CBSE Class 8 Maths: Data Handling Explained

Understanding What Data Is

At its core, data refers to raw facts and figures collected for a specific purpose. Imagine your teacher wants to know the favourite subjects of students in your class. She asks each student, and lists down their responses. This list is raw data. Each individual piece of information collected, like one student's favourite subject, is called an observation.

Raw data, when collected, is often unorganised and difficult to interpret directly. For example, if you list the marks of 30 students in a test, just looking at the raw list won't immediately tell you how many students scored above 80, or how many scored very low. To make sense of it, we need to organise this raw data. This organisation helps us to quickly grasp the main features of the data and draw meaningful conclusions. Think of it like tidying up a messy room – once everything is in its place, it's much easier to find what you're looking for!

Key Terms in Data Handling

Data

A collection of facts, figures, or information gathered for a specific purpose.

Observation

Each individual entry or piece of information in the data set.

Frequency

The number of times a particular observation occurs in a data set.

Frequency Distribution

A table that shows each observation or range of observations along with their corresponding frequencies.

Tally Marks

A quick way of counting data points by using vertical lines and crossing them with a fifth line for groups of five.

Range

The difference between the highest and lowest observation in a data set.

Class Interval

When data is grouped, each group is called a class interval (e.g., 0-10, 10-20). It helps organise large amounts of data.

Lower Limit

The smallest value in a class interval (e.g., 10 in 10-20).

Upper Limit

The largest value in a class interval (e.g., 20 in 10-20). Note: The upper limit is generally not included in that interval but in the next one for continuous data.

Class Size (or Width)

The difference between the upper and lower class limits of a class interval (e.g., 20 - 10 = 10 for 10-20).

Organising Data: Tally Marks and Frequency Tables

1. Step 1: Collect Raw Data — Gather all the information relevant to your study. For example, let's say the favourite colours of 20 students are: Red, Blue, Green, Yellow, Red, Blue, Red, Yellow, Green, Blue, Red, Green, Yellow, Blue, Red, Green, Blue, Red, Yellow, Green.
2. Step 2: List Observations and Use Tally Marks — Identify all unique observations (the different colours). Then, go through the raw data one by one and make a tally mark (a vertical bar |) next to the corresponding observation. For every fifth observation, draw a diagonal line across the previous four tally marks (like 𝄕) to make counting easier. This creates a 'group of 5'.
3. Step 3: Count Frequencies — After all observations have been tallied, count the tally marks for each observation. This count is the frequency for that observation.
4. Step 4: Create a Frequency Distribution Table — Organise the observations and their frequencies into a table with columns like 'Favourite Colour', 'Tally Marks', and 'Frequency'. Always sum up the frequencies to ensure it matches the total number of observations (20 in our example). Example Table: | Favourite Colour | Tally Marks | Frequency | |------------------|-------------|-----------| | Red | 𝄕 | 6 | | Blue | 𝄕 | 5 | | Green | 𝄕 | 5 | | Yellow | |||| | 4 | | Total | | 20 |

Visualizing Data: Bar Graphs and Histograms

• 1. Bar Graph:

  • What it is: A pictorial representation of numerical data using bars of uniform width drawn horizontally or vertically with equal spacing between them. The length of each bar represents the frequency of the data item.
  • When to use: For discrete data (data that can be counted, distinct categories like favourite colours, types of pets, marks of individual students).
  • How to draw (Example: Favourite Colours from above):
  • Step 1: Draw two perpendicular axes. Label the horizontal axis 'Favourite Colour' and the vertical axis 'Number of Students' (Frequency).
  • Step 2: Choose a suitable scale for the vertical axis (e.g., 1 unit = 1 student). This means for 'Red' (6 students), the bar will go up to 6 units.
  • Step 3: Draw bars of uniform width for each colour. The height of each bar corresponds to its frequency. Ensure there is equal space between each bar.
  • Step 4: Label the graph clearly with a title, e.g., 'Favourite Colours of Class 8 Students'.

