Chapter 6 — Data Handling

What is data? Data means pieces of information collected about events, objects or people. In mathematics we often work with numerical data that can be counted or measured. Data is the starting point for answering questions like "How many students like cricket?" or "What is the average height in a class?" Collecting and organising data properly helps us to use it for comparison and decision making.

Types of data

• Qualitative (categorical) data: These describe qualities or categories such as colours, names, types of fruit or favourite subjects. They are not numbers we can add or average; instead we count how many items fall into each category.
• Quantitative (numerical) data: These are numerical values and divide into two kinds. Discrete data are countable items like number of siblings or books. Continuous data are measurements that can take any value within a range, such as height, weight or time.

Why classify? Classification helps choose the correct method of display and analysis. For categorical data we use bar charts or pie charts. For numerical data we may list values, make frequency tables or group continuous values into class intervals. Correct classification ensures we apply appropriate formulas and graphs later.

Sources of data include observation, survey, experiment and measurement. Observational data are recorded by watching; surveys use questionnaires; experiments follow controlled steps; measurements use instruments. Each source affects accuracy and possible bias, so note how data were obtained when you interpret results.

Recording and organising begins with simple lists and tallies. Tally marks let you count quickly on the spot and later convert counts into a frequency table. For larger datasets we group similar values into class intervals, ensuring intervals are non-overlapping and cover all values. Proper recording includes noting units, sample size and date of collection.

Simple practical advice: Before collecting, decide what you want to know and choose whether to record categories or numbers. Use clear labels and consistent units. Check for missing or obvious wrong values and correct them if possible. Good initial organisation saves time when you make graphs or compute averages.

Tally marks are a fast, reliable way to count occurrences when collecting data. Each observation is recorded as a short vertical stroke. For convenience and to reduce counting errors, tally marks are grouped in fives: four vertical strokes and the fifth drawn as a diagonal across them. This visual group makes it easy to count at a glance and to convert to numerical frequency later.

Frequency is the count of how many times a particular value or category appears. After using tallies to record data, convert each group of tallies into a numeric frequency. Always check that the total of all frequencies equals the number of observations you started with.

Frequency distribution organises the raw data into an ordered table showing values or class intervals and corresponding frequencies. For few distinct values (e.g., shoe sizes 2,3,4), list each value with its frequency. For many different numerical values, group them into class intervals (e.g., 10–19, 20–29) and record frequency for each interval. Grouped frequency distributions summarise large sets of data and make patterns such as clusters or gaps visible quickly.

Choosing class intervals requires care. Intervals should be mutually exclusive and exhaustive: they must not overlap, and together they must cover all possible observations. For continuous measurement data, use class boundaries so that adjacent classes meet without gaps. Prefer equal class widths to simplify comparison and plotting; unequal widths need adjustment using frequency density for histograms.

Presenting the table usually needs clear column headings: Class (or Value), Tally, Frequency. Include a total row showing the sum of frequencies. This helps detect errors. If you start with raw data, first sort or scan through it while making tallies; then transfer counts to the frequency table for analysis and graphing. Use simple checks: recount tallies and ensure Σf equals the number of observations.

Tips for accuracy: Use a pencil for tallies so you can correct mistakes before finalising. When grouping, avoid too many classes — choose a number that reveals patterns but keeps counts meaningful. If classes are wide, consider finer grouping for detailed study. Always label intervals clearly and include class boundaries when necessary.

Bar graphs are one of the most common ways to represent categorical data. Each category (for example, fruit types) is shown on the horizontal axis and a vertical bar is drawn whose height equals the frequency of that category. Bars should be of equal width and equally spaced, unless a horizontal orientation is chosen. The vertical axis must have a clear scale starting from zero so the bar heights reflect true differences.

Construction rules for bar graphs include labelling both axes, choosing an appropriate scale, leaving a small gap between bars to show that categories are distinct (for discrete categories), and using colours or shading to increase clarity if many categories exist. Add a title that explains what data are shown and include units if required.

