale:
- 2016: 800 bicycles $\implies$ height = $800 \div 100 = 8$ units.
- 2017: 1000 bicycles $\implies$ height = $1000 \div 100 = 10$ units.
- 2018: 1200 bicycles $\implies$ height = $1200 \div 100 = 12$ units.
- 2019: 900 bicycles $\implies$ height = $900 \div 100 = 9$ units.
- 2020: 1100 bicycles $\implies$ height = $1100 \div 100 = 11$ units.
- Draw the bars with the calculated heights above the corresponding years on the horizontal axis. Ensure uniform width and equal spacing.
- Write the Title: Bicycles Manufactured in a Factory.
Example 1. The following table shows the amount of money collected from students of different classes for a charity show:
| Class | Money Collected (in $\textsf{₹}$) |
|---|---|
| VI | 250 |
| VII | 300 |
| VIII | 350 |
| IX | 400 |
| X | 450 |
Draw a bar graph to represent this data. Choose an appropriate scale.
Answer:
We will draw a vertical bar graph.
Let the horizontal axis represent 'Class' and the vertical axis represent 'Money Collected (in $\textsf{₹}$)'. Label the axes clearly.
Mark the classes VI, VII, VIII, IX, X at equal distances on the horizontal axis. Ensure uniform bar width and spacing.
Choose a suitable scale for the vertical axis. The amounts collected are 250, 300, 350, 400, 450. All these numbers are multiples of 50. A convenient scale would be "1 unit = $\textsf{₹}50$". Mark the vertical axis with divisions representing 0, 50, 100, 150, ..., up to a value slightly greater than 450 (e.g., 500). Write "Scale: 1 unit = $\textsf{₹}50$".
Calculate the height of each bar in units based on the scale:
- Class VI: $\textsf{₹}250 \implies$ height = $250 \div 50 = 5$ units.
- Class VII: $\textsf{₹}300 \implies$ height = $300 \div 50 = 6$ units.
- Class VIII: $\textsf{₹}350 \implies$ height = $350 \div 50 = 7$ units.
- Class IX: $\textsf{₹}400 \implies$ height = $400 \div 50 = 8$ units.
- Class X: $\textsf{₹}450 \implies$ height = $450 \div 50 = 9$ units.
Draw the bars with these heights above the corresponding classes on the horizontal axis. Ensure uniform width and spacing for the bars.
Give the title: Money Collected for Charity Show.
