29. [Statistics]

MM5.2 11 22 33 44 Skill 29.1 Interpreting data in column or bar graphs (1). MM6.1 11 22 33 44

Q. How many of the cost of living items are A. 3 Find the New Zealand bars in the more expensive in New Zealand than the graph. USA? Cost of Living Measure which ones are longer 1 L Milk than their USA equivalents.

1 kg Cheese Cost of Living 1 L Milk 1 way transport ticket 1 kg Cheese 1 L Petrol 1 way transport ticket Internet 2 Mbps ADSL 1 L Petrol

Cheap Restaurant Meal Internet 2 Mbps ADSL

Taxi 5 km City Cheap Restaurant Meal

0 1020304050 Taxi 5 km City AU$ 0 1020304050 USA NZ AU$ USA NZ

a) Of the trophies listed below, which sport has b) In which year did Australian car ownership the 3rd highest trophy? first exceed 50%?

Ownership of passenger vehicles - Australia Trophy heights 600 FIFA World Cup - Soccer 500 Ryder Cup - Golf 400 Wimbledon Gentlemen's - Tennis Singles Trophy 300 Davis Cup - Tennis 200 Nextel Cup - NASCAR Cars/1000 of population Cars/1000 100 0 20 40 60 80 100 120 0 Height (cm) 19991998199719961995 200520042003200220012000

c) How many of the countries listed below d) Which Australian ski resort listed below has had retail music sales in 2007 greater than an area closest to 400 hectares? US$3000 million? Value Of Total Music Retail Sales 2007 (physical & digital) Australian Ski Resorts - Skiable area 1400 AUSTRALIA 1200 CANADA 1000 GERMANY 800

FRANCE Area (hectares) 600 JAPAN 400 MEXICO 200 0 UK Selwyn USA Thredbo Mt Buller Mt HothamFalls Creek Perisher Blue Mt Baw Baw 0 2000 4000 6000 8000 10 000 12 000 Charlotte Pass Million US$

page 349 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.1 Interpreting data in column or bar graphs (2). MM6.1 11 22 33 44 e) In which year were the number of heart and f) Which city has the most wet days compared to lung transplants most similar? the amount of rainfall?

Organ transplants in Australia 2500 Rainfall in world cities 250 120 2000 200 100 1500 150 80 Av rainfall (mm) rainfall Av Av no. of wet days of wet no. Av

No. of transplants No. 1000 100 60

40 500 50

20 0 0 York Sydney Tokyo 0 London New 2001200019991998 20062005200420032002 2010200920082007 Hong Kong Melbourne Kuala Lumpur heart lung

g) Of the countries shown below which have an h) What percentage of multiple job employees increasing population in both rural and urban work 4 days a week? areas? Annual rate of Population Change (%) Australia Patterns Of Work - Australia 70 Italy 60 China 50 UK 40

USA (%) employees 30 NZ 20

Malaysia 10 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 654321 7 Rural days worked Urban single job multiple jobs

i) What percentage of Australia’s shots on goal j) Which of the African nations shown is the in the 2010 FIFA world cup were actually most densely populated? [Hint: people/area] converted to goals? Area & Population for some African Nations

1200 1200 2010 FIFA World Cup MP 1000 1000 matches played GS 800 800 goals scored GA 600 600 goals against

Area (’000 sq km)Area 400 400 G

shots on goal (millions) Population 200 200 12 10 8 6 4 02 012 4 6 8 10 2 Australia New Zealand 0 0 Cameroon Ethiopia Kenya Nigeria Tanzania

% page 350 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.2 Interpreting data in stack graphs. MM6.1 11 22 33 44 • Read such that each piece of the bar represents a percentage or proportion of the total. Example: 50% 25% 25%

Q. In which of the years shown was the price of A. 1984

bread in Australia closest to $1.00? Changing costs of bread and milk 4.00 Changing costs of bread and milk 4.00 3.50 Compare for 3.50 3.00 each bar

Price (AU$) 3.00 2.50 $1.00 2.50 2.00 2.00 1.50 $1.00 $1.00 1.50 1.00 $1.00 1.00 $1.00 0.50 0.50 0 Find the distance 1974 1984 1994 2004 0 1974 1984 1994 2004 that $1 extends Bread Milk Bread Milk

a) In which Victorian wine region does the b) In which country do mobile sales comprise amount of white wine produced equal 5%? approximately 73% of digital sales?