Pie charts are circular graphs useful to show how a whole is divided among parts. They are best when you want to emphasise proportions rather than exact counts. To draw a pie chart first compute the fraction of the total for each category by dividing the category frequency by the total frequency. Then convert this fraction to degrees by multiplying by 360°. Use a protractor to measure each central angle from the centre of the circle and draw the sectors. Shade or colour each sector and provide a legend or labels with percentages for clarity.

When to use each: Use bar graphs when exact comparisons of categories are required and categories are independent. Use pie charts when you want to show each category’s share of a complete whole; pie charts are less useful when there are many small categories or when exact comparison between two similar slices is needed.

Common mistakes to avoid include failing to start the frequency axis at zero (which exaggerates differences), using 3D effects that distort area, or drawing pie sectors without correct angles. Always check that pie sectors add to 360° and bar heights match the underlying frequency table. Be precise with measurements and show working if angles are calculated for a pie chart so the grader can follow your steps.

Histograms are used for grouped continuous data and show how data are distributed across class intervals. Unlike bar graphs for categorical data, histogram bars touch because class intervals are continuous. Each bar is drawn over the class boundaries on the horizontal axis and has a height equal to the frequency (or frequency density when class widths differ). The area of a bar, not just its height, represents the amount of data in that interval when widths vary.

Steps to construct a histogram include selecting class intervals and their boundaries, choosing a vertical scale for frequency or frequency density, drawing the horizontal axis to show class boundaries, and drawing adjoining rectangles whose bases match class widths and heights represent frequencies. If class widths are equal, the bar heights can directly equal frequencies. If widths differ, calculate frequency density = frequency / class width and plot heights using this density so areas reflect frequencies correctly.

Frequency polygons are useful alternatives to histograms when you want to compare distributions or show the overall shape more smoothly. To make a frequency polygon, mark the class mid-points on the horizontal axis and plot points whose vertical coordinates are the class frequencies. Join these points with straight lines. For a neat polygon, extend the curve to the horizontal axis at both ends by adding classes with zero frequency at left and right.

Interpreting shapes of histograms and polygons helps understand data nature. A symmetric shape around a central peak suggests a balanced distribution; a long tail to the right indicates positive skewness; multiple peaks suggest distinct groups. Histograms also reveal gaps, clusters and outliers. When comparing two datasets, overlay frequency polygons on the same axes with different line styles and a legend for clarity.

Careful details include using correct class boundaries to avoid overlapping, labelling axes and units, and ensuring totals computed from the histogram match the frequency table. Practice by drawing histograms from sample frequency tables and by converting histograms back to frequency tables to check understanding.

Ungrouped data are individual observations listed one by one. These data allow straightforward calculation of mean, median and mode because each value is known. Learning how and when to use each measure of central tendency is important for clear description of data.

Mean (Arithmetic mean) is calculated by adding all observed values and dividing by the number of observations. It is useful when values are roughly symmetric and no extreme outliers distort the average. The mean uses every value in the dataset and is therefore sensitive to very large or very small numbers.

Median is the centre value after arranging observations in ascending order. If the number of observations (n) is odd, the median is the middle term at position (n+1)/2. If n is even, the median is the average of the two middle values at positions n/2 and n/2 + 1. Median is robust against outliers and gives a better 'typical' value for skewed distributions.

Mode is the most frequently occurring value in the dataset. There may be one mode, more than one (bimodal, multimodal), or none if all values occur only once. Mode is especially useful for categorical data where mean and median do not apply, for example most common shoe size or preferred subject.

Practical steps and checks: To compute these measures, first sort the data. For mean use Σx/n. For median identify middle position(s) and compute accordingly. For mode count frequencies of values. Always state the sample size and check calculations by recomputing sums or counting frequencies. When reporting results, comment on which measure best represents the data — for example, use median for income data that is skewed by a few high earners.

