Core Maths Powerpoint - NCETM · What is happening to the World? World population today: 7,274,270,000 people Productive land today: 8,531,698,000 hectares World population 5 years

Population Change

Using Statistics

What is happening to the World?

World population today: 7,274,270,000 people Productive land today: 8,531,698,000 hectares

World population 5 years ago: 6,765,184,000 people Productive land 5 years ago : 8,629,187,000 hectares

Using these figures, what are the rates of change (i.e. the change per year) of world population and

productive land?

What is your reaction to these numbers?

What is happening to the World?

World population growth rate: 101,817,200 people per year

Productive land decay rate:

19,497,800 hectares per year

What are the main statistical measures that help us explain this population change?

http://www.worldometers.info/world-population/

The Demographic Transition Model

Here are the number of estimated births and deaths in England over the last 300 years. Use these numbers and the provided axes to draw a line graph of the births and deaths in England over the last 300 years.

Year Births

(per 1000 people) Deaths

(per 1000 people) 1700 36.2 37.8

1720 36.5 36.7

1740 36.8 36.1

1760 36.4 36.6 1780 36.3 32.2

1800 37 24.3 1820 36.9 20.6

1840 36.3 16.9

1860 36.5 13.8

1880 35.1 12.4

1900 28 12 1920 19.3 11.6

1940 13.4 10.9 1960 13.1 10.6

1980 12.9 10.3

2000 11.9 9.9 2020 10.3 10.4


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Birth & Death Rates in England

Births

Deaths


Interpret what is happening to the population of England in the following periods of time:

Stage 1: 1700 to 1760 Stage 2: 1761 to 1880 Stage 3: 1881 to 1940 Stage 4: 1941 to 2000

Stage 5: 2001 to 2020 (and beyond) Referring to the birth & death rates, try to explain why these population changes have come about.


Time Period Birth Rate Death Rate Population Change

Stage1: 1700 – 1760

High: No birth control,

High child mortality, High child dependency

High: No birth control,

High child mortality, High child dependency

Minor fluctuations

Stage 2: 1761 – 1880

High Reducing rapidly: Improved medical care

Improved way of life

Quickly increasing

Stage 3: 1881 – 1940

Reducing rapidly: Family planning,

Lowered child mortality

Low: Stabilises after fall

Increasing

Stage 4: 1941 – 2000

Low: Stabilises after fall

Low:

Mild increase

Stage 5: 2001 – 2020+

Low: Low: Minor fluctuations


Using the information in the table, plot the population of England on the same axes as the birth and death rates. Do the births and deaths correctly Interpret the change in Population?

Year Population

(million) 1700 6.4

1720 6.5 1740 6.4 1760 6.7

1780 6.9 1800 8.9

1820 12 1840 15.9

1860 20.8 1880 26 1900 32.5

1920 37.9 1940 40.6

1960 43.6

1980 47.5

2000 50 2020 51.6


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Population Change in England 1700-2020

Births

Deaths

Population

Bar Charts: Population Pyramids

Population pyramids are horizontal bar charts which depict the proportion of the population falling into each age bracket.

Continuous data: No gaps

http://www.visionofbritain.org.uk/datavalue/788090




































Bar Charts: Population Pyramids The two tables show

the percentage of the population falling into each age band for two different years in England. Produce the population pyramids for these two years. Given the shape of the graphs, can you guess which years they are?

Age Band Male Female

0 to 4 6.3% 5.7%

5 to 9 6.7% 6.0%

10 to 14 6.9% 6.3%

15 to 19 6.5% 5.9%

20 to 24 6.1% 5.9%

25 to 29 6.7% 6.6%

30 to 34 7.7% 7.6%

35 to 39 8.0% 7.8%

40 to 44 7.2% 6.9%

45 to 49 6.4% 6.2%

50 to 54 7.0% 6.8%

55 to 59 5.8% 5.6%

60 to 64 4.9% 4.8%

65 to 69 4.3% 4.5%

70 to 74 3.7% 4.2%

75 to 79 2.9% 3.8%

80 to 84 1.7% 2.8%

85 & up 1.1% 2.7%

Age Band Male Female

0 to 4 13.4% 12.8% 5 to 9 12.0% 11.4%

10 to 14 11.0% 10.4% 15 to 19 9.9% 9.7% 20 to 24 9.1% 9.5% 25 to 29 8.0% 8.4% 30 to 34 7.0% 7.2% 35 to 39 6.1% 6.1% 40 to 44 5.4% 5.4% 45 to 49 4.5% 4.4% 50 to 54 3.9% 4.0% 55 to 59 2.9% 3.0% 60 to 64 2.6% 2.8% 65 to 69 1.7% 1.9% 70 to 74 1.3% 1.5% 75 to 79 0.7% 0.9% 80 to 84 0.4% 0.5% 85 & up 0.2% 0.2%






































