Chapter 9 — Data Handling

What is data?
Data are bits of information collected about things, people or events. In everyday life we see data when we count how many students came to school, note the names of favourite games, or record marks in a test. Data can be words such as names and categories (for example, Red, Blue, Green) or numbers such as counts and measurements (for example, 5 pens, 12 books). Understanding the type of data helps decide how to organise and present it.

Raw data means the exact answers or measurements collected at first. When many people give answers, raw data can be messy. We need to organise raw data so it is easy to understand. Organisation includes grouping similar answers, counting how often each answer occurs and placing these counts in a table. Once organised, we can present the data using pictures, bars or charts which make patterns easier to spot.

Data handling is the whole process from asking a question and collecting data, to organising, displaying and interpreting the results. The process helps answer important questions. For example: Which book is most popular in the class? When do most students come to school? By following steps—decide what to ask, collect answers, record them carefully, create a table, and draw a graph—we move from raw facts to clear conclusions. This is useful in science, maths and daily planning.

Types of data: Qualitative data describe qualities or categories like colours or types of fruits. Quantitative data are numbers and tell how many or how much. Quantitative data may be discrete (countable whole numbers like number of pens) or continuous (measurements like height). In Class 6 you mostly work with counts and categories. Always think: Why are we collecting this data? A clear purpose leads to better questions and more useful results.

Ways to collect data
Collecting good data begins with a clear question. Once you know what you want to find out, choose a suitable method. Common ways are observation, surveys (questionnaires), experiments and existing records. Each method fits different situations and you must pick one that gives correct and honest information.

Observation means watching and recording what you see without asking people. It is useful when behaviour matters or when you cannot ask questions. For example, to learn how many students come to school by bicycle, you stand at the gate for a fixed time and note each bicycle. Write the results immediately to avoid forgetting. Observation gives factual counts but does not tell reasons for behaviour.

Survey or questionnaire involves asking people questions. Keep questions short and simple. Use one question at a time for class work, like 'Which game do you play most?' or 'Which fruit do you like best?'. Record each reply carefully using tally marks so you do not lose counts. Choose a clear sample: if you survey only one group, results apply to that group. If you want class-wide results, ask every student or a representative sample chosen fairly.

Records: Many useful data come from existing records such as attendance registers or library logs. These save time but check that records are complete and recent. Experiments involve changing a condition and measuring results, often used in science projects.

Important checks when collecting data: avoid leading questions, check for missing or duplicate answers, and keep privacy by not forcing personal details. Accurate, honest collection makes later organisation and graphs reliable and meaningful.

Tally marks
Tally marks are an easy, quick method to count responses as you collect them. When you record many answers, tally marks help prevent mistakes by grouping counts in fives. Use four vertical strokes and the fifth stroke diagonally or across to form a bundle of five. This lets you count by fives and reduces errors when adding many entries.

To make a tally chart, first write categories vertically in one column. As each answer is collected, put a tally mark beside the correct category. After all answers are recorded, convert each group of tallies into a number to form the frequencies. A frequency table summarises these results: it lists each category and its frequency. The frequency tells how many times that category occurred and becomes the basis for charts such as pictographs or bar graphs.

Steps to create a frequency table: decide categories, collect data using tally marks, count tallies to get numeric frequencies, and write totals. Always include a heading and check that the sum of frequencies equals the total number of respondents. If not, go back and find recording mistakes. Using tallies while collecting prevents missing or double counting replies.

Advantages of tallies and frequency tables: They are simple, quick to update during a survey and easy to convert into visual displays. For classroom use, explain tally marks clearly to everyone before starting. Use the frequency table to answer basic questions: which category is most common, which is least, and what is the total number of responses. These tables make later graph drawing straightforward and accurate.

What is a pictograph?
A pictograph presents data using pictures or symbols. Each picture stands for a fixed number of items; this fixed number is called the scale. Pictographs are useful for making information easy to see quickly, especially for small whole-number data and when categories are few. They are commonly used to show favourite foods, number of pets, or books read.

Steps to draw a pictograph: first make a frequency table from your raw data. Decide on the symbol to use for the picture and choose an appropriate scale so that the number of symbols is not too large. For example, if Mango = 10 and many categories have numbers near 10, choose scale 1 symbol = 2 students so Mango becomes 5 symbols. Write the scale clearly under the title so anyone reading the pictograph knows how to convert symbols to actual counts.

Symbols may be drawn full, half or quarter only when the scale allows fractional symbols and it is explained. Always draw neat symbols and place them in a column next to the category name. Add labels for category names and a title describing what the pictograph shows. A legend or note with the scale avoids confusion.

Reading a pictograph: count the number of symbols for a category and multiply by the scale to get the real number. Pictographs are visual and good for quick comparisons: the category with more symbols is more frequent. However, avoid using pictographs when numbers are very large because many symbols make reading hard. In such cases use bar graphs with a chosen scale. Also be careful: a pictograph can be misleading if the scale is not stated or if symbols are drawn in varying sizes.