Grouped data

Procedure for the direct method begins by writing a frequency table with class intervals and corresponding frequencies. For each class calculate the class mid-point x = (lower limit + upper limit) / 2. Multiply each mid-point by the class frequency to get fx for that class. Sum all frequencies Σf and all products Σ(fx). Finally compute the mean by dividing Σ(fx) by Σf: Mean = Σ(fx) / Σf. Present the arithmetic in a neat table to avoid errors.

Why mid-points? Mid-points serve as representative values of every observation in a class. This assumes values are evenly spread inside the class which is usually acceptable for moderate sized samples. If classes are narrow, the estimate is close to the true mean. For wide classes the estimate is cruder, so choose suitable class widths during grouping.

Step-by-step example: Suppose classes are 10–19, 20–29 and 30–39 with frequencies 5, 8 and 7. Mid-points are 14.5, 24.5 and 34.5. Compute fx: 5×14.5, 8×24.5 and 7×34.5, then Σf=20 and Σ(fx) is the sum of these products. Mean = Σ(fx)/20. Show arithmetic and reduce fractions or decimals sensibly. Check that Σf equals the number of observations recorded in raw data.

Dealing with special cases: If class widths are unequal, the mid-point method still gives a weighted average but the visual area in a histogram must use frequency density. For open-ended classes (like 60+), the mean cannot be calculated accurately without additional assumptions. If required in an exam, state assumptions. Always report the mean to a reasonable number of decimal places and explain rounding.

Short-cut (Assumed Mean) Method is an efficient way to compute the mean of grouped data when mid-points are large or when manual arithmetic needs simplification. The method reduces the size of numbers by choosing an assumed mean A, typically a mid-point near the centre of the data, and using deviations from A.

Steps to apply the method are as follows: first list classes and their frequencies and compute the mid-point x for each class. Choose an assumed mean A, often one of the mid-points. Calculate the deviation d = x − A for each class. Multiply each deviation by its class frequency to get fd. Sum all fd values to get Σfd and sum all frequencies to get Σf. Finally compute the mean as Mean = A + (Σfd / Σf). The final result is the same as the direct method but fewer long multiplications are needed because d values are smaller.

Sign and accuracy: Deviations d can be negative or positive; Σfd may therefore be negative. Ensure signs are preserved in arithmetic. Depending on the size of Σfd and Σf, you may need to give the mean to one or two decimal places. Check results by computing the direct mean if time permits.

Presenting work makes checking easier: prepare a table with columns Class, f, x, d, and fd. Write down A clearly. After summing Σfd and Σf, show substitution into the formula. The method is particularly useful in examinations to save time and reduce calculation errors.

Limits: As before, if classes are open ended or widths differ significantly, caution is needed. The method assumes mid-points represent class values adequately.

Median for grouped data

Step-by-step median: Start with a grouped frequency table and compute cumulative frequencies (cf) up to each class. Let N = Σf be the total frequency. Find N/2 and then identify the median class where cf becomes equal to or exceeds N/2. If the median class is the one whose upper cumulative frequency reaches or passes N/2, denote its lower class boundary by L, the cumulative frequency before this class by cf_prev, this class frequency by f_med and the class width by h. Apply interpolation using the formula Median = L + [(N/2 − cf_prev) / f_med] × h. This gives an estimate of the value at which half the observations lie below and half above.

Example explanation: If N is even and exact equality occurs at N/2, the median class contains the middle observations and the interpolation still applies; if counts are small, the median can be sensitive to the grouping choice. Always use class boundaries (continuous form) rather than discrete class limits when applying formulas.

Mode for grouped data

Practical advice and checks: Always list classes with true continuous boundaries. Check that f0, f1 and f2 refer to adjacent classes (if modal class is first or last, the formula cannot be used without additional assumptions). If two classes tie for highest frequency, the distribution may be bimodal and report both modes. Show full working and round answers sensibly in exams.

Range

Understanding limitations: Range depends only on two values and ignores how the rest of the data are distributed. Therefore it is sensitive to outliers: a single extreme value can greatly increase the range and give a misleading impression of variability. Because of this limitation, range is usually reported alongside measures of central tendency such as mean and median to give a fuller picture.

