GCSE 2010SP h (part) Calculate median, mean,…

mode

SP l (part) Compare distributions and

make inferences

SP u Use calculators effi ciently and

effectively, including statistical functions

N a (part) Add, subtract, multiply and

divide any number

N q Understand and use number

operations and the relationships

between them, including inverse

operations and hierarchy of operations

N u (part) Approximate to specifi ed

or appropriate degrees of accuracy

including a given power of ten, number

of decimal places and signifi cant fi gures

N v Use calculators effectively and

effi ciently

A a Distinguish the different roles played

by letter symbols in algebra, using the

correct notation

A b Distinguish in meaning between the

words ‘equation’, ‘formula’, ‘identity’

and ‘expression’

FS Process skillsRecognise that a situation has

aspects that can be represented using

mathematics

Use appropriate mathematical

procedures

Interpret results and solutions

FS PerformanceLevel 1 Apply mathematics in an

organised way to fi nd solutions to

straightforward practical problems for

different purposes

Specifi cation

ResourcesLength of string about 1.2 metres long,

tape measure

Linkshttp://nrich.maths.org/public/search.

php?search=average

http://nrich.maths.org/6267

ActiveTeach resourcesMental calculation quiz 1

Traffi c fl ows 1 video

Resources

2.1 Understanding and using numbers 2.2 Finding the mode and median 2.3 Algebra 2.4 Calculating the mean 2.5 Using the three types of average 2.6 Using calculators

Concepts and skills •• Calculate mean, mode and median. •• Recognise the advantages and disadvantages between measures of average. •• Calculate the mean of a small data set, using the appropriate key on a scientifi c calculator. •• ∑x and ∑ fx or the calculation of the line of best fi t. •• Add, subtract, multiply and divide whole numbers, negative numbers, integers, fractions

and decimals. •• Multiply and divide by any number between 0 and 1. •• Multiply and divide numbers using the commutative, associative and distributive laws

and factorisation where possible, or place value adjustments. •• Use inverse operations. •• Use brackets and the hierarchy of operations. •• Estimate answers to calculations, including use of rounding. •• Enter a range of calculations including those involving time and money. •• Know how to enter complex calculations, •• Understand and interpret the calculator display. •• Understand that premature rounding can cause problems when undertaking

calculations with more than one step. •• Calculator functions include +, –, ×, ÷, x2, √x, memory, x y, x1/y, and brackets. •• Use notation and symbols correctly. •• Write an expression. •• Select an expression/identity/equation/formula from a list.

Functional skills •• L1 Find mean and range.

Prior key knowledge, skills and concepts •• Students should already know how to add, divide and order numbers.

Starter •• Begin with the following: How long is a piece of string? (The answer is easy – it’s twice

as long as half a piece of string.) Here is a piece of string; write down, to the nearest cm, how long you think this is. Do not compare answers. Record the students’ estimates.

Main teaching and learning •• Tell the students that they are going to look at how we can summarise a set of data. •• Start by looking at the averages of the students’ guesses for the length of the piece of

string. Tell the class there are three possible averages. (Measures of central tendency). •• Arrange the students’ estimate values in order. •• Which value is most common? Explain that this number is called the mode. Tell students

that there may not always be a mode; this often happens with small samples. •• Which value is in the middle? There will be one value if there is an odd number of

values, but two if there is an even number of values. Explain that if there are two values you add them together and divide by two. In either case the value is called the median.

•• If we all had guessed the same, what would this value be if the totals of the numbers were to be the same? Tell students that you need to add all the numbers and divide by how many people there are. This is called the mean.

Enrichment •• Measure the string and compare it with the estimated mean, median and mode. Which

of the averages best describes the guesses? Discuss the circumstances under which each measure would be used.

Plenary •• Give three different sets of numbers where the median is 8 and the range is 5.

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algebraic expression average BIDMAS cube root decimal point directed numbers equation formula identity mean median mode negative number positive number power square root16

111111

GCSE 2010SP h (part) Calculate median, mean,…

mode

SP u Use calculators effi ciently and

effectively, including statistical functions

FS Process skillsUse appropriate mathematical

procedures

FS PerformanceLevel 2 Use appropriate checking

procedures and evaluate their

effectiveness at each stage

Specifi cation

CD ResourcesPowerPoint 2.7 Using frequency tables

to fi nd averages

Resources

2.7 Using frequency tables to fi nd averages

Concepts and skills

•• Calculate mean, mode and median.