England 2001 (Stage 5)

England 1851 (Stage 2)


Stage 1 Stage 2 Stage 3

Stage 4 Stage 5

Bar Charts: Population Pyramids Shown are 12 population

pyramids of 12 different nations in 2010. Which stage of the demographic transition model is each of these countries, as suggested by their pyramid. What is each country’s modal age group? What do you notice about the countries in each group?


Stage 1: F – Afghanistan (0-5) K – Niger (0-5) Stage 2: A – Haiti (5-10) B – India (0-5) C – Venezuela (10-15) L – Morocco (20-25) Stage 3: E – Saudi Arabia (25-30) H – China (20-25) Stage 4: I – United Kingdom (45-50) J – USA (45-50) Stage 5: D – Russia (30-35) G – Australia (35-40)

“3rd World”

“developing”

“westernised”

Sampling: Selecting from across the World

There are currently 196 countries in the world. In order to fully analyse the worldwide picture, we would need to use the data on every country. i.e. The population of the data. However, in order to save time and money (usually where using all the data is unfeasibly and unrealistic) we select a number of items. In this case, we would select a sample of countries. It’s important to unbiased when selecting in order to gain a fair sample.


How can we select a fair sample? Random Sampling: Data is picked at random, usually using some random generator. Systematic Sampling: Data is chosen at regular intervals (every 5th entry, for example). Cluster Sampling: Data is grouped in “clusters” and random clusters are selected. Stratified Sampling: Data is selected in proportion to “cluster” sizes. (Quota Sampling: Keep selecting data until you have enough for the given category)


You have been provided with a list of 210 countries, principalities and sovereign states that have been put into clusters by continent.

Africa The Americas

Asia Europe Oceania

In order to conclude on the population change of each continent, we will select a sample to statistically analyse.


Random Sample: Africa There are 57 countries listed in Africa. How can we randomly select 10? ▪ Assigned each country the numbers 1 to 57. ▪ Using a calculator, generate a number between 1 and

57 inclusive. RanInt#(1,57)

ALPHA . 1 SHIFT ) 5 7 ) = ▪ Repeat the process until you have 10 unique number

(press = to get a new random number) ▪ Highlight the 10 countries these number relate to in

the list.


Systematic Sample: The Americas There are 40 countries listed in The Americas. How can we systematically select 10? ▪ Select countries by highlighting one every so

many down the list. ▪ 40 ÷ 10 = 4 ▪ Choose a starting point within the first 4

countries on the list and highlight. ▪ Highlight every 4th country down the list.


Cluster Sample: Europe There are 45 countries listed in Africa. How can we cluster the countries in order to select a group of 10? ▪ The continent is already split into North,

West, East, South. ▪ Select one of these clusters that has the

appropriate number of countries in it.


Stratified Sample: Asia There are 51 countries listed in The Americas. How can we proportionally select countries across the different regions in Asia? ▪ Count how many countries are in each region

e.g. Western Asia has 18 countries ▪ Calculate how many countries should be selected from each

region by using the formula: 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑢𝑛𝑡𝑟𝑖𝑒𝑠 𝑖𝑛 𝑟𝑒𝑔𝑖𝑜𝑛

𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑢𝑛𝑡𝑟𝑖𝑒𝑠× 𝑁𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑆𝑎𝑚𝑝𝑙𝑒

e.g. 18

51× 10 = 3.53

▪ Using appropriate rounding, randomly/systematically select the calculated number of items from each group.


You choice: Oceania There are 17 countries listed in Oceania. Select one the sampling methods you have used so far to highlight a list of 10 Oceania countries. (Some will be more appropriate then others)


For each of the sampling methods you have just done, state one advantage and one disadvantage of the method.