Other dispersion ideas (conceptual): Interquartile range (IQR), variance and standard deviation provide more robust or precise measures but are introduced later. At this level, learn to use range and visual displays to judge spread. A histogram with a wide base indicates greater spread; a narrow peak indicates clustering. Box plots (conceptually) show minimum, lower quartile, median, upper quartile and maximum, giving a quick visual of spread and outliers.

Comparing distributions: Two datasets may have the same mean but different ranges. For example, two classes might both average 60 marks; one with range 10 is consistent, while another with range 40 is more varied. State both mean and range when comparing and mention any outliers. If outliers exist, median may better represent the centre.

Practical use and reporting: When computing range, list the minimum and maximum explicitly and show subtraction. If data are grouped, show which class boundaries you used. Always comment on whether the range might be affected by an outlier and whether the sample size is large enough for a reliable judgement. Encourage students to look at frequency tables or graphs along with the range to explain why spread matters in context.

Cumulative frequency

Preparing cumulative frequency: Start with a grouped frequency table. For less-than cumulative frequency, add the frequencies class by class beginning from the lowest class; the cumulative frequency for a class is the sum of all frequencies up to and including that class. For greater-than cumulative frequency, start from the highest class and add backward. The final cumulative frequency should equal the total number of observations N. It is important to decide whether to use upper class boundaries (for less-than ogive) or lower class boundaries (for greater-than ogive) before plotting.

Ogive (cumulative frequency curve) is a graph of cumulative frequency against class boundaries. To draw a less-than ogive, plot points at the upper class boundaries on the horizontal axis and their corresponding cumulative frequencies on the vertical axis. Join these points with straight lines or a smooth curve. The curve should begin at zero at the lower boundary before the first class and end at N at the upper boundary of the final class. For greater-than ogive, plot cumulative frequencies against lower class boundaries and the curve will slope downwards.

Using ogives to find median and percentiles: An ogive makes it easy to read off medians and quartiles graphically. To find the median, draw a horizontal line at N/2 on the vertical axis, find where it meets the ogive and drop a vertical line to the horizontal axis to read the median value. For quartiles, use N/4 and 3N/4 similarly. This graphical method is useful when a precise formula is not required or when the data are visualised for a quick estimate.

Careful plotting and interpretation: Always use class boundaries, label axes and include units. When estimating values from an ogive, be aware of interpolation error: the read value is an estimate. For comparisons, plot two ogives on the same graph to see which distribution tends to have higher or lower cumulative frequencies at given points. Practice drawing ogives from frequency tables and using them to estimate medians and quartiles to build confidence.

Probability

Sample space, events and outcomes: The sample space S is the set of all possible outcomes of an experiment. An event is any subset of S. For example, when rolling a fair six-sided die, S = {1,2,3,4,5,6}; the event "rolling an even number" is {2,4,6}. Listing the sample space carefully helps to count favourable outcomes correctly.

Experimental (empirical) probability is found by performing a random experiment repeatedly and observing how often the event occurs. If an event E occurs e times in n trials, the experimental probability is P(E) ≈ e/n. This method is practical when theoretical calculation is not possible or when you want to test whether a physical object (like a coin) is fair. Record each trial using tallies to avoid mistakes, and report results as fraction, decimal or percentage.

Law of large numbers (intuitive): Explain that as the number of trials n increases, the experimental probability usually becomes closer to the theoretical probability for fair experiments. Small samples can give quite different relative frequencies due to chance or biased devices. Discuss reasons why experiments might deviate from theory: biased coin, uneven surface, poor mixing of balls, or recording errors.

Practical experiment design: Choose a simple experiment such as tossing a coin, rolling a die or drawing a ball from a bag. Decide the number of trials and how you will record outcomes. After conducting trials, compute relative frequency and discuss whether the result supports theoretical expectation. Always state the number of trials n alongside the probability estimate, and comment on possible sources of error and how increasing n may improve the estimate.

