Chapter 6 — Data Handling

Definition and everyday examples. Data are facts or items of information collected to answer questions. In school, data might be students' marks, heights, favourite subjects or shoe colours. Outside school, data appear in weather reports, market prices and sports scores. Understanding what data are helps us make sense of facts instead of just remembering isolated numbers.

Importance. Data let us compare, decide and explain. For example, knowing the number of students who prefer a subject helps plan lessons; knowing daily temperatures helps dress appropriately. Learning to handle data prepares you to read newspapers, charts and results with confidence and to present your own findings clearly.

From raw facts to information. Raw data are like puzzle pieces. To get a picture you must organise and summarise them. Without arrangement a list of numbers is hard to understand. Steps include collecting data carefully, recording them accurately, counting how many times each value appears, and then making tables and graphs that show patterns.

Questions we can answer using data. Examples include: Which value is most common? What is a typical value? How spread out are the values? Are there unusual values (outliers)? These questions lead to different summaries: mode shows the most common; mean and median give a central value; range shows spread.

Practical classroom work. In Class 7 you will practise small surveys, tally marks, frequency tables and simple charts. These activities teach careful observation, neat recording and clear presentation. Good data handling also requires thinking about who was asked and how results might change with different groups — this is the start of thinking about reliability and representation.

Broad classification. Data are commonly classified as qualitative (categorical) or quantitative (numerical). Qualitative data describe qualities or categories such as eye colour, city names, or vehicle types. Quantitative data are numbers that measure amount or quantity, for example marks, heights, weights and counts.

Sub-types of quantitative data. Quantitative data are of two kinds: discrete and continuous. Discrete data take separate distinct values such as the number of brothers (0,1,2...). Continuous data can take any value within a range and are obtained by measurement, such as height measured in centimetres or time measured in seconds. The difference affects how we record and display the data.

Nominal and ordinal (for categorical data). Nominal data are categories without order, e.g. colours or names. Ordinal data are categories with a natural order, for example ratings like good, better, best or class ranks. The choice between nominal and ordinal affects the types of summary we can use: for ordinal data median-like ideas are meaningful but for nominal data only mode and frequencies make sense.

Primary vs secondary data. Primary data are collected directly by the student or researcher for a specific purpose, like a class survey. Secondary data are already available from other sources such as books, websites or reports. Primary data allow control over questions and methods, while secondary data save time but may require checks for accuracy.

Why knowing the type matters. The type of data tells you which graphs and measures are appropriate: qualitative data suit bar charts or pie charts; discrete quantitative data can use bar charts or dot plots; continuous data often need grouping and are shown with histograms or frequency polygons. Always identify the type before deciding how to handle the data.

Start with a clear question. Effective data collection begins by deciding exactly what to measure and why. A clear question guides how you collect results. For example, 'How many books did each student read last month?' is clearer than 'Do you read often?'. Precise questions give useful answers that can be counted or measured.

Design of simple surveys. Choose who to ask (sample), how many people, and whether to ask all members of a class or a subset. Keep questions short and unambiguous. Use tick-box options for categorical answers and numerical boxes for counts or measurements. Explain the purpose politely and ensure students answer honestly.

Direct measurement. For quantities like length, weight or time, use suitable measuring tools and record units (cm, kg, s). Practice reading instruments and noting values correctly. When measuring people or objects, be consistent: measure everyone in the same way and in the same units to avoid errors.

Tallying while collecting. While asking many people, use tally marks to record each response immediately. This prevents forgetting or miscounting. Tallies grouped in fives (four vertical and the fifth across) make later counting faster and more reliable.

Sample size and sampling. A larger and more varied sample gives more reliable results, but class exercises often use the whole class or a representative group. Avoid only asking friends as this produces biased results. If time or resources limit you, note the sample used so results are not wrongly generalised.

Check and clean data. After collection, check for impossible or inconsistent answers (like negative ages). Ask again or discard clearly wrong entries. Keep a record of how data were obtained — date, place, number of respondents — because this helps interpretation and reporting later.

Using tally marks correctly. Tally marks give a quick way to count responses as you collect them. Each occurrence is one vertical stroke. After four strokes, the fifth is drawn diagonally or across, making a bundle of five. This visual bundling reduces counting errors and speeds up conversion to numeric frequencies later.

From tallies to frequency tables. After collecting tallies, convert them into a frequency table. A simple frequency table has two columns: the observation (value or category) and the frequency (count). For categorical data list every category even if its frequency is zero; for numerical ungrouped data list each distinct number in increasing order.

Adding totals and checks. Always include the total frequency at the bottom of the table. Verify that the number of tally marks matches the total. This check helps catch miscounts. A correct total also gives the denominator needed to compute percentages or fractions for pie charts or mean calculations.