Victoria’s Wine Regions Digital Music Markets - Sales by channel Glenrowan USA Mornington Peninsula Japan

Gippsland Germany Australia Macedon China Rutherglen UK 0 20406080100% Canada White wine 0204060 80 100 % Red wine Online Mobile

c) In which age bracket did the highest d) Between 2000 and 2010, Roger Federer had percentage of respondents spend 3 to 4 hours played in all 4 grand slam tournaments each on habitual physical exercise? year. How many grand slam tournaments has Roger Federer won in that time? Exercise habits of Japanese males Age (years) [Hint: The grand slams are knock-out tournaments.] 70 - 79 Grand Slam performances - Roger Federer 30

60 - 69 25

50 - 59 Number of matches 15 40 - 49 10 0 20406080100 (% of males surveyed) 5 Hours spent on habitual physical exercise 0 <1 1 - 2 3 - 4 5+ 200420032002200120001999 201020092008200720062005 Year wins losses

page 351 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.3 Interpreting data in line graphs (1). MM6.1 11 22 33 44

Q. How far into the race was the Belarus team A. 5 mins + 10 seconds Convert to seconds after 5 minutes and 10 seconds? = 5 × 60 + 10 Women’s World ‘06 Double Scull: Belarus 6th = 310 2000 Women’s World ‘06 Double Scull: Belarus 6th 2000

1500 1500

1000 1000 Distance (metres) Distance (metres) 500 500 0 0 50 100 150 200 250 300 350 400 450 0 Time (seconds) 0 50 100 150 200 250 300 350 400 450 Time (seconds) 1500 m

a) In which year between 1979 and 2007 was b) Using the data below, between which 2 years the highest percentage of Australians did the percentage of unemployed Australians unemployed? increase most?

Australia - % unemployed Australia - % unemployed 12 12 11 11 10 10 9 9 percentage (%) percentage 8 (%) percentage 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 1980 1985 1990 19952000 2005 2010 1980 1985 1990 19952000 2005 2010 year year

c) In which of the years shown has there been the d) At what approximate speed, in m/s, did the greatest difference in marrying age of men and New Zealand team row? women in the USA? A) 0.5 m/s B) 2 m/s C) 5 m/s D) 10 m/s

Median age at first marriage - USA Women’s World ‘06 Double Scull: NZ 3rd 30 2000

1500 Age - years 25 1000 Distance (metres) 500 20 0 0 50 100 150 200 250 300 350 400 450 15 Time (seconds) 20001990198019701960195019401930192019101900

Males Females ......

...... page 352 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.3 Interpreting data in line graphs (2). MM6.1 11 22 33 44 e) How far into the marathon was Deena Kastor f) Approximately how much longer did it take when she began to recover from stomach Atsede than Tola to reach the 30 km mark cramps? of the 2010 Paris marathon?

2007 Boston Marathon (111th) 2010 Paris Marathon 42.195 50 40 42.195 40 30

distance (km)distance 30

Distance (km)Distance 20 20

10 10

0 0 0 33:20 1:6:40 1:40 2:10 2:46:20 0 30 60 90 120 150 Time (h:min:s) 1st: Lidiya Grigoryeva (RUS) Male winner: Tola Tadesse (ETH) 2:06:37 Time (min) Female winner: Atsede Bayisa (ETH) 2:22:02 5th: Deena Kastor (USA) km min

g) During which 5-year interval did Australia’s h) What is the cost to park for 3.5 hours? crude birth rate go most against the trend? Brisbane Central Casual Parking Rates 20 Crude Birth Rates 40 Cost ($) Cost 30 15 20 Births/ 000 people/yr

10 10

0 1975 1980 1985 1990 1995 2000 2005 2010 02143657 8 10 24 Time (h) New Zealand USA Australia Germany

i) Since 2005, which type of road user has the j) Which Australian household type had the pecentage change in road deaths most similar greatest % change in living costs for the to the ‘all road users’ average? March 2010 to June 2010 quarter? Annual Road Deaths in Australia (as a % of the number of road deaths for the year preceding Jan 2005) 7 Changes in living costs 150% 6 (Australia) 5 % change 125% 4 3 Motor cyclists 2 100% all road users Drivers 1 Cyclists 0 75% −1 Mar-06 Sep-06 Mar-07 Sep-07 Mar-08 Sep-08 Mar-09 Sep-09 Mar-10 Sep-10 Jun-06 Dec-06 Jun-07 Dec-07 Jun-08 Dec-08 Jun-09 Dec-09 Jun-10 50% Jan 05 Jan 06 Jan 07 Jan 08 Jan 09 Jan 10 Self-funded retiree Age Pensioner Employee page 353 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.4 Interpreting data in pie charts. MM6.1 11 22 33 44 • Pie graphs are circular. Consider each section of the graph as a piece of the pie. Hint: Each piece of pie represents a percentage of the total. Example: 25% 50% 75% 100%