Sampling Method

Advantage Disadvantage

Random All pieces of data equally likely to be chosen, hence it will be fair

Can be time consuming and impractical on a large scale

Systematic Unlikely to get a bias sample The process may not be random as start position and order of

data may effect selection

Cluster A quick and efficient way of selecting a sample

Open to a bias sample in the clusters aren’t similar

Stratified It’s the best way to fairly reflect the population

Can be time consuming and impractical on a large scale


You can improve the reliability of the sample in all cases by … … increasing the size of the sample

Bar Charts: Comparing Continents

What can you conclude about European countries from this graph?

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Belarus Bulgaria Czech Rep Hungary Moldova Poland Romania Russia Slovakia Ukraine

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Birth & Death Rates of a Sample of European Countries

Births

Deaths

Discrete data: Gaps

Bar Charts: Comparing Continents How does this compare to Africa? Use your sample

of African countries to construct a similar bar chart.

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Morocco Cape Verde Ghana Ethiopia Somalia South Sudan CentralAfrican Rep

DR Congo Namibia Swaziland

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Birth & Death Rates of a Sample of African Countries

Births

Deaths

Bar Charts: Comparing Continents

Each bar height represents the birth rate, whilst the height of each blue section is the corresponding death rate. What is the red region representing? Calculate the net growth rate for each country.

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Canada El Salvador Nicaragua Barbados DominicanRep

Jamacia St Lucia Bolivia Ecuador Peru

Birth rates of a Sample of American Countries

???

Deaths

Sample Averages: Analysing Rates

𝐒𝐚𝐦𝐩𝐥𝐞 𝐌𝐞𝐚𝐧: 𝐱 = 𝐱

𝐧

where 𝑥 = Sum of the data values 𝑛 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑖𝑒𝑐𝑒𝑠 𝑜𝑓 𝑑𝑎𝑡𝑎 Calculate the sample means of the birth rates and death rates for the 10 selected countries in each of the 5 continents

Sample Averages: Analysing Rates

Europe Sample Birth Rate Mean

=13 + 9 + 10 + 9 + 11 + 10 + 10 + 13 + 10 + 11

10

= 10.6

What percentage error does this sample mean have compared the population mean?

Percentage Error and Radar Charts

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𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =𝑆𝑎𝑚𝑝𝑙𝑒 𝑀𝑒𝑎𝑛 − 𝑃𝑜𝑝𝑢𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑎𝑛

𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑎𝑛× 100%

=10.6−11

11× 100%

= -3.6%


Plot the following information on the provided radial chart

Region Life Expectancy

% Urban Population Density

Africa 59 40 37

Americas 76 79 23

Europe 78 72 32

Asia 69 46 99

Oceania 77 70 5

World 71 53 53


Sample Spreads: Analysing Rates

We have met range and interquartile range as a measure of how spread out the data is. However these measures have their limitations. A more detailed measure, that considers all pieces of data, is that of standard deviation. It considers the average distance the pieces of data are from the mean.

𝒔 = (𝒙 − 𝒙 )𝟐

𝒏 − 𝟏



▪ Type equation here. European Country

1 2 3 4 5 6 7 8 9 10 Total

𝒙 = (BR) 13 9 10 9 11 10 10 13 10 11 106

𝑥 = 𝑥

𝑛 =

106

10 = 10.6



1 2 3 4 5 6 7 8 9 10 Total

𝒙 = (BR) 13 9 10 9 11 10 10 13 10 11 106

𝒙 − 𝒙 2.40 -1.60 -0.60 -1.60 0.40 -0.60 -0.60 2.40 -0.60 0.40 0

𝑥 = 𝑥

𝑛 =

106

10 = 10.6



1 2 3 4 5 6 7 8 9 10 Total

𝒙 = (BR) 13 9 10 9 11 10 10 13 10 11 106

𝒙 − 𝒙 2.40 -1.60 -0.60 -1.60 0.40 -0.60 -0.60 2.40 -0.60 0.40 0.0

(𝒙 − 𝒙 )𝟐 5.76 2.56 0.36 2.56 0.16 0.36 0.36 5.76 0.36 0.16 18.40

𝑥 = 𝑥

𝑛 =

106

10 = 10.6

𝑠 = (𝑥 − 𝑥 )2

𝑛 − 1 =

18.4

9 = 1.430 (3𝑑𝑝)


▪ Type equation here.