Theoretical probability

Counting outcomes carefully is essential. For a single fair die, total outcomes are 6. For a fair coin, total outcomes are 2. When experiments have more steps, list the combined sample space using pairs or a tree diagram. For example, tossing two coins gives S = {(H,H),(H,T),(T,H),(T,T)} with four equally likely outcomes; for two dice use 36 ordered pairs.

Compound events and independence: When outcomes result from independent steps, count combinations by multiplying possibilities. For independent events A and B, the probability of both occurring is P(A and B) = P(A) × P(B). For example, probability of getting heads on two independent coin tosses is 1/2 × 1/2 = 1/4. If events are not independent, such as drawing cards without replacement, adjust counts because the sample space changes after each draw.

Complementary events provide a handy shortcut: P(not E) = 1 − P(E). Use this when it is easier to count outcomes where E does not occur. For instance, the probability of rolling at most 5 on a die is 1 − P(6) = 1 − 1/6 = 5/6.

Link with experiments: Compare theoretical results with experimental probabilities to help students see both concepts. Small experiments may differ from theory due to chance or bias; larger trials often bring empirical results closer to theoretical values. Encourage drawing sample space diagrams, simplifying fractions and giving answers as fractions, decimals or percentages as required by the question.

Combining data

Calculations after combining include computing overall Σf, Σfx or other summaries from the combined table. For means, calculate mid-points and Σ(fx) using added frequencies; for medians compute cumulative frequencies on the combined table. Always cross-check totals against the sum of individuals to avoid arithmetic mistakes.

Comparing distributions requires measures of central tendency and spread. Compute mean, median, mode and range for each group and compare. Use histograms or frequency polygons drawn on the same axes to compare shapes visually; frequency polygons are particularly convenient because two or more distributions can be drawn together with different line styles and a legend. Look for differences in central position (which group is higher on average), spread (which group is more variable) and shape (skewness or multiple peaks).

Interpretation should be contextual. For example, if one class has a higher mean but larger range, comment on both average performance and consistency. Beware misleading impressions caused by different scales or truncated axes when comparing graphs. When combining data from groups with different sample sizes, weighted measures (using frequencies) matter; the combined mean reflects the sizes of the contributing groups.

Practical example: To compare two sections of a school, compute each section’s mean and range and then the combined histogram. Discuss which section performed better on average and which showed more consistent results, backing conclusions with numbers and graphs.

Why critical interpretation matters: Data displays can be accurate but still give a false impression if presented poorly. Learning to spot misleading elements in graphs and tables is as important as learning to draw them. Critical reading prevents wrong conclusions from reports, newspapers or charts seen online.

Common misleading practices include truncating axes so small differences appear large, using unequal scales on comparison graphs, employing 3D effects that distort areas, or choosing class widths that hide variation. Pie charts with too many tiny slices are hard to read; bar charts with inconsistent bar widths or missing labels mislead. Histograms using unequal class widths without adjusting for frequency density give false comparisons of class sizes.

How to check a display: Always read the title, axis labels, units and legend first. Check where the axes start and the scale intervals; compute actual values from the underlying data table if available. For pie charts, add up the sector angles or percentages to confirm they sum to 360° or 100%. For histograms, ensure class boundaries and widths are correct and that the areas (or heights if widths equal) represent frequencies accurately. For comparative graphs, make sure both datasets use the same scale and units; otherwise comparisons are invalid.

Examples to test: Given two bar charts claiming one category is much larger, students should recalculate percentages from raw counts or check if axes are truncated. Given a histogram with different class widths, students should compute frequency density to compare bars correctly. Teach students to ask: Who collected the data? Was the sample size adequate? Are there missing data or outliers?

Presenting fair graphs: When drawing your own graphs, label axes, start scales properly (often at zero for counts), include a title and legend, and choose appropriate class widths. If you must use a truncated axis for clarity, state it clearly on the graph. Good presentation builds trust and helps accurate interpretation.