Ordering and labelling. Organise the table logically: numerical values in increasing order or categories in a sensible sequence. Write clear column headings, such as 'Marks' and 'Frequency', and include units where needed. Neat presentation helps in drawing graphs and in answering examination questions.

Using the table for summary measures. Frequency tables are the base for calculating mean, median, mode, range and for drawing graphs. For grouped data the table will use class intervals as values. For ungrouped data the table lists distinct values. Keep raw data safely so you can re-check results if needed.

Recognising discrete data. Discrete data take separate values that are countable: 0, 1, 2, ... Examples are number of goals, number of siblings, or number of books. Discrete variables are often displayed with bar charts or dot plots where each value is a separate category and bars or dots do not touch.

Recognising continuous data. Continuous data result from measurement and can take any value within an interval, for example heights, weights, time to finish a race. Continuous data are best shown when grouped into class intervals because exact values may not be meaningful or many decimal places are impractical.

Why grouping is used. When continuous measurements vary across a range it becomes difficult to list each separate value. Grouping values into class intervals such as 120–124, 125–129 reduces complexity and reveals distribution shape. Choose equal class widths for simplicity and comparability when drawing histograms.

Choosing graph types. Use bar charts for discrete data (bars separated) and histograms for grouped continuous data (bars adjacent). The adjacency in histograms shows continuity between neighbouring intervals. Frequency polygons and ogives are further useful for comparing groups and finding medians or percentiles for grouped continuous datasets.

Practical tips and common confusions. Be careful when data are rounded: decide whether they are best treated as discrete or continuous. For example, exam marks recorded as whole numbers may be treated as discrete for simplicity, but if measurements are given to decimals treat them as continuous. Always explain your choice in answers where required.

Mean (arithmetic mean). The mean is obtained by adding all observations and dividing by their number. It uses every value and gives an average that balances values above and below it. Calculate carefully: add the list, check the total, then divide by the count. State units in the answer, and round sensibly.

When mean is suitable. Mean works best for numerical data without extreme outliers. If one or two very large or small values exist, the mean may be pulled towards the extremes and not represent a typical value.

Median (middle value). To find the median, first order the data from smallest to largest. If the number of observations is odd, the median is the middle value. If even, median is the average of the two middle values. The median resists the effect of outliers and is useful for skewed distributions or ordinal data.

Mode (most frequent value). The mode is the value that occurs most often. A dataset may have no mode (if all values differ), one mode, or many modes (multimodal). Mode is the only measure applicable for nominal categorical data—for example most common eye colour.

Comparing the three measures. Mean uses all data and is affected by extremes. Median shows the central position and resists extremes. Mode shows the most common choice. Use the measure appropriate to the question and explain any differences between them when required.

Worked practice and presentation. Show all steps: order values for median, show sum and division for mean, and a frequency count for mode. This clarity helps check calculations and is important in examinations.

Range — definition and calculation. The range is the simplest measure of spread: it equals the maximum value minus the minimum value in the dataset. Range gives a quick idea of how widely values are spread but depends only on two values and ignores the rest.

Interpretation and uses. Use range to compare variability of two small datasets. A larger range usually means the data are more spread out. However, because range is sensitive to outliers, mention if unusually large or small values affect the result.

Limitations and further ideas. Range can be misleading for skewed data. Two datasets with very different internal distributions may share the same range. At later stages you will learn variance and standard deviation that use all data points; for Class 7, understanding range plus mean and median gives a basic descriptive picture.

Range with grouped data. For grouped data approximate range using class boundaries of the first and last classes (for example 10–14 to 30–34 approximates range as 34 - 10 = 24) but remember it is an approximation because exact extremes are unknown.

Explaining results in answers. When asked to compare two groups say which has greater range and whether outliers may explain the difference. Combining range with mean or median provides a fuller answer: for instance a high mean with small range indicates values clustered above average; a large range with similar means indicates more variation.

Why group data? When many observations span a wide range it is impractical to list every value. Grouping values into class intervals simplifies the data and reveals patterns of distribution. Grouped frequency tables use ranges (classes) and a frequency for each class.

Choosing class intervals. Choose equal-width classes when possible so comparisons are straightforward. Decide class width so there are neither too many nor too few classes — typically 5 to 10 classes for moderate datasets. Classes must not overlap, and boundaries should be clear: for whole-number data you might use 10–14, 15–19; for measured data with decimals use 10.0–14.9, 15.0–19.9 to avoid confusion.

Constructing a grouped table. Create columns for class interval and frequency, and optionally cumulative frequency. Assign each observation to the correct class using the class limits. After tallying, write the numeric frequency for each class and check that the total equals the number of observations collected.