Q. Which of the years shown had the largest A. Compare the relative sizes of the sectors proportion of liver transplants? (pieces of the pie charts) for both years. Patient transplants by organ kidney 1998 2006 liver liver lung heart 1998 heart/lung 1998 2006 pancreas

a) Of the European countries listed below which b) In the Concise Oxford Dictionary, the chance had the third highest number of international of a vowel being a “U” is closest to: tourist arrivals? A) 5% B) 10% C) 15% D) 25% International tourist arrivals U Switzerland A Frequency of vowels - Concise Oxford Dictionary UK O Spain Italy I E Germany France Italy d) Which player shown below had the highest c) Which blood type accounts for closest to percentage of their match statistics as marks? 10% of the population? 2010 AFL match statistics kicks Who has which Blood Type? handballs O marks A tackles B goals behinds AB Dane Swan Chris Judd

e) Which state or territory saw a reduction in the f) Which country spends approximately 25% of number of deaths in the year to Jan 2010 its federal budget on defence? compared to the previous year? Australia’s Road deaths NSW ACT ACT VIC NSW FEDERAL BUDGETS FOR 2011 QLD NT VIC Health

NT SA QLD Defence Pensions & Welfare WA TAS SA TAS WA Other Australia USA Year to January 2009 Year to January 2010

page 354 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.5 Calculating the median of sets of data. MM6.1 11 22 33 44

Median (middle value) Set of data (even): 5, 1, 5, 3, 2, 1, 5, 2 • Write all the values in order. Ordered set: 1, 1, 2, 2, 3, 5, 5, 5 • Odd numbered set - middle value. 23+ 5 Median 25. Even numbered set - average of the 2 middle 2 ==2 values.

Q. Calculate the median of this set of data: A. 1, 2, 2, 3, 3, 3, 4, 4, 6, 8, 8, 9 order values 3, 3, 4, 2, 3, 2, 4, 6, 1, 9, 8, 8 34+ 7 find middle value Median: = = 3.5 2 2

a) Calculate the median of this set of data: b) Calculate the median of this set of data: 3, 4, 8, 5, 2, 4, 3, 6, 7 1, 3, 4, 4, 5, 2, 6, 1, 7, 9, 4 order values find middle value 2, 3, 3, 4, 4, 5, 6, 7, 8 ......

9 values so 5th value is the middle 4 ......

c) Calculate the median of this set of data: d) Calculate the median of this set of data: 1.2, 4.1, 3.2, 3, 4.1, 2.3, 2, 3.1, 2 5, 2, 3, 7, 8, 4, 6, 4

......

e) Calculate the median of this set of data: f) Calculate the median of this set of data: 12, 12, 11, 10, 11, 13, 12, 15, 12 1, 3, 1, 4, 4, 4, 2, 3, 4, 5, 2, 3 order values find middle value 10, 11, 11, 12, 12, 12, 12, 13, 15 ......

......

g) Calculate the median of this set of data: h) Calculate the median of this set of data: 2, 2, 2, 2.5, 3.5, 3.5, 4, 4.5 9, 10, 11, 10, 15, 11

......

...... page 355 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.6 Calculating the mode and range of sets of data. MM6.1 11 22 33 44

Mode (most common value) Set of data: 5, 1, 5, 3, 2, 1, 5, 2 Range Ordered set: 1, 1, 2, 2, 3, 5, 5, 5 • Write all the values in order. Mode 5 • Subtract the lowest value from the highest value. Range 5 − 1 = 4 Hint: A set of data can have more than one mode, if two or more values repeat the same number of times.

Q. Calculate the mode and range of this A. 1, 2, 2, 3, 3, 3, 4, 5, 5, 6, 8 order values set of data: Mode: 3 The value 3 is in the 1, 2, 3, 3, 4, 5, 2, 6, 8, 5, 3 Range: 8 − 1 = 7 set 3 times difference between highest and lowest

a) Calculate the mode of this set of data: b) Calculate the mode of this set of data: 2, 21, 21, 15, 16, 15, 21 3, 2, 2, 4, 5, 6, 7, 4, 5, 2, 5, 3, 4, 2 21 The value 21 is in the set 3 times

c) Calculate the mode of this set of data: d) Calculate the mode of this set of data: 18, 21, 20, 18, 22, 18, 20, 21, 22 102, 99, 98, 100, 101, 98, 102, 98

e) Calculate the range of this set of data: f) Calculate the range of this set of data: 12, 14, 16, 14, 15, 13 19, 22, 23, 15, 12, 16, 13, 15, 24, 14, 17, 18

16 − 12 = ......

g) Calculate the mode and range of this set of h) Calculate the mode and range of this set of data: data: 3, 5, 4, 8, 5, 6, 8, 6, 4, 7, 4, 7, 8, 4 31, 32, 35, 32, 34, 29, 30, 31, 33, 32

...... mode = range = mode = range =

i) Calculate the mode and range of this set of j) Calculate the mode and range of this set of data: data: 2.8, 3.1, 3.5, 3.6, 3.6, 4, 4.2, 4.5, 4.7, 4.9 14, 18, 19, 19, 24, 23, 29, 18, 28, 19

