Graphical Analysis and Representation

Categorical data (favourite colour, type of pet, brand):

• Bar chart: bars NOT touching, separated. Height = count or proportion.
• Pie chart: each slice's angle = $(\text{count} / \text{total}) \times 360°$.

Numerical data, discrete (number of pets, score out of 10):

• Bar chart: same as above. Often called a 'frequency chart'.

Numerical data, continuous (height, weight, time):

• Histogram: bars TOUCHING (representing intervals/bins). Area = frequency (or 'frequency density' for unequal widths).
• Line graph for time series.

Bivariate data (two variables per data point):

• Scatter plot: shows correlation/relationship.

Summary of distribution:

• Box plot: shows min, Q1, median, Q3, max — a 5-number summary.

Data type	Best chart
Categorical, comparing categories	Bar chart, pie chart
Categorical, proportions of whole	Pie chart
Numerical, continuous	Histogram
Time series	Line graph
Two numerical variables	Scatter plot
Distribution summary	Box plot

Worked example: pie chart. Class of 30 students. 12 walk, 8 cycle, 6 bus, 4 car. Find pie chart angles.

• Walk: $12/30 \times 360° = 144°$.
• Cycle: $8/30 \times 360° = 96°$.
• Bus: $6/30 \times 360° = 72°$.
• Car: $4/30 \times 360° = 48°$.
• Check: $144 + 96 + 72 + 48 = 360°$ ✓.

A scatter plot shows pairs of measurements (one variable on each axis). Each point represents one observation.

Correlation describes the strength and direction of any pattern:

• Positive correlation: as $x$ increases, $y$ tends to increase.
• Negative correlation: as $x$ increases, $y$ tends to decrease.
• No correlation: no clear pattern.

Strength:

• Strong: points cluster tightly around a line.
• Weak: points scattered loosely.

Line of best fit (also called trend line or regression line): a straight line drawn to best represent the pattern. It minimises the spread of points either side. (Formal calculation comes in DP — for MYP, draw by eye.)

Crucial warning. Correlation does NOT imply causation. A scatter plot might show people who eat more ice cream also drown more — but it's not because ice cream causes drowning; both correlate with hot weather. A common MYP criterion-D discussion.

A box plot (box-and-whisker plot) shows the FIVE-NUMBER SUMMARY of a dataset:

• Minimum (lowest value).
• Q1 (first quartile — 25th percentile).
• Median (Q2 — 50th percentile).
• Q3 (third quartile — 75th percentile).
• Maximum (highest value).

The 'box' goes from Q1 to Q3 (the interquartile range or IQR), with the median marked inside. The 'whiskers' extend to the minimum and maximum.

Interquartile range: $\text{IQR} = Q3 - Q1$. Measures the spread of the MIDDLE 50% of data. Less affected by outliers than the full range.

Worked example. Data: $3, 5, 7, 8, 8, 9, 11, 13, 15, 20$.

• Min: 3. Max: 20.
• Sorted: already in order. $n = 10$.
• Median: average of 5th and 6th values: $(8 + 9)/2 = 8.5$.
• Q1: median of first half $(3, 5, 7, 8, 8)$ = 7.
• Q3: median of second half $(9, 11, 13, 15, 20)$ = 13.
• IQR: $13 - 7 = 6$.

Worked example: comparing box plots. Two classes' test scores on the same exam, drawn as box plots. Class A: median 65, IQR 20. Class B: median 70, IQR 35.

• Class B scored slightly higher on average (median).
• Class B has more SPREAD (larger IQR).
• Class A is more CONSISTENT.

Box plots are great for comparing distributions on the same axis.