Estimating mean from grouped data. Exact values are not available for each observation, so estimate the mean by assuming all observations in a class lie at the class midpoint (class mark). Calculate midpoint as (lower limit + upper limit)/2. Multiply each midpoint by its class frequency to get fx. Sum all fx values and divide by total frequency n to get the estimated mean. This gives an approximate central value which is usually acceptable for grouped data used in class tests and exams.

Work carefully and present. Show the table with classes, midpoints and fx column, include the totals for fx and frequency, and write the final division clearly. State that the mean is an estimate and round the answer sensibly to required decimal places with units.

Finding median class using cumulative frequency. For grouped data the exact middle observation is not known, but we can locate the class containing the median by using cumulative frequencies. Compute cumulative frequency by adding class frequencies from the first class up to each class. Find n/2 where n is the total frequency and identify the class whose cumulative frequency first equals or exceeds n/2; that class contains the median.

Estimating the median value. Teachers may ask for the median class only, but to estimate the median value use linear interpolation: median = L + [(n/2 - cf) / f] × h, where L is the lower boundary of the median class, cf is cumulative frequency before the median class, f is the frequency of the median class and h is the class width. This formula assumes values are uniformly distributed within the class and gives a reasonable estimate.

Modal class and estimating mode. The modal class is the class with highest frequency. It indicates the interval where most observations lie. A more precise estimate of the mode for grouped data can be found by the formula mode ≈ L + [(f1 - f0) / (2f1 - f0 - f2)] × h where f1 is frequency of modal class, f0 and f2 are frequencies of previous and next classes respectively, L is lower boundary and h is class width. For Class 7 it is acceptable often to identify the modal class and explain why it represents the most common values.

Using an ogive to find median and percentiles. An ogive is a cumulative frequency curve plotted against class boundaries. To find median using an ogive draw a horizontal line at n/2 on the y-axis, read where it meets the curve and drop down to the x-axis for the median estimate. Similarly other percentiles can be read using the corresponding cumulative frequency.

Presentation and comments. State clearly when values are estimates and show cumulative frequency work. Mention if class widths are unequal as this affects interpretation and interpolation accuracy.

Bar graphs for categorical or discrete data. Bar graphs display categories on one axis and frequency on the other. Each category has a bar whose height equals its frequency. Bars are separated by gaps to show categories are distinct. Label axes, choose an appropriate scale so the bars fit the paper, and give a clear title. A double bar graph places two bars side by side for each category to compare two groups (for example boys and girls). Use a legend and different colours or patterns for clarity.

Histograms for grouped continuous data. Histograms are used when data are continuous and grouped into class intervals. Draw adjacent bars because classes touch; each bar’s base equals the class interval and its height equals the frequency. If class widths are equal you can compare heights directly; if widths differ you should use frequency density (frequency ÷ width) on the vertical axis. Always mark class boundaries and ensure no gaps between bars.

Frequency polygons. A frequency polygon is drawn by marking midpoints of each class interval at heights equal to class frequencies and joining these points with straight lines. Close the polygon to the horizontal axis at both ends by adding points at the midpoints of imaginary end classes with frequency zero. Frequency polygons are useful to compare two distributions on the same graph and to visualise the shape of the distribution smoothly.

Choosing the right graph and good practice. Use bar charts for discrete or categorical variables and histograms for continuous grouped variables. Include axis labels, units, scales starting from zero on the frequency axis, titles and legends. For exam work neatness, correct scaling and labelling are marked. When comparing groups use double bar graphs or overlaying frequency polygons to make contrasts clear.

Pie charts — parts of a whole. A pie chart represents the whole dataset as a circle. Each category occupies a sector whose central angle is proportional to its frequency. To draw a pie chart compute each angle as (frequency / total) × 360°. Use a protractor to measure each sector from a common starting line, shade or colour sectors distinctly, and include a legend or labels showing category names.

Using percentages. Converting frequencies to percentages is often helpful: Percentage = (frequency / total) × 100%. Percentages are easier to understand quickly. Ensure that the sum of angles equals 360° and the sum of percentages equals 100%. For many small categories combine very small shares into an 'Others' category for clarity.

Ogive (cumulative frequency curve). An ogive plots cumulative frequency against the upper (or lower) class boundaries. To construct, make a cumulative frequency table, then plot points at each class boundary against cumulative frequency and join them with straight lines or a smooth curve. The ogive is used to find medians and percentiles: draw a horizontal line at n/2 for median, find the intersection with the curve and read the corresponding x-value on the horizontal axis.

Comparing and interpreting. Use pie charts to show share distribution when categories form parts of a whole; use ogives when you need to read medians, quartiles or percentiles for grouped data. Always label axes, provide units and state if values are estimated. Mention any assumptions (for example linear distribution within a class when reading from ogive) when writing answers in exams.