Using grouped frequency tables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

Grouped frequency

Estimating the mean

Comparing distributions

Steps

Using grouped frequency tables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

Grouped frequency

Menu Vocabulary Opinion 1

The marks go from 1 to 50 so the range is 50.

The 31-40 group has most people .

You can’t find the range because you don’t know the exact marks.

For the median we are looking for the middle two scores.

Back Forward Cont/d Opinion 2 Answer Opinion 1 Opinion 2 AnswerQ 1 Q 2

Mr Harris has given his class a test. He has given each student a mark out of 50.Here is a tally chart to show his results.

1. Is it possible to find an exact value for the range of the marks from the table?

2. What can you say abut the median and the mode?

Grouped dataThis is used where there are many possible values for the data. It is a convenient way of handling large amounts of data. The possible values for the data are grouped (usually into equal groups). The groups are often called ‘Classes’.

Modal classThe group with the highest frequency is called the modal class.

No, you can only estimate the range.

We do not the exact data values, only which group they are in.For example, the range could be 50 – 1 = 49 at most or 41 –1 0= 31 at least.

Both opinions gave part of the correct answer.

The modal class is 31–40.The 15th and 16th student are both in the 31–40 group. The median is somewhere between 31 and 40.

Using grouped frequency tables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

Variable is the general name for what is being measured.

Continuous variables can take any value.

Menu Vocabulary Opinion 1

A < B means that number A is less than number B.

0 < t ≤ 1 means that t is bigger than zero and also less than or equal to 1.

Back Forward Cont/d Opinion 2 AnswerQ 1

1. What do the symbols < and ≤ mean?Write down some different ways of describing her groups.

Abby is doing a survey on homework.Here are her results from one of the questions on her questionnaire.

< stands for ‘is less than’ and we read it from left to right.≤ stands for ‘is less than or equal to’.Putting them together with a variable in between shows exactly the range for that group. In Abby’s survey the variable is t which is a continuous variable. t is the number of hours spent on homework.3< t ≤ 5 can be read as:• t is more than 3 and less than or equal

to 5• t can have any value from 3 to 5 but not

including 3• t must be more than 3 and not more

than 5.

Grouped frequency

Using grouped frequency tables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

Discuss different numerical variables which you might measure in a survey. Decide whether they are discrete or continuous.

Menu Vocabulary Opinion 1

Abby’s data is hours and the other is days.

The absences can only be whole numbers.

Back Forward More Opinion 2 AnswerQ 1

1. Compare Abby’s data with the marks for Mr Harris’s class. What is the difference between the types of data?

Abby is processing the data from her survey on homework.Here are her results from one of the questions on her questionnaire.Abby has written the frequency of each response in each box.

Grouped frequency

Discrete variables can only take particular values.

The number of absences and hours are both examples of variables.The number of absences can only take whole numbers. This a discrete variable. The number of hours can take any value. This is a continuous variable.

Using grouped frequency tables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

She needs to find the total number of pupils.

She could use the middle number in each group.

Do we divide by 6 because there are 6 groups?

Estimating the mean

Menu Vocabulary Opinion 1

Would it help to show the working in extra columns?

Back Forward Opinion 2 Answer Opinion 1 Opinion 2 AnswerQ 1 Q 2

1. Ms Shah needs to report the mean number of absences. She asks some pupils to work it out. Can you help them?

2. Now do the calculation for Ms Shah. Make your answer a sensible estimate.

Ms Shah is head of Year 8 at Hodder High School. She made this table showing pupils’ absences for one term.

Days absent Frequency 0–4 25 5–9 38

10–14 1615–19 420–24 225–29 1

Total  

Mid-interval valuesThese are the central values for each group. They are used to find approximate totals for each group.

Cont/d

She needs to find the total frequency. She can find an estimate of the mean by pretending that everyone scored the halfway number in the group: 2, 7, 12, 17, 22, 27. Both opinions are good.

Estimate for the mean = 647 ÷ 86= 7.523256.

This is not a sensible estimate. The number of days absent can only be a whole number.A sensible estimate is 8 days.Opinion was incorrect. You must divide by the number of pupils.

Days absent Frequency Mid-interval

valuesMidpoints × frequency Answer

0–4 25 2 2 × 25 50

5–9 38 7 7 × 38 266

10–14 16 12 12 × 16 192

15–19 4 17 17 × 4 68

20–24 2 22 22 × 2 44

25–29 1 27 27 × 1 27

Total 86 Total 647

Using grouped frequency tables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

Menu Opinion 1Back Forward Opinion 2 AnswerQ 1 Q 2

Abby is processing the data from her survey on homework.Abby has written the information in a table and has begun to calculate an estimate for the mean.Part of her calculations are shown below.

1. Complete Abby’s working for the estimated mean.

It worked out exactly so it is good.

There are 60 minutes in an hour so it is not sensible to give a decimal answer.

2. The estimated mean was given as 1.4 hours. Is that a sensible answer? If not suggest a more suitable answer.

Estimating the mean

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The mid-interval values are 1.5, 2.5, 3.5.

The table makes it easier to show the calculations.

Opinion is correct. The answer of 1 hours would be more suitable.

Opinion 1 Opinion 2 Answer

Mid-interval value = 1.5 11 × 1.5 = 16.5Mid-interval value = 2.5 5 × 2.5 = 12.5Mid-interval value = 3.5 2 × 3.5 = 7Total = 42

Mean = 42 ÷ 30 = 1.4 hours.

Both opinions are good.

Using grouped frequency tables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

Comparing distributions

Menu Opinion 1Back More Opinion 2 AnswerQ 1

1. Estimate the mean height and the modal class for each set of seedlings. Which compost performs better?

A botanist wants to compare two different seed composts, COMPO and GRO-RITE.150 seeds are sown in each compost and after three weeks, the heights of the seedlings which have germinated are measured.This table shows the heights, h mm, of each set of seedlings.

We need to find the mid-way values for each group; 5,10,15 etc.

There is no modal class because all the classes are the same size.

Ask your teacher to obtain the handspan measurements for an older or younger class. Calculate the mean and compare the results for the two classes.

Mean= 1930 ÷ 121 = 15.95 = 2285÷133 = 17.18Estimated meanCOMPO 16 mm GRORITE 17 mmModal classCOMPO 20–30 mm GRO-RITE 30–40 mm(The modal class has the highest frequency so opinion is wrong).

COMPO GRO-RITE

Height (7h)Mid-way value f h × f Height (h)

Mid-way value f h × f

0 < h ≤ 10 5 6 30 0 < h ≤ 10 5 18 9010 < h ≤ 20 10 23 230 10 < h ≤ 20 10 16 16020 < h ≤ 30 15 45 675 20 < h ≤ 30 15 20 30030 < h ≤ 40 20 36 720 30 < h ≤ 40 20 48 96040 < h ≤ 50 25 11 275 40 < h ≤ 50 25 31 775

Totals 121 1930 Totals 133 2285

Using grouped frequency tables – Developing Understanding Mastering Mathematics © Hodder and Stoughton 2014

Editable Teacher Template

Menu Vocabulary Opinion 1Back Forward More Opinion 2 Answer Opinion 1 Opinion 2 AnswerQ 1 Q 2

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