Presenting and interpreting data

Good statistical diagrams do more than display numbers. They reveal the shape, centre, spread and unusual features of a distribution. The correct display depends on what the variable means and how the data were recorded.

Prerequisites

You should be able to:

• order positive and negative numbers;
• read scales and plot coordinates;
• calculate percentages;
• find a midpoint and interpret inequalities such as $10\leq x<20$.

This lesson introduces the representations needed before averages and measures of spread and histograms.

Classifying data

A variable is a characteristic recorded for each individual or item.

Type	Meaning	Examples
Qualitative	labels or categories	eye colour, method of travel
Quantitative	numerical values with meaningful arithmetic	height, number of calls
Discrete	separate, usually countable values	number of siblings, shoe size
Continuous	any value in an interval is possible	mass, time, temperature

Quantitative data may be discrete or continuous. Qualitative data are not continuous merely because their categories have been assigned numbers. For example, coding bus, car and walk as $1,2,3$ does not make the mean code meaningful.

Data can also be described by their source:

• primary data are collected first hand for the investigation;
• secondary data were collected previously by somebody else.

And by how observations are paired:

• univariate data record one variable per individual;
• bivariate data record two variables on the same individual, such as height and arm span.

The pairing in bivariate data matters. A scatter diagram must use matched pairs $(x,y)$, not two independently sorted lists. See correlation and regression for bivariate displays.

Self-check 1

Classify each variable as qualitative, discrete quantitative or continuous quantitative.

1. The time taken to complete a race.
2. The number of emails received in a day.
3. A customer’s satisfaction category: poor, fair, good or excellent.

Answer

1. Continuous quantitative. Time can, in principle, take any value in an interval, even if the stopwatch rounds it.
2. Discrete quantitative. Emails are counted in whole numbers.
3. Qualitative. The categories are ordered, but the gaps between them have no numerical size.

Recorded values are not necessarily discrete

A mass recorded as $63.2$ kg is still continuous data. Rounding changes the recorded precision, not the underlying type of variable.

Choosing a presentation

The display must preserve the feature you want to study.

Situation	Suitable presentation	Main strength
Categories	bar chart	compares category frequencies
Small numerical data set	ordered list or stem and leaf diagram	retains every value
Discrete values with repetitions	frequency table or vertical line chart	shows exact frequencies
Grouped continuous data	histogram	compares frequency density
Ordered numerical data	cumulative frequency graph	estimates median, quartiles and percentiles
Comparing distributions	box plots on a common scale	compares centre, spread and skewness

In a bar chart, bars are separated because categories are distinct. In a histogram, class intervals form a continuous scale, so bars touch and area, not height alone, represents frequency. That distinction is developed in histograms.

Frequency tables and class intervals

A frequency table compresses repeated observations. Its frequencies must satisfy

$$\sum f=n,$$

where $n$ is the total number of observations. The relative frequency of a class is

$$\frac{f}{n},$$

and its percentage frequency is $100f/n$.

Worked example 1: constructing classes

The journey times, in minutes, are

$$7, 12, 16, 18, 21, 23, 25, 28, 31, 34, 37, 42.$$

Using classes of width $10$ gives:

Time $t$ in minutes	Frequency	Cumulative frequency
$0\leq t<10$	$1$	$1$
$10\leq t<20$	$3$	$4$
$20\leq t<30$	$4$	$8$
$30\leq t<40$	$3$	$11$
$40\leq t<50$	$1$	$12$

The convention $10\leq t<20$ includes $10$ but excludes $20$. Every possible value belongs to exactly one class, so there are no gaps or overlaps.

The cumulative frequency $8$ means that $8$ journeys took less than $30$ minutes. It does not mean that $8$ journeys lie in the class $20\leq t<30$.

Grouping loses detail. From the table we can tell that one journey took between $40$ and $50$ minutes, but not that its time was $42$ minutes.

Class boundaries for rounded data

If ages are stated as whole numbers and a class is labelled $10$ to $19$, its continuous boundaries are $9.5\leq a<19.5$. Boundaries lie halfway between possible rounded values. If classes are already written as inequalities, use the given boundaries.