𝑥 = 𝑥

𝑛 =

106

10 = 12.1

European Country

1 2 3 4 5 6 7 8 9 10 Total

𝒙 = (DR) 13 14 10 13 11 10 12 13 10 15 121



𝑥 = 𝑥

𝑛 =

106

10 = 12.1

European Country

1 2 3 4 5 6 7 8 9 10 Total

𝒙 = (DR) 13 14 10 13 11 10 12 13 10 15 121

𝒙 − 𝒙 0.90 1.90 -2.10 0.90 -1.10 -2.10 -0.10 0.90 -2.10 2.90 0.0



𝑥 = 𝑥

𝑛 =

106

10 = 12.1

𝑠 = (𝑥 − 𝑥 )2

𝑛 − 1 =

28.9

9 = 1.792 (3𝑑𝑝)

European Country

1 2 3 4 5 6 7 8 9 10 Total

𝒙 = (DR) 13 14 10 13 11 10 12 13 10 15 121

𝒙 − 𝒙 0.90 1.90 -2.10 0.90 -1.10 -2.10 -0.10 0.90 -2.10 2.90 0.0

(𝒙 − 𝒙 )𝟐 0.81 3.61 4.41 0.81 1.21 4.41 0.01 0.81 4.41 8.41 28.90


Calculate the standard deviation of the birth rates and death rates for each of the other continents using the samples you have selected. You have been provided with grids to help you. What can you conclude about the birth and deaths rates on each continent as a comparison to each other using the mean and standard deviation values?

Identifying extreme data

Approximately 95% of the data in a population lies within 2 standard deviations of the mean. The other 5% can be considered to be “outliers”.

𝒙 − 𝟐𝒔 < 𝒙 < 𝒙 + 𝟐𝒔 This data can potentially be recognised as being extreme values and some consideration should be given as whether is should be included.


Europe: Birth Rates 𝑥 = 10.6, 𝑠 = 1.43

10.6 − 2 × 1.43 < 𝑥 < 10.6 + 2 × 1.43

7.74 < 𝑥 < 13.46

Sample shows no outliers Population shows 3 outliers


Europe: Birth Rates 42 of 45 countries lie within 2 standard deviations of the mean = 93.33% Monaco (6) Kosovo (15) Ireland (15)

Can you draw conclusions on why these 3 countries may have comparatively extreme birth rates? For each continent, calculate the outlier values for birth and death rate from your samples. Identify any outliers from the population and conclude why they maybe an outlier if you can.

Finding Relationships

▪ Two sets of data may well be “correlated”, that is, there is some sort of statistical relationship between them.

▪ This is not to say that the values of one data set are causing the values of the other but a data value from one set can be used to predict a corresponding value in the other.

▪ We plot scatter diagrams with lines of best fit to achieve this.


Overall, there appears to be no relationship between birth rates and death rates, except in Africa where there appears to be a positive correlation.

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Africa

Americas

Europe

Asia

Oceania


Comparing birth rates to infant mortality rates however shows a ‘strong’ positive correlation however. This makes logical sense given that mothers are likely to want to have more children if they are dying young.

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Africa

Americas

Europe

Asia

Oceania

Note: Line of best fit doesn’t necessarily go through the origin


▪ Use the provided axes to plot scatter graphs with birth or death rates on one axis and another data set of the other.

▪ You have been provided data on population, life expectancy, percentage urban, contraception utilisation, Gross National Income and carbon emissions.

▪ Evaluate on the correlation of you’re graphs and whether there is a reason for this.


The correlation between death rate and life expectancy overall has a weak negative correlation, although Africa on its own demonstrates a strong correlation. Oceania shows to have no correlation

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Death rate

Africa

Americas

Europe

Asia

Oceania

Using Relationships

Niger has a death rate of 11 deaths per 1000 people. Niger has a life expectancy of 58. Algeria has a life expectancy of 71. Algeria has a death rate of 6.

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Death Rate

Estimate Niger to have life expectancy of 57.

Estimate Algeria to have death rate of 5.

