Tallying
Tallying is a way of adding that uses groups of five.
You can record your results on a tally chart, like the one below.
As you can see, using groups of five makes it easier to see the total.
Tally chart and frequency chart
Using the tally system to record your results is faster than writing out words or figures all the time.
If you record your findings on a tally chart, the data is already collected into groups and you will not need to group it later on.
Investigation example
To investigate the most popular colour of car passing your house over the course of an hour, it is easier to draw tally marks in one of five rows than to write red, blue, silver, silver, red, other, black, etc.
If you use a tally chart, you could note down the colour of the cars as they pass, by quickly making a tally mark, and then find the total frequencies at the end of the one-hour period.
Grouping data
When there is a large number of possible outcomes, you will usually need to group the data.
To do this, first decide on your class intervals based on the range of likely possible answers.
Investigation example
You are carrying out a survey to determine the number of magazines bought by your classmates in the past year.
The possible answers are likely to range from \({0}\) to \({100}\), so you may draw a tally chart with groupings similar to the one below.
The completed frequency table is known as a grouped frequency table.
Two-way tables
You can use a two-way table to show two sets of information.
Exercise in two-way tables
Alex carried out a survey to see how many of his classmates are left-handed.
His results are shown in the table below.
This is an example of a two-way table and it is used to show two different features in a survey.
In this case it shows:
- boys and girls
- left-handed or right-handed
You can find out lots of information from this table.
For example, you can see that there are \({2}\) left-handed boys in the class.
You can also see that there are \({13}\) girls in the class (\({1}\) left-handed and \({12}\) right-handed).
Question
Q1. How many right-handed children are there in the class?
Q2. How many children are there in total?
Answer
A1. There are \({14}\) right-handed boys and \({12}\) right-handed girls, so \(14 + 12 = 26\) in total.
A2. There are \({29}\) children in total.
This is the total of all the numbers in the table.
Continuous data
Continuous data are data which can take any values.
Examples include time, height and weight.
Because continuous data can take any value, there are an infinite number of possible outcomes.
So continuous data must be grouped before they can be represented in a frequency table or statistical diagram.
Grouping continuous dataWhen choosing groups for the data, make sure that each piece of data can be placed in one (and only one) of the groups.
Investigation example
You are investigating the length of time each member of a class spends on the internet per week.
Look at the time groupings, usually referred to as the .
Do you think they are correct?
These groups are incorrect, because the times of \({10}\) hours and \({20}\) hours can be entered into two different groups.
For example, the time \({10}\) hours can be entered into \(0 \leq time \leq 10\) (where time is less than or equal to \({10}\) hours), and also into \(10 \leq time \leq 20\) (where time is more than or equal to \({10}\) hours).
These groups are also incorrect, as the times \({10}\) hours and \({20}\) hours cannot be entered into any of the groups.
For example, the time \({10}\) hours cannot be entered either into \(0 \textless time \textless 10\) (where time is less than \({10}\) hours), or into \(10 \textless time \textless 20\) (where time is more than \({10}\) hours).
These groupings are correct.
\({10}\) hours is included in the second group, but not the first, and \({20}\) hours is included in the third group, but not the second.
Question
Cameron records the heights of \({30}\) of his classmates (in \({cm}\)).
His results are shown below:
Copy the tally chart and complete it with Cameron's results
Answer
Here is the tally chart showing Cameron's results:
Stem and leaf diagrams
A stem and leaf diagram is a method of organising data so that it can be interpreted more easily.
The data is arranged in rows, ordered from smallest to largest or sometimes from largest to smallest.
Example
The ages of people attending a clinic were as follows:
52 65 78 61 54 59 67 72 50 64 75 77 81 56 62 84 75
Draw a stem and leaf diagram to show this data.
There are 17 ages recorded and these have been put in order from the youngest (50) to the oldest (84).
They are arranged in rows with the first number being the 鈥荣迟别尘鈥 and the other numbers the 鈥榣别补惫别蝉鈥.
Key points
A stem and leaf diagram always has a key.
The values are always in order 鈥 smallest to largest.
Example
What is the median age of the people at the clinic?
52 65 78 61 54 59 67 72 50 64 75 77 81 56 62 84 75
Answer
There are 17 people at the clinic.
Once the ages are ordered in a stem and leaf diagram, it is easy to pick out the middle value (the 9th in order).
The median age is 65.
Question
What is the range of the ages of the 15 people?
Answer
The oldest person is \(84\) and the youngest is \(50\).
The range = largest 鈥 smallest.
\({84~鈥搤 50~=~34}\)
The range is \(34\).
Question
The stem and leaf diagram below shows how much \(16\) pupils spent in the school stationery shop one lunchtime.
a) What is the median amount spent?
b) How many pupils spent between \(\pounds{3}\) and \(\pounds{5}\)?
Answers
a) The median value is the middle value.
As there are \(16\) values, there are two median values, the \({8th}\) and \({9th}\).