...... mode = range = mode = range = page 356 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.7 Calculating the mean of sets of data. MM6.1 11 22 33 44 Mean (or average) Set of data: 5, 1, 5, 3, 2, 1, 5, 2 • Add all the values in the set. Mean 1 + 1 + 2 + 2 + 3 + 5 + 5 + 5 = 24 • Divide the total by the number of values in the set. 8 values so 24 ÷ 8 = 3

Q. Calculate the mean of this set of data: A. 10 + 10 + 16 + 14 + 15 = 65 10, 10, 16, 14, 15 65 ÷ 5 5 values in the set, = 13 so divide by 5

a) Calculate the mean of this set of data: b) Calculate the mean of this set of data: 6, 22, 21, 14, 18, 15 1, 3, 3, 4, 7, 9, 15

6 + 22 + 21 + 14 + 18 + 15 = 96 1 + 3 + 3 + 4 + 7 + 9 + 15 = ...... 96 ÷ 6 = 16 ......

c) Calculate the mean of this set of data: d) Calculate the mean of this set of data: 8, 8, 9, 10, 10, 10, 11, 12, 12 2.1, 2.2, 2.2, 2.5, 2.5, 2.5, 2.7, 3.3

......

e) Calculate the mean of this set of data: f) Calculate the mean of this set of data: 1, 3, 3, 4, 4, 4, 6, 7, 7, 7, 9 8, 8, 9, 10, 11, 11, 13

......

g) Calculate the mean of this set of data: h) Calculate the mean of this set of data: 2, 2, 5, 6, 8, 10, 14, 17 0, 0, 2, 1.5, 1.8, 2, 2.2, 3, 3.5, 4

......

i) Calculate the mean of this set of data: j) Calculate the mean of this set of data: 10, 12, 13, 16, 17, 18, 20, 22 3, 4, 6, 7, 8, 10, 11, 13, 16, 17

......

...... page 357 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.8 Calculating the mean, median and mode of sets of data. MM6.1 11 22 33 44 Mean (or average) Set of data: 5, 1, 5, 3, 2, 1, 5, 2 • Add all the values in the set. Mean 1 + 1 + 2 + 2 + 3 + 5 + 5 + 5 = 24 • Divide the total by the number of values in the set. 8 values so 24 ÷ 8 = 3 Median (middle value) Ordered set: 1, 1, 2, 2, 3, 5, 5, 5 Write all the values in order. • 23+ 5 Median 25. • Odd numbered set - middle value. 2 ==2 Even numbered set - average of the 2 middle values. Mode 5 Mode (most common value)

Q. Which set of data has the same mean, median A. A) Mean 1 + 2 + 4 + 4 + 4 + 6 + 7 = 28 and mode? 28 ÷ 7 = 4 A) 1, 2, 4, 4, 4, 6, 7 Median 1, 2, 4, 4, 4, 6, 7 ⇒ 4 B) 3, 5, 5, 8, 9 Mode 1, 2, 4, 4, 4, 6, 7 ⇒ 4 C) 1, 2, 2, 2, 4, 4, 6 B) Mean 3 + 5 + 5 + 8 + 9 = 30 30 ÷ 5 = 6 Median 3, 5, 5, 8, 9 ⇒ 5 Mode 3, 5, 5, 8, 9 ⇒ 5 C) Mean 1 + 2 + 2 + 2 + 4 + 4 + 6 = 21 21 ÷ 7 = 3 Median 1, 2, 2, 2, 4, 4, 6 ⇒ 2 Mode 1, 2, 2, 2, 4, 4, 6 ⇒ 2 So A) has the same mean, median and mode.

a) Which set of data has the same mean, median b) Which set of data has the same mean, median and mode? and mode? A) −2, 0, 0, 1, 2, 2, 2, 3 A) 1, 2, 3, 3, 3, 4, 5 B) 10, 10, 11, 11, 11, 12, 12 B) 5, 5, 6, 7, 9, 10

......

c) Which set of data has the same mean, median d) Which set of data has the same mean, median and mode? and mode? A) 8, 8, 9, 10, 11 A) 29, 30, 30, 32, 34 B) −1, −1, 1, 1, 1, 3, 3 B) 6, 6, 7, 9, 9, 9, 10 C) 2, 3, 3, 4, 5, 7 C) −2, 2, 3, 3, 3, 5, 7

......

...... page 358 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.9 Interpreting histograms. MM6.1 11 22 33 44

Q. Using this histogram, how many times did the A. 1 + 2 + 2 + 2 USA engage in military conflict prior to 1950? = 7 Military conflicts in US history 10 10 8

8 6

6 4 Add each value

4 2 No. of US Military conflictsNo. 1 222 2 0

No. of US Military conflictsNo. 1750 - 1799 1800 - 1849 1850 - 1899 1900 - 1949 1950 - 1999 0 Year 1750 - 1799 1800 - 1849 1850 - 1899 1900 - 1949 1950 - 1999 Year

a) How many quasars have been discovered with b) How many cities in the European Union have a redshift of 5 or greater? a population greater than 2 million people?