Using Relationships

▪ You have been given 11 scatter diagrams detailing data from a sample of 10 African countries (plus in most graphs, the African average)

▪ By sketching and using a line of best fit on each one, complete the data tables for the 5 given African countries.

▪ You are given 1 piece of data on each country, you must estimate the others from the appropriate graphs.

Using Relationships

How well does your data connect up? Would your birth rate and GNI data fit well on this graph?

Ignoring the Trend

Sometime times pieces of data don’t follow the trend of the other pieces of data. These can be considered as outliers. There may well be reason for this exception to ignore the trend (Mayotte is a very small island)

Mayotte has a death rate of 31 And a infant mortality of 4.

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Finding and Using Frequencies

Data can be presented in a frequency table, being populated from the given raw data. The table below shows the tallied birth rates of the 44 European countries, along with their corresponding frequencies.

Birth Rate (per 1000 people)

Tally Frequency

6 I 1

7 0

8 IIII 4

9 IIIII II 7

10 IIIII IIIII IIII 14

11 IIIII III 8

12 IIIII I 6

13 III 3

14 0

15 II 2


These frequencies can summed down the table to arrived at the cumulative frequencies. The cumulative frequencies can be plotted in order to create a cumulative frequency curve.


Tally Frequency Cumulative Frequency

6 I 1 1

7 0 1 + 0 = 1

8 IIII 4 1 + 4 = 5

9 IIIII II 7 5 + 7 = 12

10 IIIII IIIII IIII 14 12 + 14 = 26

11 IIIII III 8 26 + 8 = 34

12 IIIII I 6 34 + 6 = 40

13 III 3 40 + 3 = 43

14 0 43 + 0 = 43

15 II 2 43 + 2 = 45


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European Birth Rates

Finding and Using Frequencies Using the provided table and axes, calculate and plot the

cumulative frequencies of the European death rates.

Death Rate (per 1000 people)


3 I 1 1

4 I 1 2

5 0 2

6 I 1 3

7 IIIII I 6 9

8 IIIII I 6 15

9 IIIII IIII 9 24

10 IIIII II 7 31

11 III 3 34

12 II 2 36

13 III 4 40

14 IIII 4 44

15 I 1 45


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European Death Rates

Grouping Data

Some data, usually that which has a higher spread, is better represented having been grouped. The groups do not have a subscribed width and can indeed vary in size. The groups should be selected to present the data appropriately. For example, birth rates in Africa range from 11 to 50 and I may wish to have 6 groups…

Grouping Data

▪ The model group is 35 < 𝑥 ≤ 40 births per 1000. ▪ This doesn’t necessarily mean that the mode of

the raw data lies within this group. ▪ 34 birth per 1000 is the actual mode value.



𝟏𝟎 < 𝒙 ≤ 𝟐𝟎 IIIII 5 5

2𝟎 < 𝒙 ≤ 𝟑𝟎 IIIII IIIII I 11 16

3𝟎 < 𝒙 ≤ 𝟑𝟓 IIIII IIIII III 13 29

𝟑𝟓 < 𝒙 ≤ 𝟒𝟎 IIIII IIIII IIIII 15 44

𝟒𝟎 < 𝒙 ≤ 𝟒𝟓 IIIII III 8 52

𝟒𝟓 < 𝒙 ≤ 𝟓𝟎 IIIII 5 57

Plot the max of each group against the cumulative frequency. The CF at this point is zero

Grouping Data

Group the African death rates into appropriate groups and plot the corresponding cumulative frequency curve.

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African Birth Rates

Grouping Data

Example grouping for death rates for African countries.

Modal group is 8 < 𝑥 ≤ 10.



𝟐 < 𝒙 ≤ 𝟔 IIIII IIII 9 9

𝟔 < 𝒙 ≤ 𝟖 IIIII IIIII 10 19

𝟖 < 𝒙 ≤ 𝟏𝟎 IIIII IIIII III 13 32

𝟏𝟎 < 𝒙 ≤ 𝟏𝟐 IIIII IIIII 10 42

𝟏𝟐 < 𝒙 ≤ 𝟏𝟒 IIIII IIII 9 51

𝟏𝟒 < 𝒙 ≤ 𝟐𝟏 IIIII I 6 57

Grouping Data

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African Death Rates

Comparing Data Graphically

What can we say about how European birth rates compared to their death rates?

