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Steps Grouped frequency Estimating the mean Comparing distributions Mastering Mathematics © Hodder and Stoughton 2014 Using grouped frequency tables – Developing Understanding Grouped frequency 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? The marks go from 1 to 50 so the range is 50. You can’t find the range because you don’t know the exact marks. The 31-40 group has most people . For the median we are looking for the middle two scores. Menu Back Forward Cont/d Vocabulary Mastering Mathematics © Hodder and Stoughton 2014 Q1 data No, can Grouped only estimate Bothyou opinions gave part of the the range. correct This is used where there are many answer. possible values for the data. It is aonly We do not the exact data values, convenient way handling large which groupclass theyofisare in. The modal 31–40. amounts of data. possible values For theth The range could be 50in– 1 The example, 15th and 16 student are both for the are grouped (usually into = 49 at data most or 41 –1 0= 31 atisleast. the 31–40 group. The median equal groups). The groups are often somewhere between 31 and 40. called ‘Classes’. Modal class The group with the highest frequency is called the modal class. Opinion 1 Opinion 2 Answer Q2 Opinion 1 Opinion 2 Answer Using grouped frequency tables – Developing Understanding Grouped frequency Abby is doing a survey on homework. Here are her results from one of the questions on her questionnaire. 1. What do the symbols < and ≤ mean? Write down some different ways of describing her groups. 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. Menu Back Forward Cont/d Vocabulary Mastering Mathematics © Hodder and Stoughton 2014 Q1 < stands for ‘is less than’ and we read it from left to right. Variable is the general name for what is ≤being stands for ‘is less than or equal to’. measured. Putting them together with a variable in between shows exactly thetake range that Continuous variables can anyfor value. 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. Opinion 1 Opinion 2 Answer Using grouped frequency tables – Developing Understanding Grouped frequency 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. 1. Compare Abby’s data with the marks for Mr Harris’s class. What is the difference between the types of data? Abby’s data is hours and the other is days. The absences can only be whole numbers. Discuss which Discretedifferent variablesnumerical can only variables take particular you might measure in a survey. Decide values. whether they are discrete or continuous. 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. Menu Back Forward More Vocabulary Mastering Mathematics © Hodder and Stoughton 2014 Q1 Opinion 1 Opinion 2 Answer Using grouped frequency tables – Developing Understanding Estimating the mean Ms Shah is head of Year 8 at Hodder High School. She made this table showing pupils’ absences for one term. 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. Shewe needs to find Do divide by 6 the totalthere number because are of 6 pupils. groups? Would it helpuse to the show She could themiddle working in extra number in each group. columns? She needs to Mid-interval find the total values frequency. She can These find an are estimate the central of thevalues mean for by pretending each group.that They are used everyone scored to the findhalfway approximate number totals in the for each group. group: 2, 7, 12, 17, 22, 27. Both opinions are good. Menu Back Forward Cont/d Vocabulary Mastering Mathematics © Hodder and Stoughton 2014 Q1 Days absent Frequency Days Mid-interval Midpoints × 0–4 25 Frequency Answer absent values frequency 5–9 38 0–4 25 2 × 25 50 10–14 2 16 5–9 38 7 × 38 266 15–19 7 4 10–14 16 12 × 16 192 20–24 12 2 15–19 4 17 × 4 68 25–29 17 1 20–24 22 22 × 2 44 Total2 25–29 1 Total 86 27 27 × 1 Total 27 647 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. Opinion 1 Opinion 2 Answer Q2 Opinion 1 Opinion 2 Answer Using grouped frequency tables – Developing Understanding Estimating the mean 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. 2. The estimated mean was given as 1.4 hours. Is that a sensible answer? If not suggest a more suitable answer. The midIt worked interval out exactly values so it is are 1.5, good.2.5, 3.5. Menu Back Forward Mid-interval value = 1.5 Mid-interval value = 2.5 Mid-interval value = 3.5 Total = 42 Mean = 42 ÷ 30 = 1.4 hours. Both opinions are good. Opinion is correct. The answer of 1 hours would be more suitable. The table it There are makes 60 easier to in show minutes an the calculations. hour so it is not sensible to give a decimal answer. More Mastering Mathematics © Hodder and Stoughton 2014 11 × 1.5 = 16.5 5 × 2.5 = 12.5 2 × 3.5 = 7 Q1 Opinion 1 Opinion 2 Answer Q2 Opinion 1 Opinion 2 Answer Using grouped frequency tables – Developing Understanding Comparing distributions Ask your teacher to obtain the handspan GRO-RITE measurements for an older or younger class. Mid-way Mid-way Height (7h) value the f Height (h) the value results f h×f h×f Calculate mean and compare 0 < h ≤ 10 6 30 0 < h ≤ 10 5 18 90 for the two5 classes. 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. COMPO 10 < h ≤ 20 20 < h ≤ 30 30 < h ≤ 40 40 < h ≤ 50 10 15 20 25 Totals 23 45 36 11 121 230 675 720 275 1930 Mean = 1930 ÷ 121 = 15.95 1. Estimate the mean height and the modal class for each set of seedlings. Which compost performs better? 10 < h ≤ 20 20 < h ≤ 30 30 < h ≤ 40 40 < h ≤ 50 10 15 20 25 Totals 16 20 48 31 133 160 300 960 775 2285 = 2285÷133 = 17.18 Estimated mean COMPO 16 mm GRORITE 17 mm Modal class COMPO 20–30 mm GRO-RITE 30–40 mm (The modal class has the highest frequency so opinion is wrong). 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. Menu Back More Mastering Mathematics © Hodder and Stoughton 2014 Q1 Opinion 1 Opinion 2 Answer Using grouped frequency tables – Developing Understanding Editable Teacher Template Information Vocabulary 1. Task – fixed More 2. Task – appears Q1 Opinion 1 Q1 Opinion 2 Q1 Answer Q2 Opinion 1 Q2 Opinion 2 Q2 Answer Menu Back Forward More Vocabulary Mastering Mathematics © Hodder and Stoughton 2014 Q1 Opinion 1 Opinion 2 Answer Q2 Opinion 1 Opinion 2 Answer Using grouped frequency tables – Developing Understanding