•• Calculate the mean of a small data set, using the appropriate key on a scientifi c calculator.

•• ∑x and ∑£x… .

Functional skills

•• L2 Use and interpret statistical measures, tables and diagrams, for discrete and continuous data… .

Prior key knowledge, skills and concepts

•• Students should already have basic number skills.

Starter

•• What would you call an eight-foot-high, thickset person covered in long hair – a Wookiee, perhaps? Who has heard of a creature called a yeti or abominable snowman? Explain that no one has seen one of these creatures, but large footprints have been seen in the snow on mountains.

Main teaching and learning

•• Tell students there have been large footprints seen in the snow on the school fi eld, so they are going to look at shoe sizes.

•• Fill in the frequency of each size of shoe in the class on PowerPoint 2.7, slide 1.

•• Tell the students that PowerPoint 2.7, slide 2 shows the frequencies for a larger sample of shoes.

•• What size of shoe has the highest frequency? (8) Explain that this number is called the mode.

•• Which number is in the middle? Explain that the total frequency is 60, so the middle two will be the 30th and the 31st values. There are 27 values that are size 7 or less and 45 that are size 8 or less. The 30th and 31st values will both be size 8. This value is the median.

•• Now demonstrate how to fi nd the mean. Explain that there are 4 people with size 5 shoes, so these added together come to 5 + 5 + 5 + 5 = 4 × 5 = 20. To fi nd the total for each size you multiply the size, x, by the frequency, f. Show students the extra column that has been added to the frequency table on PowerPoint 2.7, slide 3. ∑ f × xThe mean = ––––– . ∑ f

Enrichment

•• Ask students to fi nd the mode, median and mean of the class’s shoe sizes. Discuss which of the measures best describes the shoe size of the class. Is there any evidence as to who has made the large footprints?

Plenary

•• Divide the class into two groups (e.g. boys and girls). Draw a tally chart showing number of children in a family. Ask each member of one group for the number of children in their family. Complete the frequency table. Repeat with a new frequency table for the second group.

For each group answer the following:(a) What is the range?

(b) What is the total number of children in the families?

(c) What are the median and the mean?

(d) What is the mean of the two groups combined? (A common error will be to give the mean of the two means, which is only true if the group sizes are equal.)

frequency table

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GCSE 2010SP h (part) Calculate median, mean,…

mode and modal class

FS Process skillsUse appropriate mathematical

procedures

FS PerformanceLevel 2 Use appropriate checking

procedures and evaluate their

effectiveness at each stage

Specifi cation

CD ResourcesPowerPoint 2.8 Grouped data

Linkshttp://www.statistics.gov.uk/StatBase/

Expodata/Spreadsheets/D3687.xls

Resources

2.8 Modal class and median of grouped data

2.9 Estimating the mean of grouped data

Concepts and skills

•• Calculate modal class and the interval which contains the median. •• Estimate the mean of grouped data using the mid-interval value. •• Find the median… for large data sets with grouped data. •• Estimate the mean for large data sets with grouped data. •• Understand that the expression ‘estimate’ will be used where appropriate, when fi nding

the mean of grouped data using mid-interval values.

Functional skills

•• L2 Use and interpret statistical measures, tables and diagrams, for discrete and continuous data…

Prior key knowledge, skills and concepts

•• Students should already have basic number skills.

Starter

•• Discuss the things students have to cover with their mobile phone charges (texts, calls to other mobiles, etc). Charges are likely to vary according to what they have to cover. Ask the students to write down the monthly phone charges they pay to the nearest pound.

Main teaching and learning

•• Create a grouped frequency table for phone charges by fi lling in column 1 on PowerPoint 2.8, slide 1.

•• Ask students to come up and fi ll in the tally and frequency columns on PowerPoint 2.8, slide 1.

•• Ask students to identify which class has the highest frequency. Explain that this is the modal class.

•• Can we fi nd the median value? Explain that, without the individual values, we can only fi nd the class interval in which the median lies.

∑x •• Show students how to work out ––– . Explain that this is the middle value. Ask students

2to identify in which class this lies. Tell them that this is the class in which the median lies.

•• Can we fi nd an exact value of the mean? Explain that we can actually only fi nd an estimate for the mean.