These are \(\pounds{4.84}\) and \(\pounds{4.96}\).
We find the mean of these two values.
(\(\pounds{4.84}\) + \(\pounds{4.96}\)) \(\div{2} = \pounds{4.90}\).
b)
Six pupils spent between \(\pounds{3}\) and \(\pounds{5}\).
Back-to-back stem and leaf diagrams
Back-to-back stem and leaf diagrams are used to show and compare data for two different groups.
Example
The stem and leaf diagram below shows the resting heart rates (in beats per minute) of thirteen men and thirteen women from a sports club.
What is the median heart rate for the women?
The median value of \(13\) is the \({7}{th}\) value.
Looking at the women鈥檚 side of the diagram the \({7}{th}\) value is \(68\) counting from the stem and along the leaf.
Question
What is the range of heart rates for the men?
Answer
Looking at the men鈥檚 side of the diagram, the lowest heart rate is \(49\) and the highest is \(76\).
The range is \({76~鈥搤49} = {27}\) beats per minute.
Question
The stem and leaf diagram shows the times taken for \(20\) office workers to drive to work on a particular day.
i) What is the median time taken to drive to work?
a) \(26\) minutes
b) \(26.5\) minutes
c) \(27.5\) minutes
ii) How many office workers took less than 30 minutes to drive to work?
a) \(9\)
b) \(11\)
c) \(10\)
Answers
i) The correct answer is c) \(27.5\) minutes.
ii) The correct answer is b) \(11\).
Question
The stem and leaf diagram below shows the heart rates, in beats per minute, of \(12\) PE students before and after exercise.
i) What is the difference between the median heart rate of the students before and after exercise?
a) \({13.5~bpm}\)
b) \({14.5~bpm}\)
c) \({14~bpm}\)
ii) The range for the heart rates is larger before exercise than after exercise.
a) True
b) False
c) You can't tell
Answers
i) The correct answer is a) \({13.5~bpm}\)
ii) The correct answer is b) False
Test section
Question 1
Following an overnight trip to an outdoor pursuit centre, students were asked:
Look at the three options and decide why this is a poor question.
a) There are no evening activities at the centre
b) It doesn't ask about daytime activities
c) There's no opportunity to say 'None'
Answer
The correct answer is c) There's no opportunity to say 'None'
The options should offer every possible answer to the person responding to the questionnaire.
Question 2
Here's a question that was asked to children at a school:
Why is this a poor question?
a) There's nowhere to say that you are \({10}\) years old
b) There's nowhere to say that you are \({9}\) years old
c) There's nowhere to say that you are \({13}\) years old
Answer
The correct answer is a) There's nowhere to say that you are \({10}\) years old
The section "under \({10}\) years old" doesn't include \({10}\) itself.
Question 3
Use this two-way table to find out how many boys are in the class.
Answer
There are \({16}\) boys in the class.
Question 4
Use this two-way table to find out how many children in the class are left-handed.
Answer
\({3}\) children in the class are left-handed.
Question 5
You need to construct a table to show continuous data going from \({0}\) to \({10}\).
What sign should replace the asterisk *?
\({0}\leq{weight}~\textless~{5}\)
\({5}~{*}~{weight}\leq{10}\)
Should it be replaced with:
a) \(\textless\)
b) \(\leq\)
c) \(\textgreater\)
Answer
The correct answer is b) \(\leq\)
You need to include \({5}\) itself in this group.
Question 6
You need to construct a table to show continuous data going from \({20}\) to \({100}\).
What sign should replace the asterisk *?
\({20}\leq{length}~\textless~{50}\)
\({50}\leq{length}~{*}~{70}\)
\({70}\leq{length}\leq{100}\)
Answer
The correct answer is \(\textless\)
\({70}\) belongs to the next group, so the 'less than' sign is what is needed here.
Question 7
What's wrong with these continuous groups?
\({time}\textless{20}\)
\({20}\leq{time}\leq{50}\)
\({50}\leq{time}~\textless~{70}\)
\({70}\leq{time}\textless{100}\)
\({time}\geq{100}\)
Answer
The sign before \({50}\) in the second group needs to be changed to \(\textless\).
So, it should be \({20}\leq{time}\textless{50}\).
Question 8
Here are the heights of pupils in the class:
\({150},~{167},~{141},~{156},~{160},~{177}\)
\({169},~{166},~{159},~{153},~{163},~{165}\)
\({164},~{171},~{176},~{155},~{161},~{165}\)
How many pupils' heights are in the group \({160}~\textless~{height}\leq{165}\)?
Answer
The correct answer is \({5}\)
The group doesn't include \({160}\), but does include \({165}\).
Question 9
According to the diagram, how many children scored \({40}\) or more in the test?
Answer
There are \({5}\) scores between \({40}\) and \({49}\) but you have to include the scores that are higher than that too.
So the correct answer is \({7}\) children.