Presentation of data AI HL

IB Mathematics: Applications and Interpretation HL · Unit 4

Lesson 4.2 · Presentation of data

Organise data in tables and graphs, then use cumulative frequency and box-and-whisker diagrams to interpret distributions.

A living hope

Peter writes that God has given believers “new birth into a living hope through the resurrection of Jesus Christ from the dead, and into an inheritance that can never perish, spoil or fade. This inheritance is kept in heaven for you.” 1 Peter 1:3–4 (NIV)

Graphs help us organise information so that patterns become easier to see. The Bible’s promise of an inheritance kept in heaven gives Christians a hope that does not fade. Our confidence in eternal life rests on God’s promise through the resurrection of Jesus.

Learning objectives

By the end of this lesson, you should be able to:

organise data in a frequency table and represent it with a histogram;
construct and interpret a cumulative frequency table and graph;
estimate the median, quartiles, percentiles, range and interquartile range;
draw and interpret a box-and-whisker diagram; and
compare distributions using their centre, spread, shape and outliers.

1. Frequency tables and grouped data

A frequency table records how often values or groups of values occur. For continuous data, values are often organised into class intervals.

For example, the interval $10<x\leq20$ includes values greater than 10 and up to and including 20. When class intervals are written as inequalities, check carefully which boundary values are included.

Worked example · Reading a frequency table

The lengths of salmon caught by a fisherman over one month are grouped into equal intervals. Reading the bar heights from the histogram gives the following frequencies:

Salmon length, $l$ (cm)	Frequency
$0<l\leq10$	0
$10<l\leq20$	42
$20<l\leq30$	33
$30<l\leq40$	52
$40<l\leq50$	60
$50<l\leq60$	25
$60<l\leq70$	38
$70<l\leq80$	14
$80<l\leq90$	7

The greatest frequency is 60, so the class interval with the most salmon is $40<l\leq50$ cm.

Frequencies read from a drawn graph may be approximate if the bar heights do not fall exactly on grid lines.

2. Histograms

A histogram represents grouped numerical data using adjoining bars. The horizontal axis shows the class intervals, and the vertical axis shows frequency for equal-width classes.

Draw the bars so that they touch; the data are continuous.
Label both axes and include units where appropriate.
Use a consistent scale and show each class interval clearly.
For this syllabus section, histograms use equal class intervals.

Worked example · Drawing a histogram

Use the table below to draw a histogram. The class intervals all have width 5, so use frequency as the height of each bar.

Class interval	Frequency
$5\leq x<10$	5
$10\leq x<15$	10
$15\leq x<20$	12
$20\leq x<25$	8

Draw four adjoining bars over the intervals, with heights 5, 10, 12 and 8. Label the horizontal axis $x$ and the vertical axis “Frequency.”

Important: A histogram is used for grouped numerical data. A bar chart is often used for categories, and its bars are separated by gaps.

3. Cumulative frequency

Cumulative frequency is the running total of frequencies up to and including each class. It tells us how many data values are below a particular class boundary.

Worked example · Cumulative frequency table

A class of 30 students has marks grouped as shown. Add each frequency to the total of all previous frequencies.

Marks, $m$	Frequency	Cumulative frequency
$20\leq m<30$	2	2
$30\leq m<40$	3	5
$40\leq m<50$	5	10
$50\leq m<60$	7	17
$60\leq m<70$	6	23
$70\leq m<80$	4	27
$80\leq m<90$	2	29
$90\leq m<100$	1	30

For example, the cumulative frequency at 70 marks is 23. This means that approximately 23 students scored below 70 marks.

Drawing a cumulative frequency graph

Calculate the cumulative frequency for each class.
Plot each cumulative frequency against the upper boundary of its class.
Begin at the lower boundary of the first class with cumulative frequency 0.
Join the points with a smooth increasing curve.

For the table above, plot $(20,0),(30,2),(40,5),(50,10),(60,17),(70,23),(80,27),(90,29),(100,30)$ .

4. Finding the median, quartiles and percentiles

On a cumulative frequency graph, read a data value from the horizontal axis after locating the required cumulative frequency on the vertical axis.

For a data set of size $n$ :

$\text{Lower quartile }Q_1:\frac{n}{4} \qquad \text{Median }Q_2:\frac{n}{2} \qquad \text{Upper quartile }Q_3:\frac{3n}{4}$

The $p$ th percentile is found at cumulative frequency $\dfrac{p}{100}n$ . For example, the 75th percentile is located at cumulative frequency $0.75n$ , which is also the upper quartile.