•• Complete the table by fi lling in the remaining columns. ∑ fx

•• Now show students how to fi nd an estimate of the mean by working out: ––– . ∑ f

Enrichment

•• Ask students to collect data on house prices in the local area. They could perhaps restrict the data they collect to terraced houses. Local papers can give this information. They can then create a grouped frequency table and fi nd the modal class, class into which the median lies and an estimate of the mean price.

Plenary

•• The table shows the number of visitors under 21 to a sweet shop in one day.

Age (years) 1 to 5 6 to 10 11 to 15 16 to 20

Frequency 20 46 58 29

(a) How many people visited altogether? (153)(b) Which is the modal group? (11 to 15 inclusive)(c) Which is the median class? (11 to 15)(d) Joan says that an estimate for the age is 21.5. Why must this answer be wrong?

(Greater than the highest value.)

modal class

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GCSE 2010SP h (part) Calculate … range, quartiles

and interquartile range

SP l Compare distributions and make

inferences

FS Process skillsUse appropriate mathematical

procedures

Interpret results and solutions

FS PerformanceLevel 2 Apply a range of mathematics to

fi nd solutions

Specifi cation

CD resourcesResource sheet 2.10

PowerPoint 2.10 Quartiles, range and

interquartile range

Linkshttp://www.metoffi ce.gov.uk/education/

teachers/historic_weather_data.html

ActiveTeach resourcesThe Extreme 2 video

Median quiz

Interquartile range animation

RP KC Averages and range knowledge

check

RP PS Averages and range problem

solving

Resources

2.10 Range, quartiles and interquartile range

Concepts and skills

•• Calculate range.

•• Find the… quartiles and interquartile range for large data sets….

•• Compare distributions and make inferences, using … measures of spread, including …quartiles.

•• Compare the mean and range of two distributions, or median and interquartile range, as appropriate.

Functional skills

•• L2 Use and interpret statistical measures …

Starter

•• Discuss with students recent weather and climate change. It is possible to get historical information about the weather for various parts of the country dating back to 1853 from the Met Offi ce website.

Main teaching and learning

•• Show students Resource sheet 2.10, which provides weather information about an area of England. Look at the maximum temperature for each of the 11 months.

•• Now explain to the students that they are going to summarise this statistical information about the maximum temperature. Explain that the fi rst step is to take the temperatures and put them in order. Show students PowerPoint 2.10, slide 1, which has the maximum temperature data unordered and then ordered.

•• Explain that the highest temperature minus the lowest temperature is called the range. Show students PowerPoint 2.10, slide 2.Now explain to the students that they can divide the numbers into four parts.

n + 1You know how to fi nd the median value: remember the median is the –––– th value. 2In this case n = 11. This is also called the second quartile (Q2). Show students PowerPoint 2.10, slide 2. n + 1We can fi nd the lower quartile (Q1). We fi nd this by working out the –––– th value. 4 In this case the 3rd value. Show students PowerPoint 2.10, slide 3. 3(n + 1)We can fi nd the upper quartile (Q3). We fi nd this by working out the ––––––– th value. 4In this case the 9th value. Show students PowerPoint 2.10, slide 3. The interquartile range is Q3 – Q1. Show students PowerPoint 2.10, slide 3.

Enrichment

•• Ask students to use the Met Offi ce site to fi nd weather data for their local area.

Plenary

•• What are the range, median, lower quartile, upper quartile and interquartile range for these numbers? 1, 2, 4, 5, 6, 6, 7, 8, 8, 11, 12 (11, 6, 4, 8, 4)

•• What are the positions of the upper and lower quartiles of 99 pieces of data listed in order? (75th and 25th values)

•• The data shows the number of days that some students in Years 10 and 11 were absent from school last year. Compare the mean and range of the two sets:

4 6 3 2 7 1 3 2 5 3 4 1 6 2 4 (Mean is 3.5; range is 6.)

5 1 0 2 8 2 4 3 6 3 3 5 6 2 1 (Mean is 3.4; range is 8.)

distribution interquartile range lower quartile quartiles range upper quartile

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Resource sheet 1 2.10The resource sheet table gives some information about the weather in an area of the United Kingdom.

Month Max Temp

(deg C)

Min Temp.

(deg C)

Rain

(mm)

Sunshine

(hrs)

Jan 10 5 73 80

Feb 10 4 81 67

March 12 4 37 165

April 18 5 2 210

May 17 7 135 165

June 20 11 79 150

July 21 12 38 209

Aug 21 12 38 210

Sep 19 10 17 142

Oct 15 8 70 100

Nov 11 4 56 87

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