50 European Union 12 Cities with population >1 000 000 40 10

30 8

Number of Quasars 6 20 cities of Number 4 10 2 0 0 2-31-2 >43-4 4.0 4.5 5.0 5.5 6.0 population (million) Redshift (z) - distance from Earth

10 Add each value

...... 0 32 11 2 11 4.5 5.0 5.5 6.0 Redshift (z) - distance from Earth

3 + ......

c) On February 7th, 2011 how many departures d) The best approximation for the number of did Jetstar have out of Brisbane after 6:00 pm? finishers in the 2010 New York Marathon is:

A) 30 000 B) 45 000 C) 60 000 Brisbane Departures: Jetstar - Feb 7th 2011 7 New York Marathon 2010 15000 6

Finishers 12000 4 9000 No. of departures No. 3

2 6000 1 0 3000 Time

6:00 - 8:599:00 - 11:59 0 12:00 - 14:5915:00 - 17:5918:00 - 20:5921:00 - 23:59 <20 20 - 29 30 -39 40 - 4950 - 5960 - 69 70+ Age group

...... page 359 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.10 Interpreting stem-and-leaf plots (1). MM6.1 11 22 33 44 To complete a stem-and-leaf plot from a given set of data: • Write the values from the data set - each unit digit is a leaf beside its corresponding tens (or hundreds) digit, which is a stem. Hint: tens value units value hundreds & STEM LEAF tens values units value 0 2 = 2 STEM LEAF 1 57 = 15 and 17 23 7 = 237

To calculate values from a stem-and-leaf plot: Data set of 13 elements: Mode (most common value) • Find the leaf digit that repeats most. • Read the number resulting from the { 13, 18, 18, 19, 20, 21, 21, 22, 22, 22, 29, 30, 31 } corresponding stem and leaf. mode = 22 Median (middle value) median (7th element) = 21 • Count the number of leaves. range If an odd number of leaves: stem leaves lowest value = 13 range = high − low • Count from the top left leaf until you = 31 − 13 reach the middle leaf. 1 3898 median = 21 = 18 • This digit is the unit and must be put 2 0 122291 with the corresponding stem. 310 mean = 286 ÷ 13 mode = 22 If an even number of leaves: = 22 • Count from the top left leaf until you highest value = 31 reach the two middle leaves. • Read the digits with their corresponding stems. • Find the average of the 2 middle numbers. Range • Subtract the lowest number (top left leaf) from the highest number (bottom right leaf).

Q. This back-to-back stemplot shows Richmond A. 22 scores for each team ⇒ and North Melbourne scores during the 2010 median = average of 11th and 12th scores AFL home and away season. Find the difference North Melbourne: between the medians of the two sets of data. 84+ 90 174 median = = = 87 North Melbourne Richmond 2 2 9 3 Richmond: 4 58 77+ 78 155 75 5334 69 median = = = 77.5 330 6 47 2 2 middle leaves 28 7 378 middle leaves 422 8 069 710 945 difference = 87 − 77.5 = 9.5 4 0103 50 9063 211 9 6123

a) Complete the stem-and-leaf plot for this data: b) Complete the stem-and-leaf plot for this data: 3, 12, 16, 17, 20, 21, 32, 35, 35, 37, 39, 43, 48 202, 204, 207, 210, 223, 223, 226, 228, 229, 230, 231, 232, 236 stem leaves 0 3 stem leaves 1 2 76 20 2 01 21 3 22 4 23 page 360 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.10 Interpreting stem-and-leaf plots (2). MM6.1 11 22 33 44 c) The stem plot shows a set of scores obtained d) The stem-and-leaf plot shows a set of IQ by a year 9 Maths class. Find the median and scores obtained by 30 year 10 students. Find range of the data. the median and mode of the data. stem leaves STEM LEAF 19 scores 8 7 2 0362 9 0 234457899 3 2 3675 10 1 2 225799 4 698 nine leaves 11 4 5 79 46 - middle score 5 74 + middle leaf 12 11 2 4 6 8 6 2 + nine leaves 13 7 398 14 1 8 5 median = median = ...... range = 85 − 20 = mode = ...... median = range = median = mode =