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European Birth/Death Rates

Births

Deaths

Comparing Data Graphically

What can we say about how African birth rates compared to their death rates?

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African Birth/Death Rates

Births

Deaths

Analysing Cumulative Curves

African Birth Rates can be analysed more closely finding the median and quartiles of the data.

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African Birth Rates

𝟓𝟕 × 𝟑

𝟒= 𝟒𝟐. 𝟕𝟓

𝟓𝟕

𝟐= 𝟐𝟖. 𝟓

𝟓𝟕

𝟒= 𝟏𝟒. 𝟐𝟓

Upper Quartile

Median

Maximum

Minimum

𝟓𝟕

𝟏𝟎

𝟐𝟗 𝟑𝟒 𝟑𝟗. 𝟓 𝟓𝟎

Upper Quartile

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African Death Rates

Analysing Cumulative Curves

Use your curve to find an estimate for the African Death Rates’ median and quartiles.

Low Quartile

𝟓𝟕 × 𝟑

𝟒= 𝟒𝟐. 𝟕𝟓

𝟓𝟕

𝟐= 𝟐𝟖. 𝟓

𝟓𝟕

𝟒= 𝟏𝟒. 𝟐𝟓

Upper Quartile

Median

Maximum

Minimum

𝟓𝟕

𝟐

7.5 𝟗. 𝟓 12. 𝟓 𝟐𝟏

Creating and Comparing Boxplots

Construct box-plots of African birth and death rates using the 5 values you have found from your cumulative frequency curves. Compare the 2 plots.

Minimum Median Maximum Lower Quartile

Upper Quartile

25% 25% 25% 25%

Creating and Comparing Boxplots

▪ On average, the birth rates are much higher than the death rates (median of 34 compared to 9.5)

▪ The spread of the birth rates is also higher: Interquartile range: Births = 39.5 – 29.5 = 10

Deaths = 12.5 – 7.5 = 5 Range: Births = 50 – 10 = 40 Deaths = 21 – 2 = 19

Birth Rates

Death Rates

Comparing Areas instead of Heights

The bar chart shows the frequencies of the groups of birth rates for Africa. Although the heights correctly represent the frequencies, why could the graph be misleading? These 2 groups are seen at “bigger”


As a more appropriate method of comparing the groups, we use the area of each bar to represent the frequencies.

Frequency = Class Width x “Frequency Density” Area = Width x Height Frequency Density is placed on the y axis and valves for this can be calculated by using the formula:

𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝑫𝒆𝒏𝒔𝒊𝒕𝒚 =𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚

𝑪𝒍𝒂𝒔𝒔 𝑾𝒊𝒅𝒕𝒉



Class Width Frequency Frequency Density

𝟏𝟎 < 𝒙 ≤ 𝟐𝟎 10 5 𝟓 ÷ 𝟏𝟎 = 𝟎. 𝟓

2𝟎 < 𝒙 ≤ 𝟑𝟎 10 11 11÷ 𝟏𝟎 = 𝟏. 𝟏

3𝟎 < 𝒙 ≤ 𝟑𝟓 5 13 𝟏𝟑 ÷ 𝟓 = 𝟐. 𝟔

𝟑𝟓 < 𝒙 ≤ 𝟒𝟎 5 15 𝟏𝟓 ÷ 𝟓 = 𝟑

𝟒𝟎 < 𝒙 ≤ 𝟒𝟓 5 8 𝟖 ÷ 𝟓 = 𝟏. 𝟔

𝟒𝟓 < 𝒙 ≤ 𝟓𝟎 5 5 𝟓 ÷ 𝟓 = 𝟏


The height of the 2 bars have now been reduced as the areas represent the frequencies. As the these bars are twice as wide, they are now half the height they were in the bar chart.

Frequency = Area of bar = 10 x 0.5 = 5 Frequency = Area of bar

= 5 x 3 = 15




𝟐 < 𝒙 ≤ 𝟔 9

𝟔 < 𝒙 ≤ 𝟖 10

𝟖 < 𝒙 ≤ 𝟏𝟎 13

𝟏𝟎 < 𝒙 ≤ 𝟏𝟐 10

𝟏𝟐 < 𝒙 ≤ 𝟏𝟒 9

𝟏𝟒 < 𝒙 ≤ 𝟐𝟏 6

Complete the table below and sketch the associated histogram for African death rates.




