Chapter 3 — Data Handling

What is data? Data are facts or information collected for analysis. Examples: scores of students, daily temperatures, number of cars passing a junction.

Why handle data? To organize information so we can recognise patterns, make comparisons, and draw conclusions for decision making.

Basic steps in data handling

• Collect data (survey, observation, experiment).
• Record data (lists, tally marks).
• Organise data (frequency table, class intervals for grouped data).
• Represent data visually (pictograph, bar graph, histogram, pie chart).
• Interpret results (look for trends, highest/lowest, averages).

Types of data

• Qualitative (categorical): names or categories (favourite colour, type of fruit).
• Quantitative (numerical): numbers (marks, heights). Quantitative data can be discrete (countable values like 0,1,2...) or continuous (measured values like 12.4 cm).

Organising data

Small sets of numerical data can be listed and converted into a frequency table using tally marks. For large numerical data, use class intervals (grouped data) and record frequency of each class.

Example of a simple frequency table (marks out of 10)

How visual representation helps: Pictures and graphs summarise data so we can easily compare categories, see trends, and estimate proportions.

What is Recording Data?

Recording data means writing down the observations you collect in an organised way so they can be understood, analysed and presented. Good recording converts raw observations into tables or charts that show how often each value or category occurs.

Steps to record data

1. Decide the question and what you will measure (the variable).
2. Choose appropriate units (e.g., number of students, kg, °C).
3. Collect observations carefully (survey, measurement, experiment).
4. Use tally marks to count repeated observations as you collect them.
5. Make a frequency table from tallies (list each category/value and its frequency).
6. Use graphs (bar graph, pictograph, histogram, pie chart) to present the frequency table visually.
7. Analyse the recorded data (look for patterns, highest/lowest frequency, trends).

Types of data relevant while recording

• Qualitative / Categorical: Names or categories (e.g., favourite colour, type of pet). Recording uses categories and frequencies.
• Quantitative: Numerical data. It can be discrete (countable values: number of books) or continuous (measurements: height, weight).

Tally marks

Tally marks group counts in fives for quick recording: |||| = 4, ||||/ = 5 (four vertical and one diagonal through them). They help avoid recounting mistakes.

Frequency table

A frequency table lists each category or interval and the number of observations (frequency). For grouped (continuous) data you create class intervals and count observations in each interval.

Important points when recording

• Use clear categories or class-intervals that do not overlap.
• Keep units consistent and note them.
• Record raw data first (in a notebook) and then make tallies and tables.
• Check totals: sum of frequencies must equal number of observations.

What is organising data? Organising data means arranging raw observations so we can see patterns easily. A frequency table is a compact way to show how often each value or range of values occurs.

Types of data and tables

• Ungrouped frequency table – used for discrete data with a small number of distinct values (eg. outcomes of a dice, shoe sizes). You list each value and its frequency (how many times it occurs).
• Grouped frequency table – used when data take many values or form a continuous range (eg. marks, ages, temperatures). Values are put into class intervals (groups) and you record the frequency for each interval.

Key parts of a frequency table

• Class interval – a group or range of values (e.g., 10–19, 20–29).
• Frequency (f) – number of observations in that class or for that value.
• Cumulative frequency – running total of frequencies up to a class.
• Relative frequency – frequency divided by total number of observations (can be given as a fraction, decimal, or percent).
• Class width – size of the interval, usually constant for all classes in a grouped table.
• Class mark (midpoint) – (lower limit + upper limit) / 2, used for some graphs like frequency polygons.

How to make a good grouped frequency table (steps)

1. Find the smallest and largest values (range = max − min).
2. Decide the number of classes (often 5 to 10 for school-level data) and choose a suitable class width so that classes cover the entire range without overlap.
3. Write non-overlapping, equal-width class intervals.
4. Count observations in each class (use tally marks for ease) and record frequencies.
5. Calculate cumulative and relative frequencies if needed.

Tips: Classes should be continuous (no gaps) and non-overlapping. For continuous data you must decide inclusive/exclusive limits (for Class 10–19, next class 20–29 etc.).

Worked example (small dataset)

Data: number of books read by 30 students in a month:

0,1,3,2,5,6,7,4,3,2,5,6,8,9,7,6,5,4,3,2,1,0,2,3,4,5,6,7,8,9

Choose classes of width 2: 0–1, 2–3, 4–5, 6–7, 8–9. Construct the grouped frequency table:

This table summarises the raw list into useful groups and makes further analysis (graphs, averages) easier.

What is a pictograph?
A pictograph (or pictogram) is a way of representing data using pictures or symbols. Each picture (or symbol) stands for a fixed number of items. Pictographs make numerical information easy to understand at a glance.

Key ideas:

• Every symbol represents a specific number of units (called the scale), shown in a legend. Example: 1 star = 5 students.
• To show a category’s value, you draw as many whole symbols as fit into the value; fractions of a symbol (like half a symbol) can be used when necessary.
• Read values by multiplying the number of symbols by the scale.

How to draw a pictograph — step by step

1. Collect and tabulate the data clearly.
2. Choose a simple, repeatable symbol (e.g., circle, star, apple).
3. Decide the scale so the number of symbols stays reasonable (not too many or too few).
4. Draw a legend: say what 1 symbol = how many units.
5. Draw the symbols for each category, label the categories and give numerical values if needed.
6. Include a title and, if possible, use color or spacing to improve readability.

Reading and interpreting
If the legend says 1 symbol = k units, and a category has n symbols, the actual value = n × k. If a category shows 2 and a half symbols and 1 symbol = 4 units, the value = 2.5 × 4 = 10 units.