  2. Histogram:

  • What it is: A type of bar graph used for grouped frequency distribution. Here, the class intervals are shown on the horizontal axis, and the frequencies on the vertical axis. Unlike bar graphs, there are no gaps between the bars, as it represents continuous data.
  • When to use: For continuous data (data that can take any value within a range, like heights, weights, ages, marks grouped into intervals).
  • **How to draw (Example: Marks of students in a test out of 50, grouped data):

  Class Interval: 0-10, 10-20, 20-30, 30-40, 40-50
  Frequency: 3, 7, 12, 8, 5**

  • Step 1: Draw two perpendicular axes. Label the horizontal axis 'Marks' (showing class intervals) and the vertical axis 'Number of Students' (Frequency).
  • Step 2: Choose a suitable scale for the vertical axis (e.g., 1 unit = 1 student). For the horizontal axis, mark the limits of the class intervals (0, 10, 20, 30, 40, 50).
  • Step 3: Draw rectangular bars. The width of each bar will be the class size (10 in this case). The height corresponds to the frequency. Since the data is continuous, the bars are drawn adjacent to each other without any gaps. If the first interval doesn't start from 0 (e.g., 20-30), use a 'kink' or 'zig-zag' mark on the horizontal axis near the origin to indicate that the scale has been broken.
  • Step 4: Label the graph with a title, e.g., 'Marks of Class 8 Students in a Test'.

Exam Tip: Avoiding Common Mistakes

When dealing with Data Handling questions, students often make a few common errors. Firstly, always remember the key difference between a bar graph and a histogram: bar graphs have spaces between bars and are for discrete data, while histograms have no spaces between bars and are for continuous data (grouped frequency distribution).

Secondly, ensure your axes are properly labelled with units and a clear scale is chosen for the frequency axis; an inconsistent scale can lead to misrepresentation. When constructing frequency tables for grouped data, pay close attention to the class intervals. For example, in an interval like 10-20, the observation '10' belongs to this interval, but '20' usually belongs to the next interval (20-30) to avoid double-counting. Always double-check your tally marks and frequency counts to avoid calculation errors. A small mistake in counting can lead to an entirely wrong graph or conclusion.

Practice Questions with Solutions

• Q: The marks obtained by 15 students in a Math test (out of 10) are: 6, 7, 5, 8, 9, 6, 5, 7, 8, 7, 6, 5, 9, 8, 7. Construct a frequency distribution table using tally marks.
  A: Step 1: Identify the unique observations. The marks are 5, 6, 7, 8, 9.
  Step 2: Go through the data and make tally marks for each mark.
  Mark 5: |||
  Mark 6: |||
  Mark 7: ||||
  Mark 8: |||
  Mark 9: ||
  Step 3: Count the tally marks to find the frequency.
  Step 4: Construct the frequency distribution table.
  | Marks | Tally Marks | Frequency |
  |-------|-------------|-----------|
  | 5 | ||| | 3 |
  | 6 | ||| | 3 |
  | 7 | |||| | 4 |
  | 8 | ||| | 3 |
  | 9 | || | 2 |
  | Total | | 15 |
  Final answer: The frequency distribution table is as shown above.