Stem and leaf diagrams

A stem and leaf diagram is a compact ordered list. It shows the distribution while retaining each observation. It needs:

1. leaves in ascending order;
2. every observation, including repeated values;
3. a key that fixes the place value.

Worked example 2: constructing a stem and leaf diagram

For the data

$$18, 22, 27, 21, 35, 30, 22, 41, 39, 26,$$

use the tens digits as stems and sort the units digits within each row:

1 | 8
2 | 1 2 2 6 7
3 | 0 5 9
4 | 1

Key: $2\mid 6$ represents $26$.

There are $10$ leaves, confirming that all $10$ observations are present. The repeated leaf $2$ on stem $2$ is essential because $22$ occurs twice.

The ordered values can now be read directly. The fifth and sixth are $22$ and $26$, so the median is

$$\frac{22+26}{2}=24.$$

Back to back stem and leaf diagrams

Two small data sets can share stems. Leaves for one set appear on the left and leaves for the other on the right. Left hand leaves are written so that values increase away from the stem to the left, while right hand leaves increase away from the stem to the right.

Group A       Group B
  8 4 1 | 1 | 2 5 9
9 7 3 0 | 2 | 1 1 6 8
    5 2 | 3 | 0 4

Key: $1\mid 2\mid 1$ represents $21$ in Group A and $21$ in Group B.

Always make the key unambiguous. A key such as $3\mid 7=3.7$ differs by a factor of $10$ from $3\mid 7=37$.

Self-check 2

The leaves on stem $4$ are $0,3,3,8$, and the key says $4\mid3$ represents $4.3$. Write the four values and state their mode.

Answer

The values are $4.0,4.3,4.3,4.8$. The mode is $4.3$.

Cumulative frequency graphs

For grouped data, cumulative frequency is plotted against the upper class boundary. The points represent statements of the form

$$\text{number of observations below }x.$$

Include the lower boundary of the first class with cumulative frequency $0$, then join the points with a smooth increasing curve or straight segments, according to the expected convention.

Worked example 3: plotting cumulative frequency

For the table in Worked example 1, plot

$$(0,0), (10,1), (20,4), (30,8), (40,11), (50,12).$$

Notice that $(30,8)$ uses the upper boundary $30$ and cumulative frequency $8$. Plotting the class midpoint $25$ would change the meaning and give incorrect quartile estimates.

For $n$ observations, common A level graph estimates use cumulative frequency levels

$$Q_1:\frac n4,qquad Q_2:\frac n2,qquad Q_3:\frac{3n}{4}.$$

Draw horizontally from each frequency level to the curve, then vertically to the data axis. Because grouped data do not reveal the positions within a class, graphically read quartiles are estimates.

Worked example 4: interpolating a median

Suppose $40$ values are grouped and the cumulative frequencies at $x=20$ and $x=30$ are $14$ and $26$. The median is at cumulative frequency

$$\frac{40}{2}=20.$$

Assuming values are evenly distributed through this class, the required fraction across it is

$$\frac{20-14}{26-14}=\frac{6}{12}=\frac12.$$

Therefore the estimated median is

$$20+\frac12(30-20)=25.$$

This is linear interpolation. It is an estimate based on an even spread within the class, not a claim that an observed value was exactly $25$.

Percentiles and proportions

The $p$th percentile is the value below which approximately $p\%$ of observations lie. For $n=200$, the $85$th percentile is read at cumulative frequency

$$0.85\times200=170.$$

To estimate the number above a value $a$, read the cumulative frequency $F(a)$ and calculate

$$n-F(a).$$

Read the direction carefully

Cumulative frequency gives the number below a value. If a question asks for values greater than $a$, subtract your reading from the total.

Self-check 3

A cumulative frequency curve represents $80$ measurements. At $x=15$, the cumulative frequency is approximately $52$.

1. Estimate how many measurements are below $15$.
2. Estimate how many are at least $15$.
3. At what cumulative frequency should $Q_3$ be read?