Worked example · Estimating from the cumulative frequency table

There are 30 students, so use cumulative frequencies 7.5 for $Q_1$ , 15 for the median, 22.5 for $Q_3$ , and 22.5 for the 75th percentile. Reading from the cumulative frequency graph gives estimates. Using linear interpolation within the classes:

$Q_1$ lies in the $40$ to $50$ class: $Q_1\approx45$ marks;
the median lies in the $50$ to $60$ class: median $\approx57.1$ marks;
$Q_3$ lies in the $60$ to $70$ class: $Q_3\approx69.2$ marks;
the 75th percentile is the same as $Q_3$ , approximately 69.2 marks.

These are estimates because the original marks are grouped into intervals.

5. Range and interquartile range

The range describes the full spread of the data. The interquartile range (IQR) describes the spread of the middle 50% of the data.

$\boxed{\text{Range}=\text{maximum}-\text{minimum}} \qquad \boxed{\text{IQR}=Q_3-Q_1}$

Worked example · Using quartiles

Suppose $Q_1=45$ marks and $Q_3=69.2$ marks for a group of students.

$\text{IQR}=69.2-45=\boxed{24.2\text{ marks (approximately)}}$

The middle half of the students’ marks span approximately 24.2 marks.

6. Box-and-whisker diagrams

A box-and-whisker diagram, or box plot, summarises a data set using five values:

minimum;
lower quartile, $Q_1$ ;
median, $Q_2$ ;
upper quartile, $Q_3$ ; and
maximum.

The box extends from $Q_1$ to $Q_3$ , with a line at the median. The whiskers extend to the minimum and maximum. Always use a labelled, consistent numerical scale.

Worked example · Five-number summary

Consider the ordered data set:

$17,\ 27,\ 34,\ 52,\ 54,\ 59,\ 61,\ 68,\ 78,\ 82,\ 85,\ 87,\ 91,\ 93$

There are 14 values. The median is halfway between the 7th and 8th values. The lower and upper quartiles are the medians of the lower and upper halves.

Minimum = 17
$Q_1=52$
Median $=\dfrac{61+68}{2}=64.5$
$Q_3=85$
Maximum = 93

$\text{Range}=93-17=\boxed{76} \qquad \text{IQR}=85-52=\boxed{33}$

7. Comparing distributions

When comparing two distributions using box plots, comment on the context and compare the following features:

Median: which group typically has the higher value?
IQR: which group has more spread in its middle 50%?
Range: which group has the greater overall spread?
Shape: are the distributions roughly symmetric or skewed?
Outliers: are any unusually low or high values shown?

A median near the centre of the box with whiskers of similar length may suggest approximate symmetry. A longer whisker on one side or a median off-centre in the box may suggest skewness.

Write comparisons in context. For example: “Group A has the higher median mark, while Group B has the larger IQR, so Group B’s middle half of marks is more spread out.”

IB-style practice · Skill questions

The frequency table below gives values for $x$ .

Class interval $5\leq x<10$ $10\leq x<15$ $15\leq x<20$ $20\leq x<25$

Frequency 5 10 12 8

State the axes labels and bar heights needed to draw a histogram. [3 marks]

Show solution

Label the horizontal axis $x$ and the vertical axis Frequency. Draw adjoining bars over the four class intervals with heights 5, 10, 12 and 8, respectively.
A data set has 50 values. State the cumulative frequencies at which you would read the lower quartile, median and upper quartile from a cumulative frequency graph. [3 marks]

Show solution

$Q_1:\frac{50}{4}=12.5,\qquad \text{Median}:\frac{50}{2}=25,\qquad Q_3:\frac{3(50)}{4}=37.5$
The ordered data are $30,45,55,60,65,65,70,75,75,110,120,125$ . Find the range, median, lower quartile, upper quartile and IQR. [5 marks]

Show solution

Range $=125-30=\boxed{95}$ . Median $=(65+70)/2=\boxed{67.5}$ . The lower half is $30,45,55,60,65,65$ , so $Q_1=(55+60)/2=\boxed{57.5}$ . The upper half is $70,75,75,110,120,125$ , so $Q_3=(75+110)/2=\boxed{92.5}$ . Therefore IQR $=92.5-57.5=\boxed{35}$ .
For the data set in Question 3, determine whether 30 is an outlier using the $1.5\times\mathrm{IQR}$ rule. [3 marks]

Show solution

Lower fence $=Q_1-1.5(\mathrm{IQR})=57.5-1.5(35)=5$ . Since $30>5$ , it is not a low outlier. The upper fence is $92.5+52.5=145$ , so 30 is not an outlier.

