IB MYP 3 Mathematics 9.3 Frequency Tables, Grouped Data, Stem-and-Leaf Plots and Histograms Study Notes - New Syllabus

9.3 – Frequency Tables, Grouped Data, Stem-and-Leaf Plots and Histograms

When a data set contains many values, it can be difficult to see patterns by looking at the raw data alone. A useful way to make the data easier to understand is to organise it into frequency tables or graphical representations.

In this subtopic, we will learn how to organise numerical data using:

• frequency tables;
• grouped frequency tables;
• class intervals;
• modal classes;
• stem-and-leaf plots; and
• histograms.

Frequency Tables

A frequency table records how many times each value occurs. The number of times a value occurs is called its frequency.

For example, suppose the number of books read by a group of students is:

\(2,\ 3,\ 1,\ 4,\ 2,\ 3,\ 2,\ 5,\ 1,\ 3\)

A frequency table can be created by counting how many times each value occurs.

The total of all frequencies must equal the total number of observations.

\( \text{Total frequency}=\text{Total number of observations} \)

Tally and Frequency Tables

A tally column can be used before writing the frequency. Tally marks make it easier to count large amounts of data accurately. Groups of five are usually represented by four vertical marks crossed by a fifth mark.

 Finding the Mode from a Frequency Table

The mode is the value with the greatest frequency. Therefore, a frequency table can be used to find the mode quickly.

For example, if the highest frequency in a table is \(3\) and it belongs to the value \(6\), then:

\( \text{Mode}=6 \)

Grouped Data

When a numerical data set contains many different values, listing every individual value in a frequency table may not be practical. Instead, the values can be placed into class intervals.

A class interval is a range of values grouped together.

Grouping the data makes large data sets easier to organise and interpret.

Constructing a Grouped Frequency Table

Consider the following data representing the number of visitors to a park over several days:

\(12,\ 18,\ 24,\ 27,\ 31,\ 35,\ 39,\ 42,\ 45,\ 47,\ 51,\ 54,\ 58,\ 63,\ 68\)

We can group the values into intervals of width \(10\).

The frequencies add to \(15\), which matches the number of original observations.

Class Interval and Class Width

The class width describes the size of each interval. For example, the intervals \(10\text{–}19\), \(20\text{–}29\), \(30\text{–}39\) have the same width.

For whole-number data, the interval \(10\text{–}19\) contains:

\(10,\ 11,\ 12,\ldots,\ 19\)

The next interval begins at \(20\), so no value is counted twice and no value is missed.

 Modal Class

For grouped data, we cannot usually identify the exact mode because the individual values inside each interval are not known. Instead, we identify the modal class.

The modal class is the class interval with the highest frequency.

For the grouped table above, the highest frequency is \(3\). The modal classes are:

\(30\text{–}39,\quad 40\text{–}49,\quad 50\text{–}59\)

There are three modal classes because they have equal highest frequencies.

 Stem-and-Leaf Plots

A stem-and-leaf plot organises numerical data by separating each value into a stem and a leaf.

For two-digit numbers, the tens digit is usually the stem and the units digit is the leaf.

For example:

• \(44\rightarrow 4|4\)
• \(49\rightarrow 4|9\)

Here:

• \(4\) is the stem;
• \(4,9\) is the leaf.

Therefore, the notation \(4|9\) represents the number \(49\).

Constructing a Stem-and-Leaf Plot

Consider the data:

\(12,\ 15,\ 17,\ 21,\ 23,\ 24,\ 24,\ 28,\ 31,\ 35,\ 37\)

Separate the tens and units digits:

The leaves should be written in ascending order.

Key: \(1|2=12\)

 Ordered Stem-and-Leaf Plot

An ordered stem-and-leaf plot has the leaves arranged from smallest to largest. This makes it easier to identify the:

• minimum;
• maximum;
• median;
• mode;
• range; and
• overall distribution of the data.

A major advantage of a stem-and-leaf plot is that it shows the actual data values while also showing how the values are distributed.

Reading a Stem-and-Leaf Plot

Consider:

Using the key \(2|1=21\), the data values include:

\(21,\ 24,\ 24,\ 27,\ 30,\ 32,\ 35,\ 38,\ 41,\ 43,\ 46\)

The minimum is \(21\), the maximum is \(46\), and the range is:

\(46-21=25\)

Histograms

A histogram is a graph used to display the distribution of numerical data, especially when the data has been grouped into intervals.

A histogram looks similar to a bar chart, but there are important differences.

Constructing a Histogram

Suppose the grouped frequency table is:

To draw the histogram:

1. Put the numerical intervals on the horizontal axis.
2. Put frequency on the vertical axis.
3. Choose a suitable scale.
4. Draw a bar for each interval.
5. Make the bars touch.
6. Label both axes and include a clear title.

Histogram Scale and Frequency

For equal-width intervals, the height of each bar represents the frequency. For example, if the interval \(20\text{–}29\) has frequency \(8\), its bar reaches \(8\) on the vertical axis.

At MYP 3 level, histograms with equal-width intervals can be interpreted directly using frequency as the bar height.

 Comparing Frequency Tables, Stem-and-Leaf Plots and Histograms

What Can We Learn from These Representations?

Once numerical data has been organised, we can look for important features such as:

• the most common value or interval;
• the smallest and largest values;
• the range;
• where most observations are concentrated;
• whether values are spread out or clustered; and
• possible unusual values or outliers.

A stem-and-leaf plot is especially useful when the exact values are important. A grouped table or histogram is often more useful when there are many observations.

Common Mistakes