Advantages: Easy to understand, visually appealing, good for small data sets and comparisons. Limitations: Not precise for large numbers or many categories; can be misleading if scale or symbols are not clear.

A bar graph is a visual way to represent categorical or discrete numerical data using rectangular bars. Each bar’s length (or height) shows the value (usually frequency or count) for a category. Bar graphs help compare values at a glance.

Key features:

• Two perpendicular axes — usually a horizontal axis (x-axis) for categories and a vertical axis (y-axis) for values or frequencies.
• Bars of equal width; space/gaps between bars to show distinct categories.
• A clear title, labels for both axes, and a scale on the value axis.
• A legend when more than one data set is shown (for example, in a double bar graph).

How to draw a bar graph (step-by-step):

1. Collect the data and make a frequency table (categories vs. values).
2. Choose which axis will show categories and which will show values.
3. Decide a suitable scale on the value axis so the largest value fits comfortably (for example, 1 cm = 5 units).
4. Draw and label the axes, mark the scale, and write the category labels evenly along the category axis.
5. Draw a bar for each category whose height (or length) corresponds to its value using the chosen scale. Keep bar widths equal and leave small gaps between bars.
6. Add a title and legend (if needed). Check labels and correctness.

Types of bar graphs and related notes:

• Vertical bar graph — bars rise vertically from the category axis (most common in textbooks).
• Horizontal bar graph — bars extend horizontally (useful when category names are long).
• Double/Grouped bar graph — compares two sets (e.g., boys vs. girls) side by side for each category; use different colours and a legend.
• Stacked bar graph — parts of a whole shown stacked in one bar per category (useful to show composition).
• Difference from histogram: histograms show continuous interval data with adjacent bars (no gaps); bar graphs show discrete/categorical data with gaps.

What is a comparative bar graph?
A comparative bar graph is a pictorial way to represent and compare two or more related sets of data by placing bars for each set side by side for each category. When two sets are compared it is called a double bar graph. When more than two sets are compared it is called a multiple bar graph.

Purpose
Comparative bar graphs make it easy to compare quantities across categories (for example, marks of two students in different subjects or sales of different products across months) and to spot trends, differences and similarities at a glance.

Main components

• Title — describes what the graph shows.
• Axes — horizontal axis (categories) and vertical axis (values).
• Scale — numerical increment on the vertical axis (choose a suitable scale so bars fit the graph).
• Bars — for each category there are two or more bars, one for each data set; bars for the same category are grouped together.
• Legend / Key — shows which colour or pattern corresponds to which data set.

How to draw a comparative bar graph (step-by-step)

1. Decide your categories (x-axis) and the quantity to measure (y-axis).
2. Choose an appropriate scale for the y-axis so the largest value fits comfortably.
3. For each category, draw side-by-side bars (same width) — one for each data set — and keep equal spacing between category groups.
4. Use different colours or patterns for each data set and include a legend.
5. Label axes, write the title, and if needed add gridlines to read values easily.

Interpreting a comparative bar graph
Compare bar heights for the same category to see which data set has a larger or smaller value. To quantify differences, subtract the heights (or values) of the bars. To compare overall patterns, look across categories for trends (e.g., which data set consistently has higher bars).

Important points / cautions

• Use the same scale on the vertical axis for all data sets so comparisons are valid.
• Bars representing the same category must be grouped together and have equal width.
• Keep colours/patterns distinct and include a clear legend.
• If units differ between data sets, convert them to the same unit before plotting.

Interpreting data and drawing conclusions means reading organised information (tables, charts, graphs) carefully, extracting meaningful facts, and making reasoned statements based on evidence. It involves noticing patterns, comparing quantities, calculating summary measures, checking for unusual values (outliers), and answering specific questions using the data.

Key steps to interpret data:

• Read the title, labels, units and scale to understand what is being measured.
• Observe the overall pattern: increasing, decreasing, steady, cyclic (repeating) or random.
• Compare categories or groups to see which is largest, smallest or equal.
• Compute simple summary measures (mean, median, mode, range) to describe center and spread.
• Look for outliers that may affect averages or suggest special causes.
• Answer the question asked using numerical evidence and state any assumptions or limitations (sample size, bias, measurement units).

When drawing conclusions be careful: correlation (two things happening together) doesn’t always mean causation (one thing caused the other). Always check if the sample is large and representative, and avoid making strong claims unsupported by data.

Practical tips:

• Use percentages or relative frequencies when group sizes differ, so comparisons are fair.
• Summarise data with a small number of statistics and a clear graph to support your statement.
• State the conclusion clearly and give the evidence (e.g., “Class A’s average score is 6 points higher than Class B’s.”).

What it means: Practical skills and presentation in Data Handling means collecting data carefully, organising it into tables (using tally marks and frequency), and presenting it clearly using suitable graphs so others can read and interpret the information easily.

Key steps:

• Plan the data collection: decide what to measure, how many observations, and the sampling method.
• Record raw data neatly (use tally marks for quick counting).
• Create a frequency table. If values are many or continuous, group them into class intervals.
• Choose the best presentation: pictograph, bar graph, double bar graph, histogram or pie chart, depending on the type of data.
• Draw the graph properly: include a clear title, labelled axes (with units), an appropriate scale, and a legend if needed.
• Check for readability: equal-width bars (for bar graphs), no misleading scales, and consistent symbols for pictographs.

Practical tips:

• Use tally marks (|||| then \/ ) to avoid counting mistakes.
• Start the vertical axis of a bar graph at zero to avoid distortion.
• Choose class intervals of equal width for histograms; list interval boundaries clearly.
• For pie charts compute each sector angle = (frequency / total) × 360°.
• Always include total frequency (sample size) when presenting percentages or relative frequencies.