##### Frequency tables

Frequency tables show the total of the tally marks. Some frequency tables include the tallies.

This frequency table is the same as the table It has another column added with the totals frequencies of the tallies. The frequency table has a space to write a total at the bottom of the frequency column. This helps you to know how many pieces of data were collected. In this example the student recorded the colors of 157 cars. Most frequency tables will not include tally marks. Here is a frequency table without tallies. It shows how many people were treated for different diseases in one weak. Tip! Before you could draw any meaningful conclusions about what type of illness is most common at a clinic, you would need to know where this data was collected. The frequency of different diseases would be different in different parts of the world.

### So what is the frequency column?

The frequency column tells you how often or how frequently each result speared in the data and the data is discrete.

### Grouping data in class intervals

Sometimes numerical data needs to be recorded in different groups. For example, if you collected test results for 40 students you might find that students scored between 40 and 84 (out of 100). If you recorded each individual score ( and they could be all different) you would get a very large frequency table that is difficult to manage. To simplify things, the collected data can be arranged in groups called intervals. A frequency table with results arranged in class intervals is called a grouped frequency table.

### Example: The range of scores (40-48) has been divided into class intervals. Notice that the class intervals do not overlap so it is clear which data goes in what class.

In this example, the test does not allow for fractions of a mark, so all test scores are integers and the data is discrete.

Note! All this tables we are going to use it to construct bar charts and other frequency diagrams. These diagrams give a clear, visual impression of the data.

### Steam and leaf diagrams

A steam and leaf diagram is a special type of table that allows you to organize and display grouped data using the actual data values. When you use a frequency table to organize grouped data you cannot see the actual data values, just the number of data items in each group. Steam and leaf diagrams are useful because when you keep the actual values, you can calculate the range and averages for the data.

In a steam and leaf diagram each data item is broken into two parts: a steam and a leaf. The final digit of each value is the leaf and the previous digits are the stem. The stems are written to the left of a vertical line and the leaves are written to the right of the vertical line.For example a score of 13 would be shown as: In this case, the tens digits in the steam and the units digit is the leaf.

A larger data value such as 259 would be shown as: In this case, the steam represents both the tens and the hundreds digits while the units digit is the leaf.

To be useful, a stem and leaf diagram should have at least 5 stems. If the number of stems is less than that, you can split the leaves into 2 (or sometimes even 5) classes. If you do this, each stem is listed twice and the leaves are grouped into a lower and higher class. For example, if the stem in tens and the leaves are units, you would make two classes like this: Values from 10 to 14 (leaves 0 to 4) are included in the first class, values from 15 to 19 (leaves 5 to 9) are included in the second class.

Steam and leaf diagrams are easier to work with if the leaves are ordered from smallest to greatest.

Example 1: This data set shows the ages of customers using an internet coffee.

Draw a stem and leaf diagram to display this data. Group the ages in intervals of ten, 10-19; 20-29 and so on. These are two digit numbers, so the tens digit will be the stem. List the stems in ascending order down the left of the diagram. Work through the data in the order it is given, writing the units digits (the leaves) in a row next to the appropriate stem. Space the leaves to make it easier to read them. If you need to work with the data, you can redraw the diagram, putting the leaves in ascending order. From the re-organized stem and leaf diagram you can quickly see that:

• The youngest person using the internet café was 17 years old (the first data item)
• The oldest person was 55 (the last data item)
• Most users were in the age group 30-39 ( the group with the largest number of leaves) A back to back stem and leaf diagram is used to show two sets of the data. The second set of data is plotted against the same stem, but the leaves are written to the left.

This steam and leaf plot compares the battery life of two different brands of mobile phone. ### Two-way tables

A two-way table shows the frequency of certain results for two or more sets of data. Here is a two way table showing how many men and woman drivers were wearing their seat bells when they passed a check point. The headings at the top of the table give you information about wearing seat belts. The headings down the side of the table give you information about gender.

You can use the table to find out:

• How many men were wearing seat belts
• How many women were wearing seat belts
• How many men were not wearing seat belts
• How many women were not wearing seat belts

You can also add the totals across and down to work out:

• How many men were surveyed
• How many women were surveyed
• How many people (men + women) were wearing seat belts or not wearing seat belts.

Two examples of two-way tables:

Drinks and crisps sold at a school tuck shop during lunch break How often male and female students use Facebook 