e) Complete the back-to-back stem-and-leaf plot f) This back-to-back stemplot shows the for the following two sets of data representing numbers of sausages and soft drinks sold at the ages of the teachers at the local high school. the school fundraisers in one year. Find the Find which set has the greater median. difference between the medians of the two sets Male: 27, 28, 33, 36, 39, 40, 47, 47, 48, 49, 50 of data. 52, 55 Female: 22, 23, 26, 27, 29, 29, 34, 36, 38, 38, 41 sausages soft drinks 43, 44, 44, 45, 48, 49, 50, 56, 59, 61 14 3 4 432 15 2488 male female 211 16 0 2 3 2 6 17 7 3 99765 18 0 2 4 5 6

g) The back-to-back stemplot below shows the h) This back-to-back stemplot shows Collingwood English and Maths scores of a year 10 class. and Fremantle scores during the 2010 AFL Find the difference between the medians of home and away season. Find the difference the two sets of scores. between the medians of the two sets of data. Collingwood Fremantle MATHS ENGLISH 3 9 1 4 2 1 4 0 5 8 4 0 5 887 6 5 3 0 5 1 3 07 679 99855 5 3 2 1 6 4 6 9 9 6 76 68120 27 6 6 5 0 7 2 2 5 57999 65 9 3367 2 8 0 4 6 7 55 0100 8 9 4 5 3 111 23458 9 73 5120 5 133 9 14 7 155 2016 page 361 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.11 Interpreting dot plots. MM6.1 11 22 33 44

Q. What is the median number of weeks in which A. 5 Check the data for the number of The Beatles songs were at number one weeks. on the US Billboard charts? It is displayed on the bottom axis. Find the median of this data. BEATLES SINGLES at number 1 position on the US Billboard BEATLES SINGLES at number 1 position on the US Billboard

1 2 3 4 5 6 7 8 9 Number of weeks 1 2 3 4 5 6 7 8 9 data is Number of weeks number of weeks

a) How many countries won more than b) Estimate to the nearest hour, the most 7 medals at the 2010 Winter Olympics? common number of hours of sleep required by a mammal. 2010 Winter Olympics - Vancouver Average Daily Sleep - 48 mammals

0 5 10 15 20 Number of countries of Number 0 21 43 567 8 9 10 14 gold medals Hours of sleep per day h

c) Complete the dot plot and find the median of d) Complete the dot plot and find the median of the following data: the following data: 5, 9, 7, 6, 9, 8, 8, 8, 7, 6, 10, 11 21, 18, 21, 23, 22, 19, 17, 22, 20, 17, 19, 21 Frequency Score Frequency Score 567891011 17 18 19 20 21 22 23

e) This dot plot shows the age of a Pope at his f) This dot plot shows the number children election to office. What is the median number Henry VIII had with his 6 wives and of the distribution? 3 mistresses. What is the median number of Popes (1800 - present day) the distribution? Henry VIII’s wives and known mistresses X X 50 60 70 80 X X Age at election (years) X X XX X 0 1 253 46 Number of children

page 362 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.12 Interpreting frequency tables. MM6.1 11 22 33 44 Mode (most common value) • Find the highest number in the frequency row. Median (middle value) • Add all the frequencies to know how many scores are in total. If an odd numbered total: • Add from the left on the frequency line till you reach the middle score. If an even numbered total: • Add from the left on the frequency line till you reach the two middle scores. • Find the average of the 2 middle scores. Mean (or average) • Multiply each score by its frequency. • Add the results. • Divide by the total number of frequencies (scores). Range • Subtract the lowest score from the highest score. Q. How many scores are there of 6 or less in the A. There are 2 lots of 6 and 3 lots of 5 scored. following distribution? 2 + 3 Score 59876 = 5 How many times Frequency 31342 the score occurs

a) How many scores are there of more than 13 in b) How many scores are there of 8 or more in the the following distribution? following distribution? 14 > 13 Score 10 11 141312 Score 411086 2 Frequency 45638 Frequency 35417

c) Find the median and range of the following d) Find the mean and mode of the following distribution. distribution. Score 15 19181716 Score 26543 Frequency 33725 20 scores Frequency 42112 10 scores

17 18 35 2432415162×+×+×+×+× 35 median + 17.5 mean = = = = 10 = 10 ...... 2 2 ...... range = 19 − 15 = 4 mode = ...... median = range = mean = mode =

e) Find the median and range of the following f) Find the mean and mode of the following distribution. distribution. Score 04321 Score 04321 Frequency 5511 146 Frequency 42356

median = mean = ...... mode range = = ...... median = range = mean = mode = page 363 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.13 Calculating the median, range, upper quartile (UQ), lower MM6.1 11 22 33 44 quartile (LQ) and interquartile range (IQR) for box-and-whisker plots. • Find the median or middle value of the set of data. • Divide the data into an upper half and a lower half. • Find the median of the upper values of the set of data, or the upper quartile (UQ). • Find the median of the lower values of the set of data, or the lower quartile (LQ). • Find the interquartile range (IQR) of the set of data by subtracting the LQ from the UQ. Example: lower half upper half Data set of 16 elements: {1, 2, 2, 2, 3, 4, 4, 4, 6, 8, 8, 8, 8, 8, 9, 9}

interquartile range IQR = UQ − LQ IQR = UQ − LQ lower quartile upper= 8 − quartile 2.5 = 8 − 2.5 LQ = 2.5 median = 5 = UQ5.5 = 8 = 5.5 median of the lower half median of the upper half minimum value1222 88 maximum value 3444 6888 99

