Representation of data Revision Notes

Frequency tables are a fundamental way to organise data by listing each distinct value or category alongside the number of times it occurs (its frequency). This summarises large datasets into a clear format, making it easier to identify common values and overall distribution.

For example, consider the ages of 12 students: 14, 15, 14, 16, 15, 15, 14, 16, 17, 15, 14, 16. A frequency table lists each age and counts how many students are that age:

14 years: 4 students 15 years: 4 students 16 years: 3 students 17 years: 1 student This table quickly shows that ages 14 and 15 are most common. Frequency tables can also include relative frequencies (proportions or percentages) to compare categories more easily.

• Frequency tables summarise data by counting occurrences of each value.
• They help identify the most common or rare values in a dataset.
• Relative frequencies can be added to show proportions or percentages.

To create a frequency table: first list all distinct data values or categories in one column. Then, count how many times each value appears in the dataset and record this in the frequency column. Optionally, calculate relative frequencies by dividing each frequency by the total number of data points.

For grouped or continuous data, class intervals are used instead of individual values. For example, test scores might be grouped into intervals like 0–9, 10–19, etc., with frequencies counting how many scores fall into each interval.

Ensure that all data points are included and that intervals do not overlap. The sum of frequencies should equal the total number of observations.

• List distinct values or class intervals clearly.
• Count and record frequencies accurately.
• Calculate relative frequencies by dividing frequency by total data count.
• Check that frequencies sum to total observations.

Two-way tables display frequencies for combinations of two categorical variables, allowing analysis of how one variable relates to another. Each row represents categories of one variable, and each column represents categories of the other.

For example, a survey of 30 students records their favourite sport (Football, Basketball) and gender (Male, Female). A two-way table might look like:

| | Football | Basketball | Total | |-----------|----------|------------|-------| | Male | 8 | 7 | 15 | | Female | 5 | 10 | 15 | | Total | 13 | 17 | 30 | This table helps identify patterns, such as basketball being more popular among females in this group.

• Two-way tables compare two categorical variables simultaneously.
• Rows and columns represent categories of each variable.
• They reveal associations or differences between variables.
• Totals help check data completeness and calculate proportions.

Interpreting frequency tables involves identifying the most or least frequent categories, spotting patterns, and understanding the distribution shape (e.g., uniform, skewed).

For two-way tables, interpretation focuses on relationships between variables. For example, comparing row percentages can show if one category is more common within a subgroup.

Care must be taken not to infer causation from association alone. Tables summarise data descriptively and support further analysis or hypothesis testing if needed.

• Identify common and rare categories in frequency tables.
• Look for patterns or trends in data distribution.
• In two-way tables, compare frequencies across rows and columns to find associations.
• Avoid assuming causation from observed associations.

When collecting data, ensure categories are mutually exclusive and collectively exhaustive to avoid ambiguity in tables.

Use clear labels and consistent units to make tables easy to understand.

Check for errors by verifying that totals match the number of observations and that frequencies are non-negative integers (for counts). If relative frequencies are used, they should sum to 1 or 100%.

• Define categories clearly and avoid overlap.
• Label tables and columns precisely.
• Verify totals and frequency counts for accuracy.
• Use relative frequencies to compare groups of different sizes.

Pictograms use repeated images or symbols to represent data quantities, making it easier to compare values visually. Each symbol corresponds to a fixed number of items, such as one symbol representing 5 people or 10 units sold.

For example, if a class has 25 students who prefer football, and one football icon represents 5 students, then 5 football icons would be drawn.

Pictograms are useful for small datasets and when communicating data to audiences who benefit from visual cues. However, they can be less precise than numerical charts and are not suitable for very large or complex data. Drawing skills are not required for exams, but understanding how to interpret pictograms is essential.

• Each symbol represents a fixed quantity.
• Number of symbols = data value ÷ quantity per symbol.
• Good for simple, categorical data comparisons.

Bar charts display data using rectangular bars where the length or height corresponds to the data value. They are versatile and can represent different data types and comparisons.

Simple bar charts show frequencies or counts for categories. Sectional bar charts divide bars into segments representing subgroups, useful for showing parts of a whole within categories.

Percentage sectional bar charts convert these segments into percentages, allowing comparison of relative proportions across categories regardless of total size. Multiple bar charts place bars for different groups side by side for direct comparison across categories.

• Simple bar chart: one bar per category.
• Sectional bar chart: bars divided into parts representing subgroups.
• Percentage sectional bar chart: segments shown as percentages of total bar height.
• Multiple bar chart: bars for different groups placed side by side.
• Bars should be equally spaced and labelled clearly.

Pie charts represent data as slices of a circle, where each slice's angle corresponds to the proportion of that category relative to the total.

To construct a pie chart, calculate each category's percentage of the total, then convert this to an angle by multiplying by 360°.