Bar graph
A bar graph uses bars to represent frequencies of categories. The length or height of each bar matches the frequency. Bar graphs help compare categories clearly and are preferred when numbers are large or precise comparison is needed. They are simple to draw and read when axes, labels and scale are chosen correctly.

To draw a bar graph first prepare a frequency table. Decide which axis will show categories (usually the horizontal axis) and which will show frequency (vertical axis). Choose a scale for the frequency axis so that the highest frequency fits comfortably on your paper; for instance, 1 cm = 2 items or 1 square = 1 item. Mark and label equal scale intervals on the frequency axis. On the category axis write each category at equal distances and draw bars of equal width for each category. Leave a small equal gap between bars so each category is separated and easy to compare.

Label both axes clearly and give the graph a meaningful title. If you use colours or patterns, include a small legend to explain them. Bars can be vertical or horizontal: vertical bars are common but horizontal bars are useful when category names are long. Make sure bars start from zero on the frequency axis; starting from a number other than zero can mislead the reader about differences.

To read a bar graph, look at the top of a bar and use the scale to find its frequency. You can answer questions such as which category has the highest frequency, which has the lowest, and by how much one category differs from another. For classroom problems, carefully show calculations for any differences or totals you find from the bar graph.

Double bar graph
A double bar graph lets you compare two related sets of data across the same categories. For example, you may compare the number of boys and girls who like each sport. Each category will have two bars placed side by side: one bar for the first group and another for the second group. Place equal width bars close together for each category to make comparison simple.

How to draw: prepare two frequency tables (one for each group). Choose the category axis and frequency axis and decide a single scale for the frequency axis that fits both sets of data. For each category draw a pair of bars — left bar for the first group and the adjacent bar for the second group. Use distinct colours or patterns for each group and add a legend to explain which colour shows which group. Title the graph and label the axes so the reader knows what is being compared.

When reading a double bar graph, look at bar pairs category by category. Note which group has the taller bar and by how much. To find precise differences, read the heights using the scale and subtract the smaller frequency from the larger. Double bar graphs are especially useful for comparing changes over time (for example, sales in January and February) or comparing subgroups within categories (for example, boys versus girls in subjects).

Careful drawing matters: equal width bars and equal spacing keep the graph neat. Also check that the scale begins at zero and that the bars do not touch unless intentionally shown as stacked bars. For class exams, draw neat bars, label colours and include a small written answer showing calculations when you compare values from the graph.

How to read graphs
Graphs are a visual summary of data. To interpret any graph correctly, read the title first to know the subject. Then check the labels on the axes and the units used. Examine the scale carefully: count how many units each division on the axis represents. Know whether the graph uses pictographs, single or double bars, or grouped intervals. This information guides how you read values and answer questions.

When reading a pictograph, multiply the number of symbols by the scale to find the actual value. For a bar graph read the top of the bar and use the vertical axis scale to get the frequency. For double bar graphs compare bars in each pair and use the legend to identify groups. For grouped data with class intervals, you often need to work with totals in ranges rather than exact values.

Common tasks when interpreting graphs include finding the highest and lowest categories, calculating the difference between two categories, finding the total for several categories, and comparing groups. Always show the calculation steps: for example, if one bar represents 18 students and another 11, write difference = 18 − 11 = 7. Use addition for totals or averaging methods for mean where needed.

Watch for misleading features: scales that do not start at zero, different bar widths, or symbols drawn with different sizes can give a false impression. Also check labels and the scale note; a missing scale makes accurate reading impossible. When answering questions, write complete sentences and include units such as 'students' or 'books' to make answers clear. Practise with many graphs so you become quick at reading values and spotting important information.

Why group data?
When many numerical observations vary over a range, grouping them into class intervals makes the data easier to view and analyse. Class intervals are ranges such as 10–19, 20–29, and so on. Instead of listing each number, we count how many observations fall in each interval. This produces a grouped frequency table which is easier to use for seeing where numbers cluster.

How to make class intervals: first find the smallest and largest values to know the data range. Then decide how many intervals you need; for classroom data five to eight intervals are common but choose what keeps counts simple. Pick equal width intervals that cover all values without overlapping. Use a consistent rule for endpoints, for example include the lower limit and exclude the upper limit (10–19 means 10 ≤ x < 20) so each value falls into exactly one class.

After choosing intervals, go through the data and tally each value into the appropriate interval. Convert tallies to frequencies and write totals. Grouped tables show how many observations lie in each range and help answer questions like which range has most observations or what proportion falls into a given interval. For example, if pupil heights are grouped into intervals, you can quickly see which height group is common.