0123456789

Q. For this box-and-whisker plot, find the median A. median = middle value and upper quartile (UQ). median = 176 UQ is the median of the upper scores ⇒ UQ = 186

150 160 170 180 190 200 150 160 170 180 190 200 Height (cm) Height (cm)

a) For this box-and-whisker plot, find the median b) For this box-and-whisker plot, find the lower and lower quartile (LQ). quartile (LQ) and upper quartile (UQ).

15 1617 1819 2021 22 23 2425 2627 2829 30 20 22 2426 28 30 32 34 36 test scores age (years) median = 30 ...... median = 30 LQ = LQ = UQ =

c) What is the median and upper quartile (UQ) of d) What is the median and interquartile range the set of ages of the 19th century American (IQR) of the number of medals won by the presidents when they were first elected? USA at each of the Olympics between 1908 and 2004, as shown on this boxplot?

40 50 60 70 Age 50 70 90110 130 150 170 190 medals

...... median = UQ = median = IQR = page 364 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.14 Interpreting scatter plots. MM6.1 11 22 33 44

Q. In what year did Bekele (Ethiopia) set the A. 2005 world record time of 26:17.53 for the 10 000 m event? World Athletics 10 000 metre times - top 25 27 minutes World Athletics 10 000 metre times - top 25 27 minutes 50 50 40 seconds 40

seconds 30 Bekele 30 20

20 10

10 26 minutes 1994 1996 1998 2000 2002 2004 2006 2008 26 minutes 1994 1996 1998 2000 2002 2004 2006 2008

a) Select the most appropriate trend line for this b) How many of the mammals tested have less scatter plot. [Hint: The sums of the distances from the than 5 g of fat and less than 2 g of protein points above and below the line, to the line, are approximately in any 100 g of their milk? equal.] Fat and protein composition of milk from different mammalian species Area and population of continents (per 100 g of fresh milk) 20 12 B 11 Whale 10 Seal Mouse 15 9

protein (g) protein 8 7 10 6 5 Elephant Water Buffalo Area (million sq. miles) (million sq. Area A 4 Cow 5 3 Donkey Goat 2 Rhesus Monkey 1 Human 0 0 0 500 1000 1500 2000 2500 3000 3500 0 5 10 15 20 25 30 35 40 45 50 population (millions) fat (g)

c) Select the most appropriate trend line for this d) The difference in milk consumption between scatter plot. [Hint: The sums of the distances from the 1960 and 1990 is: points above and below the line, to the line, are approximately A) < 25 litres/head/year equal.] B) = 25 litres/head/year A The planets C) 25 litres/head/year 30 > Milk per capita consumption 25 175 (earth = 1) 20 150

15 125 Distance from the Sun Distance from litres/head/year 10 100

B 5 75

0 -300 -200 -100 0 100 200 300 400 500 190019101920193019401950196019701980199020002010 Average surface temperature (˚C)

page 365 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 Skill 29.15 Interpreting frequency histograms. MM6.1 11 22 33 44

Q. Find the range, median and mean of the A. The data for the number of tails is: distribution. 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5 Tossing 5 coins 6 first 10 terms last 10 terms 5 range = last term − first term = 5 − 0 = 5 4 median = average of 10th and 11th term = 3 3 mean = sum of all numbers ÷ number of terms Frequency 2 = 65 ÷ 20 1 = 3.25 0 0 321 54 Number of tails

a) Find the range and median of the distribution. b) Find the range and median of the distribution. Symphony Composers (1950 - 2000) 13 Wimbledon singles titles 12 10 11 9 10 8 9 7 8 6 7 6 5 5 4 4

3 Number of composers Number of players 3 2 2 1 1 0 0 3 54 6987 1 32 4965 7108 Number of titles Number of symphonies

...... range = median = range = median =

c) Find the range and mean of the distribution. d) Find the median and mean of the distribution. [Round the mean to the nearest whole number.] Words in a paragraph 7 Maths Test Scores 6 6 5 5 4 4

frequency 3 3 2 Frequency 2 1 1 0 0 54321 807060 10090 number of letters Scores

......