• Q: The number of vehicles passing a school gate in 30 minutes are given below: Car, Bike, Auto, Bus, Bike, Car, Auto, Bike, Car, Bike, Bus, Car, Auto, Bike, Bike, Car, Auto, Bus, Bike, Car, Bike, Auto, Bike, Car, Bus, Auto, Bike, Car, Bike, Auto. Construct a frequency distribution table and then draw a bar graph to represent this data.
  A: Step 1: Create a frequency distribution table using tally marks.
  | Vehicle | Tally Marks | Frequency |
  |---------|-------------|-----------|
  | Car | 𝄕 ||| | 8 |
  | Bike | 𝄕 𝄕 || | 12 |
  | Auto | 𝄕 ||| | 8 |
  | Bus | ||| | 3 |
  | Total | | 30 |
  Step 2: Draw two perpendicular axes. Label the horizontal axis 'Vehicle Type' and the vertical axis 'Number of Vehicles'.
  Step 3: Choose a scale for the vertical axis, e.g., 1 unit = 2 vehicles.
  Step 4: Draw bars of uniform width for each vehicle type, with equal spacing between them. The height of the bars will be:
  Car: 8 units (up to 8 on scale)
  Bike: 12 units (up to 12 on scale)
  Auto: 8 units (up to 8 on scale)
  Bus: 3 units (up to 3 on scale)
  Step 5: Give the graph a title: 'Vehicles Passing School Gate in 30 Minutes'.
  Final answer: The frequency table is created, and the bar graph represents this data with distinct bars for each vehicle type, showing their frequencies.

• Q: The heights (in cm) of 25 students are given: 145, 150, 148, 152, 155, 147, 151, 149, 153, 146, 150, 154, 148, 152, 156, 149, 151, 155, 147, 153, 148, 150, 152, 146, 154. Group the data into class intervals of 145-150, 150-155, 155-160 and prepare a frequency distribution table. Then, draw a histogram for this data.
  A: Step 1: Determine the class intervals. (Given: 145-150, 150-155, 155-160). Remember, the upper limit is excluded from the current interval and included in the next.
  Step 2: Tally the data for each interval.
  145-150 (includes 145, 146, 147, 148, 149): 145, 148, 147, 149, 146, 148, 147, 148, 146. Tally: 𝄕 𝄕 |||| = 9
  150-155 (includes 150, 151, 152, 153, 154): 150, 152, 155(next), 151, 153, 150, 154, 152, 151, 153, 150, 152, 154. Tally: 𝄕 𝄕 𝄕 || = 13
  155-160 (includes 155, 156, 157, 158, 159): 155, 156, 155. Tally: ||| = 3
  Step 3: Create the frequency distribution table.
  | Height (cm) | Tally Marks | Frequency |
  |-------------|-------------|-----------|
  | 145-150 | 𝄕 𝄕 |||| | 9 |
  | 150-155 | 𝄕 𝄕 𝄕 || | 13 |
  | 155-160 | ||| | 3 |
  | Total | | 25 |
  Step 4: Draw two perpendicular axes. Label the horizontal axis 'Height (cm)' with class intervals and the vertical axis 'Number of Students' (Frequency).
  Step 5: Choose a scale for the vertical axis, e.g., 1 unit = 1 student. Mark the class boundaries (145, 150, 155, 160) on the horizontal axis.
  Step 6: Draw rectangular bars of widths equal to the class size (5 cm) and heights corresponding to the frequencies. The bars should be adjacent with no gaps.
  145-150: height 9 units
  150-155: height 13 units
  155-160: height 3 units
  Step 7: Title the histogram 'Heights of 25 Students'.
  Final answer: The grouped frequency table is made, and a histogram is drawn showing the continuous distribution of heights.

• Q: A survey found the favourite fruit of 20 children: Mango, Apple, Banana, Mango, Orange, Apple, Mango, Banana, Mango, Apple, Orange, Mango, Banana, Mango, Apple, Orange, Mango, Banana, Apple, Mango. Which fruit is the most popular, and which is the least popular based on this data?
  A: Step 1: Organise the raw data into a frequency table.
  | Fruit | Tally Marks | Frequency |
  |--------|-------------|-----------|
  | Mango | 𝄕 𝄕 || | 12 |
  | Apple | 𝄕 | | 5 |
  | Banana | |||| | 4 |
  | Orange | ||| | 3 |
  | Total | | 20 |
  Step 2: Identify the fruit with the highest frequency and the lowest frequency.
  Most popular (highest frequency): Mango (12 children)
  Least popular (lowest frequency): Orange (3 children)
  Final answer: Mango is the most popular fruit, and Orange is the least popular fruit among the children surveyed.

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