Data Representation and Averages

Choosing the right chart:

Chart type	Best for
Bar chart	Comparing discrete or categorical data frequencies
Pictogram	Simple comparative data (each symbol = fixed quantity)
Pie chart	Showing proportions of a whole (one dataset)
Vertical line chart	Ungrouped discrete numerical data (e.g. number of siblings)
Time series (line graph)	Data recorded at regular time intervals
Frequency table	Organising raw data before drawing charts

Pie chart — calculating sector angles:

$\text{Angle for category}=\frac{\text{frequency}}{\text{total}}\times 360°$

Worked example — 40 students chose a sport: 15 football, 10 basketball, 8 tennis, 7 swimming.

Football: $\frac{15}{40}\times 360=135°$; Basketball: $90°$; Tennis: $72°$; Swimming: $63°$. Sum: $360°$ ✓

Interpreting time series: the overall trend (rising, falling, stable) and any seasonal patterns are the key things to comment on.

Mode: the most frequent value (for grouped data: the modal class).

Median: the middle value when data is ordered. For $n$ values, the median is the $\frac{n+1}{2}$th value.

Mean: $\bar{x}=\frac{\sum fx}{\sum f}$ (where $f$ is frequency and $x$ is the value/midpoint).

Range: largest value − smallest value (a measure of spread).

Worked example — from a frequency table:

Goals ($x$)	Frequency ($f$)	$fx$
0	3	0
1	8	8
2	5	10
3	4	12

$n=20$; $\sum fx=30$; Mean $=30/20=1.5$ goals ✓

Median: $\frac{20+1}{2}=10.5$th value → between 10th and 11th. Count: 0s (3), 1s (8) → 11 values up to 1. Both 10th and 11th are 1. Median $=1$ ✓

Outliers: values much larger or smaller than the rest. The mean is sensitive to outliers; the median is resistant.

When data is grouped, use the midpoint of each class interval as an estimate for the mean.

Worked example — estimate the mean from:

Height (cm)	Frequency	Midpoint	$f\times$ mid
$140\le h<150$	5	145	725
$150\le h<160$	12	155	1860
$160\le h<170$	8	165	1320

$\sum f=25$; $\sum f\times \text{mid}=3905$; Estimated mean $=3905/25=156.2$ cm ✓

Modal class: the class with the highest frequency → $150\le h<160$ ✓

Median class: the class containing the $\frac{25+1}{2}=13$th value. Cumulative: 5, then 17. The 13th value is in $150\le h<160$ ✓

Histograms are used for continuous grouped data. The vertical axis shows frequency density, not frequency.

$\text{Frequency density}=\frac{\text{frequency}}{\text{class width}}$

Equal class widths: bars are equally wide; areas are proportional to frequencies.

Unequal class widths: you must use frequency density so that area still represents frequency correctly.

Worked example — class $20\le t<30$, frequency 15, width 10. Frequency density $=15/10=1.5$ ✓

Cumulative frequency graphs: plot cumulative frequency against the upper class boundary. Use to read off the median (at $n/2$), quartiles (at $n/4$ and $3n/4$), and percentiles.

Median from cumulative frequency graph — for $n=60$ values: read off the value at cumulative frequency 30 on the vertical axis.

A box plot (box-and-whisker diagram) displays the five-number summary:

$\text{Minimum}{Q}_1\text{Median}{Q}_3\text{Maximum}$

Interquartile range (IQR): $IQR=Q_3-Q_1$ — the spread of the middle 50% of data.

Reading quartiles from ordered data or cumulative frequency:

• $Q_1$: value at position $\frac{n}{4}$ (use $\frac{n+1}{4}$ for small discrete datasets — both methods are accepted)
• $Q_3$: value at position $\frac{3n}{4}$ (use $\frac{3(n+1)}{4}$ for small discrete datasets)

Worked example — ordered data: 3, 5, 7, 9, 11, 14, 16, 20 ($n=8$).

$Q_1$: position $\frac{8}{4}=2$nd value $=5$. $Q_3$: position $\frac{3\times 8}{4}=6$th value $=14$.

Median: mean of 4th and 5th values $=(9+11)/2=10$. $IQR=14-5=9$. ✓

Comparing distributions: use the median (centre) and IQR (spread) to compare two datasets. A higher median means a higher typical value; a higher IQR means more variability.

Statistics drawn from a sample are used to infer properties of the population. Key principles:

• A larger sample gives more reliable estimates.
• A representative (unbiased) sample is essential — convenience samples may be skewed.
• Statistical measures describe the sample; they are used to make inferences about the population.

Using statistics to describe: state the measure (mean, median, IQR) and interpret it in context.

"The mean journey time is 34 minutes, suggesting the typical commuter spends just over half an hour travelling."

Misleading statistics: the mean can be pulled by outliers; presenting only the mean without the spread can misrepresent a dataset. A median with IQR gives a more complete picture.

1. Histogram — plotting frequency instead of frequency density

If class widths differ, plotting frequency makes wider bars appear more frequent than they are. Calculate frequency density $=$ frequency / class width and plot that.

2. Mean from grouped data — using class boundaries instead of midpoints

Use the midpoint of each class interval (e.g. $145$ for $140\le h<150$), not the boundary values.

3. Box plot — confusing IQR with the full range

The IQR is $Q_3-Q_1$ (the box width). The range is maximum − minimum (the full whisker span).

4. Median — not ordering data first

The median is the middle value of the ordered dataset. Finding the median of unordered data gives a wrong answer.

Mistake	Correction
"Modal class is the class with the largest upper boundary"	Modal class has the highest frequency (or frequency density in a histogram)
"Cumulative frequency: plot against midpoint"	Plot against the upper class boundary
"IQR = Q3 × Q1"	$IQR=Q_3-Q_1$ (subtract, not multiply)