For example, if 30 out of 120 students prefer basketball, the angle for basketball is (30/120) × 360° = 90° (a quarter circle). Comparative pie charts place two or more pie charts side by side to compare distributions between groups.

• Calculate percentage = (category frequency ÷ total) × 100.
• Angle for slice = (percentage ÷ 100) × 360°.
• Use a protractor to draw accurate slices.
• Comparative pie charts highlight differences in proportions between groups.

Venn diagrams use overlapping circles to represent sets and their relationships, such as intersections (common elements) and unions (all elements combined).

For example, if 20 students like football, 15 like basketball, and 5 like both, two overlapping circles can show these numbers: 15 only football, 10 only basketball, and 5 in the overlap.

Venn diagrams help analyse categorical data involving membership in multiple groups and can clarify complex relationships visually.

• Each circle represents a set.
• Overlap shows elements common to sets.
• Non-overlapping parts show elements unique to each set.
• Useful for problems involving unions, intersections, and complements.

Box-and-whisker diagrams summarise numerical data by showing the minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum values.

The box spans from Q1 to Q3, with a line at the median. Whiskers extend to the minimum and maximum data points.

For example, if a dataset has Q1=12, median=15, Q3=20, minimum=8, and maximum=25, the box is drawn from 12 to 20 with a line at 15, whiskers from 8 to 12 and 20 to 25 respectively. Outlier identification is not required at this level but understanding the spread and central tendency is important.

• Shows data spread and central tendency.
• Box covers middle 50% of data (interquartile range).
• Median divides data into two halves.
• Whiskers show range from minimum to maximum.
• Useful for comparing distributions between groups.

Stem-and-leaf diagrams display numerical data by splitting each value into a 'stem' (leading digits) and a 'leaf' (last digit), preserving original data while showing distribution.

For example, data values 23, 25, 27, 31, 33 would be represented as:

2 | 3 5 7 (stem 2, leaves 3,5,7) 3 | 1 3 (stem 3, leaves 1,3) Back-to-back stem-and-leaf diagrams place two datasets back-to-back on a common stem, facilitating comparison of distributions between two groups.

• Stem = leading digits, leaf = last digit.
• Data remains visible while showing shape of distribution.
• Back-to-back diagrams compare two datasets side by side.
• Useful for small to moderate sized datasets.

Bar charts represent categorical data using rectangular bars with lengths proportional to the values they represent. They are ideal for comparing frequencies or amounts across different categories.

Advantages include clear visual comparison between categories and ease of interpretation. They can be drawn vertically or horizontally and can display grouped or stacked data.

However, bar charts are not suitable for continuous data or showing trends over time. Bars must be equally spaced to avoid misleading impressions.

• Good for categorical data comparison.
• Easy to read and interpret.
• Not suitable for continuous data or trends.

Histograms display frequency distributions of continuous data by grouping data into intervals (bins) and representing the frequency of data points in each bin with adjacent bars.

They are useful for showing the shape of data distribution, such as skewness or modality.

Unlike bar charts, histogram bars touch each other to indicate continuous data. However, choosing inappropriate bin widths can distort the data representation, either hiding details or exaggerating variability.

• Show frequency distribution of continuous data.
• Bars touch to indicate continuity.
• Bin width choice affects accuracy and clarity.

Pie charts show proportions of categories as slices of a circle, with each slice’s angle proportional to the category’s percentage of the total.

They provide an immediate visual impression of relative sizes and are useful for showing parts of a whole.

However, pie charts can be misleading if there are many categories or slices with similar sizes, making it hard to compare accurately. They also do not show exact values and are less effective than bar charts for precise comparisons.

• Good for showing parts of a whole.
• Visually intuitive for proportions.
• Can be misleading with many or similar-sized categories.
• Not precise for exact comparisons.

Line graphs connect data points with lines, typically used to show trends over time or continuous change.

They are effective for illustrating increases, decreases, and patterns in data series.

However, line graphs are not suitable for categorical data without a natural order and can be misleading if data points are sparse or irregularly spaced.

• Show trends and changes over time.
• Useful for continuous data series.
• Not suitable for unordered categorical data.

Stem-and-leaf diagrams organise numerical data to show distribution while retaining original data values.

They allow quick identification of data shape, central tendency, and spread.

However, they are less effective for very large datasets and can be difficult to interpret for complex data.

• Display data distribution and retain actual values.
• Useful for small to moderate datasets.
• Less practical for large datasets.

Box-and-whisker plots summarise data using five key statistics: minimum, lower quartile, median, upper quartile, and maximum.

They clearly show data spread, central tendency, and symmetry.

While they do not show individual data points, they are excellent for comparing distributions across groups. Outlier identification is possible but not required at this level.

• Summarise data distribution with quartiles.
• Good for comparing groups.
• Do not show individual data points.