Class interval	$5\leq x<10$	$10\leq x<15$	$15\leq x<20$	$20\leq x<25$
Frequency	5	10	12	8

IB-style practice · Cumulative frequency and box plots

Application 1 · Examination marks

A class of 30 students has marks grouped in the table below.

Marks, $m$	Frequency	Cumulative frequency
$20\leq m<30$	2
$30\leq m<40$	3
$40\leq m<50$	5
$50\leq m<60$	7
$60\leq m<70$	6
$70\leq m<80$	4
$80\leq m<90$	2
$90\leq m<100$	1

Complete the cumulative frequency column. [3 marks]
State the cumulative frequencies to use to estimate the median and quartiles. [3 marks]
Use linear interpolation to estimate $Q_1$ , the median and $Q_3$ . [6 marks]
Estimate the IQR. [2 marks]

Show worked solution · Application 1

(a) Cumulative frequencies: 2, 5, 10, 17, 23, 27, 29, 30.

(b) With $n=30$ , use 7.5 for $Q_1$ , 15 for the median and 22.5 for $Q_3$ .

(c)

$Q_1\approx40+\frac{7.5-5}{5}\times10=\boxed{45}$

$\text{Median}\approx50+\frac{15-10}{7}\times10 \approx\boxed{57.1}$

$Q_3\approx60+\frac{22.5-17}{6}\times10 \approx\boxed{69.2}$

(d) IQR $\approx69.2-45=\boxed{24.2\text{ marks}}$ .

Application 2 · Distances travelled by cars

A company records the distance travelled by each of its 50 cars. The cumulative frequency graph shows 35 cars have travelled up to 4,000 km, and 46 cars have travelled up to 6,000 km.

Estimate how many cars have travelled between 4,000 km and 6,000 km. [2 marks]
State the cumulative frequencies used to estimate the median, lower quartile and upper quartile. [3 marks]
What percentage of the cars have travelled a distance greater than the upper quartile? [2 marks]

Show worked solution · Application 2

(a) $46-35=\boxed{11}$ cars.

(b) For $n=50$ : lower quartile at CF $12.5$ , median at CF $25$ , upper quartile at CF $37.5$ .

(c) One quarter of the data lie above the upper quartile, so $\boxed{25\%}$ .

Application 3 · Comparing two groups

Two classes take the same examination. Their box plots show:

Class	Minimum	$Q_1$	Median	$Q_3$	Maximum
A	40	55	63	74	89
B	35	50	68	78	92

Which class has the higher median? [1 mark]
Calculate the IQR for each class. [2 marks]
Compare the spread of the middle 50% of marks. [2 marks]

Show worked solution · Application 3

(a) Class B, with median 68.

(b) Class A: $74-55=19$ . Class B: $78-50=28$ .

(c) Class B has the larger IQR, so its middle 50% of marks are more spread out.

Reflection · An inheritance kept in heaven

Data displays help us understand what has been observed. Christian hope looks beyond what we can see now: through Jesus’ resurrection, believers are promised an inheritance that will never perish or fade. May this hope encourage us to live faithfully, care for others and give thanks to God.

1 Peter 1:3–4 (NIV) — God “has given us new birth into a living hope through the resurrection of Jesus Christ from the dead, and into an inheritance that can never perish, spoil or fade.”

Lesson 4.2 · Presentation of data

A living hope

Learning objectives

1. Frequency tables and grouped data

2. Histograms

3. Cumulative frequency

Drawing a cumulative frequency graph

4. Finding the median, quartiles and percentiles

5. Range and interquartile range

6. Box-and-whisker diagrams

7. Comparing distributions

IB-style practice · Skill questions

IB-style practice · Cumulative frequency and box plots

Application 1 · Examination marks

Application 2 · Distances travelled by cars

Application 3 · Comparing two groups

Reflection · An inheritance kept in heaven

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IB Mathematics: Applications and Interpretation HL 6 Sections80 LessonsLifetime

Curriculum

Presentation of data AI HL

Lesson 4.2 · Presentation of data

A living hope

Learning objectives

1. Frequency tables and grouped data

2. Histograms

3. Cumulative frequency

Drawing a cumulative frequency graph

4. Finding the median, quartiles and percentiles

5. Range and interquartile range

6. Box-and-whisker diagrams

7. Comparing distributions

IB-style practice · Skill questions

IB-style practice · Cumulative frequency and box plots

Application 1 · Examination marks

Application 2 · Distances travelled by cars

Application 3 · Comparing two groups

Reflection · An inheritance kept in heaven

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