𝟐 < 𝒙 ≤ 𝟔 4 9 𝟗 ÷ 𝟒 = 𝟐. 𝟐𝟓

𝟔 < 𝒙 ≤ 𝟖 2 10 𝟏𝟎 ÷ 𝟐 = 𝟓

𝟖 < 𝒙 ≤ 𝟏𝟎 2 13 𝟏𝟑 ÷ 𝟐 = 𝟔. 𝟓

𝟏𝟎 < 𝒙 ≤ 𝟏𝟐 2 10 𝟏𝟎 ÷ 𝟐 = 𝟓

𝟏𝟐 < 𝒙 ≤ 𝟏𝟒 2 9 𝟗 ÷ 𝟐 = 𝟒. 𝟓

𝟏𝟒 < 𝒙 ≤ 𝟐𝟏 7 6 𝟔 ÷ 𝟕 = 𝟎. 𝟖𝟔

Complete the table below and sketch the associated histogram for African death rates.



Use the 2 histograms to complete the given tables by calculating the frequencies. Make a comparison of European birth rates to deaths rates.



Class Width Frequency Density

Frequency

𝟓 < 𝒙 ≤ 𝟖 3 1.67 3 x 1.67 = 5

𝟖 < 𝒙 ≤ 𝟏𝟎 2 10.5 2 x 10.5 = 21

𝟏𝟎 < 𝒙 ≤ 𝟏𝟐 2 7 2 x 7 = 14

𝟏𝟐 < 𝒙 ≤ 𝟏𝟓 3 1.67 3 x 1.67 = 5



Frequency

𝟐 < 𝒙 ≤ 𝟔 4 0.75 4 x 0.75 = 3

𝟔 < 𝒙 ≤ 𝟖 2 6 2 x 6 = 12

𝟖 < 𝒙 ≤ 𝟏𝟎 2 8 2 x 8 = 16

𝟏𝟎 < 𝒙 ≤ 𝟏𝟐 2 2.5 2 x 2.5 = 5

𝟏𝟐 < 𝒙 ≤ 𝟏𝟓 3 3 3 x 3 = 9

Comparing Data in Different Formats

Interpret the following to graphs of birth rates in The Americas and Asia by extracting appropriate data and making comparisons between the two.


Comparing Areas instead of Heights American Birth Rates

(per 1000 people) Cumulative Frequency Frequency

𝟏𝟎 < 𝒙 ≤ 𝟏𝟐 5 5

𝟏𝟐 < 𝒙 ≤ 𝟏𝟒 15 10

𝟏𝟒 < 𝒙 ≤ 𝟏𝟖 21 6

𝟏𝟖 < 𝒙 ≤ 𝟐𝟎 30 9

𝟐𝟎 < 𝒙 ≤ 𝟐𝟒 36 6

𝟐𝟒 < 𝒙 ≤ 𝟑𝟏 40 4

Asian Birth Rates (per 1000 people)


Frequency

𝟕 < 𝒙 ≤ 𝟏𝟏 4 1.75 7

𝟏𝟏 < 𝒙 ≤ 𝟏𝟓 4 2.25 9

𝟏𝟓 < 𝒙 ≤ 𝟏𝟖 3 2.67 8

𝟏𝟖 < 𝒙 ≤ 𝟐𝟏 3 1.67 5

𝟐𝟏 < 𝒙 ≤ 𝟐𝟑 2 4 8

𝟐𝟑 < 𝒙 ≤ 𝟐𝟕 4 1 4

𝟐𝟕 < 𝒙 ≤ 𝟑𝟎 3 1.33 4

𝟑𝟎 < 𝒙 ≤ 𝟑𝟕

7 0.86 6


▪ Median: The Americas = 17.5, Asia = 20. ▪ On average, birth rates are higher in Asia than in

The Americas. ▪ Range: The Americas = 20, Asia = 29. ▪ IQR: The Americas = 7.5, Asia = 10. ▪ Asia has a greater spread of birth rates than in

The Americas

The Americas

Asia

Note: Raw data values Used for the boxplots

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