Answer

1. Approximately $52$.
2. Approximately $80-52=28$.
3. At $3(80)/4=60$.

Box plots

A box plot displays the five number summary:

$$\text{minimum},quad Q_1,quad \text{median},quad Q_3,quad \text{maximum}.$$

The box runs from $Q_1$ to $Q_3$, with a line at the median. Whiskers extend to the extremes, unless a question uses a stated outlier convention. The box length is the interquartile range

$$\operatorname{IQR}=Q_3-Q_1.$$

Worked example 5: drawing and interpreting a box plot

A data set has

$$\min=4,qquad Q_1=11,qquad Q_2=15,qquad Q_3=23,qquad \max=30.$$

Draw all five values on one linear scale. The box extends from $11$ to $23$, its median line is at $15$, and the whiskers end at $4$ and $30$.

The IQR is

$$23-11=12.$$

Each interval from the minimum to $Q_1$, from $Q_1$ to the median, from the median to $Q_3$, and from $Q_3$ to the maximum contains approximately $25\%$ of the data. Their unequal physical lengths show differing concentration. The interval $15$ to $23$ is relatively long, so that quarter of the observations is more spread out.

The longer upper half of the box and longer upper whisker suggest positive skew. This is evidence about shape, not a proof of an exact model.

Width is not frequency

A longer section of a box plot does not contain more observations. Each quartile section contains about one quarter of them. Greater length means greater spread.

Outliers on box plots

A common rule labels values as outliers if

$$x<Q_1-1.5\operatorname{IQR} \quad\text{or}\quad x>Q_3+1.5\operatorname{IQR}.$$

If this rule is specified, whiskers usually end at the most extreme non-outlying observations and outliers are plotted separately. Do not assume a convention that the question has not given. Learn how to investigate suspicious observations in outliers and cleaning data.

For the values above, the fences are

$$11-1.5(12)=-7,qquad 23+1.5(12)=41.$$

Neither $4$ nor $30$ is an outlier under this rule.

Comparing distributions

A complete comparison uses context and normally addresses both:

1. location, usually the median;
2. spread, usually the IQR or range.

Shape and outliers may add useful evidence. Use comparative language such as greater, smaller, more consistent or more positively skewed.

Worked example 6: writing a comparison

Two delivery services have these summaries, in minutes:

Service	Median	IQR	Range
A	$28$	$8$	$31$
B	$31$	$5$	$18$

A strong interpretation is:

Service A is typically faster because its median delivery time is $3$ minutes lower. Service B is more consistent because both its IQR and range are smaller.

Do not write only “A is better”. Faster and more consistent are different properties, and the data do not say which matters more to a customer.

If two box plots are being compared, they must use the same scale. Otherwise physical box lengths are not comparable.

Common misconceptions

• A cumulative frequency point goes at the midpoint. It goes at the upper class boundary because it counts all observations below that boundary.
• The four quarters of a box plot have equal lengths. They contain roughly equal numbers of observations, but their lengths show spread.
• A large range proves most data are variable. The range may be caused by one extreme value. The IQR describes the middle $50\%$.
• Joining bar chart bars makes a histogram. A histogram uses a continuous scale and frequency density, particularly when class widths differ.
• A precise calculator answer makes grouped data exact. Grouped representations have already lost information, so interpolated statistics remain estimates.

Final self-check

The five number summaries for test scores are:

$$\begin{array}{c|ccccc} &\min&Q_1&\text{median}&Q_3&\max\\ \hline \text{Class P}&18&42&56&68&91\\ \text{Class Q}&25&48&53&61&82 \end{array}$$

1. Find the IQR for each class.
2. Which class has the higher typical score?
3. Which class is more consistent by IQR?
4. What proportion of Class P scored between $42$ and $68$?

Answer

1. Class P: $68-42=26$. Class Q: $61-48=13$.
2. Class P, because its median is $56$, compared with $53$ for Class Q.
3. Class Q, because its IQR is smaller.
4. Approximately $50\%$, because $42=Q_1$ and $68=Q_3$.

A concise comparison is: Class P achieved a slightly higher typical score, but Class Q’s scores were more consistent.

Next steps

• Use summaries from data displays in averages and measures of spread.
• Learn why unequal class widths require histograms.
• Decide how data should be gathered in sampling.
• Study paired variables in correlation and regression.