...... range = mean = median = mean = page 366 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 SkillSkill 229.9.1616 DrawingDrawing bbox-and-whiskerox-and-whisker pplots.lots. MM6.1 11 22 33 44 • Order the given data set. • Find the median, lowest and greatest values, lower quartile and upper quartile. • Mark the maximum and minimum values with whiskers. • Mark the median of all values with a vertical line. • Mark the upper quartile and lower quartile with the box edges as shown below.

whisker whisker box median

Q. Draw a box-and-whisker plot for this set of A. 7, 8, 13, 15, 20, 22, 24, 27, 30, 32 data: maximum value = 32 7, 8, 13, 15, 20, 22, 24, 27, 30, 32 minimum value = 7 22 20 median = + = 21 2 upper quartile = 27 median of upper half lower quartile = 13 median of lower half 0 5 1015 20 25 30 35

0 5 1015 20 25 30 35

a) Draw a box-and-whisker plot for this set of b) Draw a box-and-whisker plot for this set of data: data: 15, 21, 21, 23, 25, 27, 32, 36, 39 34, 47, 11, 15, 57, 24, 20, 11, 19, 50, 28, 37 LQ UQ 15 21 2534 39

10 20 30 40 0102030 4050 60 70

c) Draw a box-and-whisker plot for this set of d) Draw a box-and-whisker plot for the set of data: data whose lowest value is 104, greatest value 51, 17, 25, 39, 7, 49, 62, 41, 20, 6, 43, 13 is 158, median is 136, lower quartile is 116 and upper quartile is 142.

0102030 4050 60 70 100 120 140 160

e) Draw a box-and-whisker plot for the set of f) Draw a box-and-whisker plot for the set of data whose lowest value is 0, greatest value is data whose lowest value is 15, greatest value 11.8, median is 4.1, lower quartile is 1.2 and is 60, median is 35, lower quartile is 25 and upper quartile is 8.5 upper quartile is 40.

0 123456711 8910 12 0102030 40 50 60 page 367 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29 MM5.2 11 22 33 44 SkillSkill 229.9.1717 CalculatingCalculating tthehe mmedian,edian, uupperpper qquartileuartile ((UQ),UQ), llowerower qquartileuartile ((LQ)LQ) MM6.1 11 22 33 44 a andnd iinterquartilenterquartile rrangeange ((IQR)IQR) fforor ffrequencyrequency ttablesables aandnd sstem-and-leaftem-and-leaf plots.plots. • Find the median or middle value of the set of data. • Divide the data into an upper half and a lower half. • Find the median of the upper values of the set of data, or the upper quartile (UQ). • Find the median of the lower values of the set of data, or the lower quartile (LQ). • Find the interquartile range (IQR) of the set of data by subtracting the LQ from the UQ. (see skill 29.13, page 346)

Q. Calculate the median and upper quartile (UQ) A. 50 scores altogether ⇒ th th for the data displayed in this frequency table. median = average of 25 and 26 scores th th ⇒ Athens, 2004 - bronze medal winning countries 25 and 26 scores are 2 } Number of medals 152 3 4 6 7 8 9 10 median = 2 Frequency 1513 7 6 2 241 0 0 UQ is the median of the 25 upper scores ⇒ th UQ = 13 score counting from the top score: scores 1 to 15 are all 1 scores 16 to 28 are all 2 9, 9, 9, 9, 7, 6, 6, 5, 5, 4, 4, 4, 4 ⇒ 13th score from UQ = 4 the top

a) For this stem-and-leaf plot, find the median b) Calculate the median and lower quartile (LQ) and the lower quartile (LQ). for the data displayed in this frequency table. stem leaves Athens, 2004 - silver medal winning countries Number of 13 scores 152 3 4 6 7 8 9 10 1 3898 ⇒ medals median = 7th term 2 0 11 2229 Frequency 1511 6 5 2 441 0 0 310 18+ 19 37 LQ = = = ...... 2 2 ...... median = 21 LQ = 18.5 median = LQ =

c) Calculate the median and upper quartile (UQ) d) Calculate the median and lower quartile (LQ) for the data displayed in this frequency table. for the data displayed in this frequency table. Adam Scott - PGA, 2006 rounds 72 or less Score 6364 65 66 67 68 69 70 71 72 Score 596 7 8 10 11 12 13 14 Frequency 151 4 4 51111 7 9 Frequency 203 1 4 105 0 12 3

...... median = UQ = median = LQ =

e) Find the interquartile range (IQR) for the set f) This stem-and-leaf plot shows the ages of the of data shown in this stem-and-leaf plot. 19th century American presidents when they STEM LEAF were first elected. Find the interquartile range 1 1 12344 (IQR) for the set of data. 1 7 89 STEM LEAF 2 0 224 4 7 89 25 5 689 5 0 001224 3 0 1 2 234 5 5 5678889 3 6 6789 62 4 112334 6568 ...... IQR IQR ...... = ...... = page 368 www.mathsmate.co.nz © Maths Mate 5.2/6.1 Skill Builder 29