Drawbacks: grouped data lose information about individual values; you cannot tell exact numbers from class intervals. However, grouped data are useful for large sets and form the basis of histograms and frequency polygons studied later. Always label intervals clearly and include the total number of observations at the end of the table. This helps avoid mistakes and allows checking that all values are accounted for.

Mode
The mode is the most frequently occurring value in a data set. It tells us which item or number appears most often. Mode is useful for both numerical and categorical data. For example, in a list of favourite colours, the colour that appears most is the modal category. In marks, the mark value that occurs most often is the mode.

To find the mode from raw data, count how many times each value appears and choose the value with the highest count. From a frequency table simply pick the category with the largest frequency. A data set can have one mode (unimodal), two modes (bimodal) or more than two modes (multimodal). If all values occur only once or all have the same frequency, we say there is no mode.

Mode for grouped data: when data are in class intervals, the interval with the highest frequency is called the modal class. For class 6 you will mainly identify the modal class rather than calculate an exact modal value. Note that mode does not use all data; it depends only on the highest frequency. This makes mode less affected by extreme values but also means it may ignore the overall shape of the data.

Applications: Mode is helpful to find the most common size, colour or preference. For example, a shopkeeper may check the modal shoe size to order stock. In classroom problems always show the frequency counts that lead to the mode and state if there are multiple modes. Writing the mode with units or category name makes answers clear and complete.

Median
The median is the middle value of an ordered list of numbers. It divides the data so that half of the observations are below it and half are above it. Finding the median first requires arranging the numbers in order, from smallest to largest (or largest to smallest). This order is essential because the middle depends on the positions of values, not on their original order.

How to find the median: if the number of observations n is odd, the median is the value at position (n+1)/2 in the ordered list. If n is even, the median is the average of the values at positions n/2 and n/2 + 1. For example, with five numbers the median is the third value; with six numbers the median is the average of the third and fourth values.

The median is useful because it is not affected by very large or very small outliers. For example, in incomes or marks, a few extreme values can change the mean but the median stays central to the bulk of the data. For grouped data, you can say which class interval is the median class by finding the cumulative frequencies and locating the middle position; exact calculation for grouped median is taught later.

When answering exam questions, show the ordered list and the position calculation or the two middle values used to compute the median. Write the numerical result clearly and include units when appropriate. Practice with odd and even sets builds confidence and helps choose the correct method when numbers must be ordered first.

Mean (Arithmetic mean)
The mean or average is a number that represents the centre of a data set. It is found by adding all observation values and dividing by the number of observations. The mean uses every value, so it reflects the contribution of all data points. When you want a single number to describe typical performance, such as average marks, the mean is useful.

Steps to calculate mean: write down all observations clearly, add them to find the total sum, count how many numbers there are (n), and divide the sum by n. If the result is not a whole number, you may leave it as a fraction or round it as needed for the problem. For example, three numbers 5, 8 and 10 have sum 23, mean = 23/3 ≈ 7.67.

Remarks about mean: unlike median, the mean is affected by very large or very small values (outliers). For example, if most students score around 60 but one student scores 100, the mean will increase more than the median. For this reason, compare mean with median to understand data spread. In grouped data, mean can be estimated using class mid-points and frequencies; this is taught in higher classes.

When presenting answers in exams, show addition clearly and the division step. Include units if the data measure length, marks or money. Practise with small sets of numbers and check calculations by multiplying the mean by n to see if you get back the total sum. This check helps find arithmetic mistakes before final submission.

From data to answers
After collecting and organising data, the main purpose is to answer questions. Typical tasks include finding totals, comparisons, averages and percentages, and drawing conclusions. Read the title and labels of the table or graph first. Then identify which values you need and perform the correct arithmetic operations—addition for totals, subtraction for differences, and division for averages or percentages. Always write steps so the method is clear.

Example questions: Ask which category is the largest or smallest, how many more items one category has than another, and what is the total or average. When you use pictographs, remember to multiply symbol counts by the scale. For bar graphs read heights using the scale on the axis. When comparing two categories from a double bar graph, subtract the smaller frequency from the larger and state the units such as 'students' or 'books'.

Basic probability
Probability helps describe how likely an event is to happen. In simple class problems we use terms: certain (will happen), impossible (cannot happen), likely, unlikely and equally likely. For experiments with equally likely outcomes, theoretical probability is the number of favourable outcomes divided by total possible outcomes. For example, a fair six-sided die has six equally likely outcomes; probability of getting 4 is 1/6. A coin has two outcomes; probability of Head = 1/2.

Experimental probability is found by performing the experiment many times and recording how often the event occurs: experimental probability = number of times event occurs ÷ total trials. With more trials the experimental probability usually becomes closer to the theoretical probability. When answering probability questions, list possible outcomes clearly (for example, {1,2,3,4,5,6} for a die) and count favourable outcomes carefully. Use fractions and, when asked, simplify or convert to percentages. Combining data handling and probability helps you make reasonable predictions from observations and understand chance in simple everyday events.